JavaScript/JavaScript within HTML

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    <title>JavaScript/JavaScript within HTML</title>
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      <contributor>
        <username>~2026-24052-01</username>
        <id>3577584</id>
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      <comment>typo fix</comment>
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      <text bytes="8897" sha1="lbj6zzb5216qi9ff6tfjwh36qoymvvz" xml:space="preserve">&lt;noinclude&gt;{{JavaScript}}&lt;/noinclude&gt;

&lt;div class=&quot;nonumtoc&quot;&gt;
__TOC__
&lt;/div&gt;

The language JavaScript was originally introduced to run in browsers and handle the dynamic aspects of user interfaces, e.g., validation of user input, modifications of page content (DOM) or appearance of the user interface (CSS), or any event handling. This implies that an interconnection point from HTML to JS must exist. The HTML element &lt;code&gt;&lt;script&gt;&lt;/code&gt; plays this role. It is a regular HTML element, and its content is JS.

The &lt;code&gt;&lt;script&gt;&lt;/code&gt; element may appear almost anywhere within the HTML file, within &lt;code&gt;&lt;head&gt;&lt;/code&gt; as well as in &lt;code&gt;&lt;body&gt;&lt;/code&gt;. There are only a few criteria for choosing an optimal place; see [[#Location_of_&lt;script&gt;_elements|below]].

== Internal vs. external JavaScript ==
The &lt;code&gt;&lt;script&gt;&lt;/code&gt; element either contains JS code directly, or it points to an external file resp. URL containing the JS code through its &lt;code&gt;src&lt;/code&gt; attribute. The first variant is called ''Internal JavaScript'' or ''Inline JavaScript'', the second ''External JavaScript''.

In the case of ''Internal JavaScript'' the &lt;code&gt;&lt;script&gt;&lt;/code&gt; element looks like:
&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;script&gt;
  // write your JS code directly here. (This line is a comment in JS syntax)
  alert(&quot;Hello World!&quot;);
&lt;/script&gt;
&lt;/syntaxhighlight&gt;

''Internal scripting'' has the advantage that both your HTML and your JS are in one file, which is convenient for quick development. This is commonly used for temporarily testing out some ideas, and in situations where the script code is small or specific to that one page.

For the ''External JavaScript'' the &lt;code&gt;&lt;script&gt;&lt;/code&gt; element looks like:

&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;!--  point to a file or to a URL where the code is located. (This line is a comment in HTML syntax) --&gt;
&lt;script src=&quot;myScript.js&quot;&gt;&lt;/script&gt;
&lt;script src=&quot;js/myScript2.js&quot;&gt;&lt;/script&gt;
&lt;script src=&quot;https://example.com/dist/js/externallib.js&quot;&gt;&lt;/script&gt;
&lt;script src=&quot;https://example.com/dist/js/externallib.min.js&quot;&gt;&lt;/script&gt;

&lt;!-- although there is nothing within the script element, you should consider that the HTML5 spec  --&gt;
&lt;!-- doesn't allow the abbreviation of the script element to: &lt;script src=&quot;myScript.js&quot; /&gt;         --&gt;
&lt;/syntaxhighlight&gt;

=== Separate Files for Javascript Code ===
Having your JS in a separate file is recommended for larger programs, especially for such which are used on multiple pages. Furthermore, such splits support the pattern of [[w:Separation of Concerns|Separation of Concerns]]: One specialist works on HTML, and another on JS. Also, it supports the division of the page's content (HTML) from its behavior (JS).

Overall, using ''External scripting'' is considered a [[w:best practice|best practice]] for software development.

=== Remote Code Injection vs. Local Library ===
With the example &lt;syntaxhighlight lang=&quot;html&quot;&gt;&lt;script src=&quot;https://example.com/dist/js/externallib.min.js&quot;&gt;&lt;/script&gt;&lt;/syntaxhighlight&gt; you can inject remotely maintained code from the server &lt;code&gt;https://example.com&lt;/code&gt; in your local web project. Remote code updates may break your local project or unwanted code features may be injected into your web project. On the other hand, centralized maintained and updated libraries serve your project due to bugfixes that are automatically updated in your project when the library is fetched again from the remote server.

=== Minified vs. Non-Minified Code ===
Minified Javascript code compresses the source code e.g. by shorting comprehensive variables like &lt;tt&gt;vImage&lt;/tt&gt; into a single character variable &lt;tt&gt;a&lt;/tt&gt;. This reduces significantly the size of the library and therefore reduces network traffic and response time until the web page is ready. For development and learning it might be helpful to have the uncompressed libraries locally available.

== External JavaScript ==
For more detailed information you can refer to MDN &lt;ref&gt;MDN: [https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script The script element]&lt;/ref&gt;.

=== The &lt;code&gt;src&lt;/code&gt; attribute ===
Adding &lt;code&gt;src=&quot;myScript.js&quot;&lt;/code&gt; to the opening &lt;code&gt;script&lt;/code&gt; tag means that the JS code will be located in a file called ''myScript.js'' in the same directory as the HTML file. If the JS file is located somewhere else, you must change the &lt;code&gt;src&lt;/code&gt; attribute to that path. For example, if it is located in a subdirectory called ''js'', it reads &lt;code&gt;src=&quot;js/myScript.js&quot;&lt;/code&gt;.

=== The &lt;code&gt;type&lt;/code&gt; attribute ===
JS is not the only scripting language for Web development, but JS is the most common one on client-side (PHP runs on server-side). Therefore it's considered the default script type for HTML5. The formal notation for the type is: &lt;code&gt;type=&quot;text/javascript&quot;&lt;/code&gt;. Older HTML versions know a lot of other script types. Nowadays, all of them are graded as ''legacy''. Some examples are: &lt;code&gt;application/javascript&lt;/code&gt;, &lt;code&gt;text/javascript1.5&lt;/code&gt;, &lt;code&gt;text/jscript&lt;/code&gt;, or &lt;code&gt;text/livescript&lt;/code&gt;.

In HTML5, the spec says that - if you use JS - the &lt;code&gt;type&lt;/code&gt; attribute should be omitted from the script element &lt;ref&gt;WHATWG: [https://html.spec.whatwg.org/dev/scripting.html#attr-script-type The type attribute]&lt;/ref&gt;, for ''Internal Scripting'' as well as for ''External Scripting''.

&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;!-- Nowadays the type attribute is unnecessary --&gt;
&lt;script type=&quot;text/javascript&quot;&gt;...&lt;/script&gt;

&lt;!-- HTML5 code --&gt;
&lt;script&gt;...&lt;/script&gt;
&lt;/syntaxhighlight&gt;

=== The &lt;code&gt;async&lt;/code&gt; and &lt;code&gt;defer&lt;/code&gt; attributes ===
Old browsers use only one or two threads to read and parse HTML, JS, CSS, ... . This may lead to a bad [[w:user experience|user experience (UX)]] because of the latency time when loading HTML, JS, CSS, images, ... sequentially one after the next. When the page loads for the first time, the user may have the impression of a slow system.

Current browsers can execute many tasks in parallel. To initiate this parallel execution with regards to JS loading and execution, the &lt;code&gt;&lt;script&gt;&lt;/code&gt; element can be extended with the two attributes &lt;code&gt;async&lt;/code&gt; and &lt;code&gt;defer&lt;/code&gt;.

The attribute &lt;code&gt;async&lt;/code&gt; leads to asynchronous script loading (in parallel with other tasks), and execution as soon as it is available.
&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;script async src=&quot;myScript.js&quot;&gt;&lt;/script&gt;
&lt;/syntaxhighlight&gt;

&lt;code&gt;defer&lt;/code&gt; acts similar. It differs from &lt;code&gt;async&lt;/code&gt; in that the execution is deferred until the page is fully parsed.
&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;script defer src=&quot;myScript.js&quot;&gt;&lt;/script&gt;
&lt;/syntaxhighlight&gt;

== Location of &lt;code&gt;&lt;script&gt;&lt;/code&gt; elements ==
The &lt;code&gt;script&lt;/code&gt; element may appear almost anywhere within the HTML file. But there are, however, some best practices for speeding up a website &lt;ref&gt;Yahoo: [http://developer.yahoo.com/performance/rules.html Best practices for speeding up your website]&lt;/ref&gt;. Some people suggest to locate it just before the closing &lt;code&gt;&lt;/body&gt;&lt;/code&gt; tag. This speeds up downloading, and also allows for direct manipulation of the Document Object Model (DOM) while it is rendered. But a similar behavior is initiated by the above-described &lt;code&gt;async&lt;/code&gt; and &lt;code&gt;defer&lt;/code&gt; attributes.

&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;!DOCTYPE html&gt;
&lt;html&gt;
&lt;head&gt;
  &lt;title&gt;Example page&lt;/title&gt;
&lt;/head&gt;
&lt;body&gt;
  &lt;!-- HTML code goes here --&gt;
  &lt;script src=&quot;myScript.js&quot;&gt;&lt;/script&gt;
&lt;/body&gt;
&lt;/html&gt;
&lt;/syntaxhighlight&gt;

== The &lt;code&gt;&lt;noscript&gt;&lt;/code&gt; element ==
It may happen that people have deactivated JS in their browsers for security or other reasons. Or, they use very old browsers which are not able to run JS at all. To inform users in such cases about the situation, there is the &lt;code&gt;&lt;noscript&gt;&lt;/code&gt; element. It contains text that will be shown in the browser. The text shall explain that no JS code will be executed.

&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;!DOCTYPE html&gt;
&lt;html&gt;
&lt;head&gt;
  &lt;title&gt;Example page&lt;/title&gt;
  &lt;script&gt;
    alert(&quot;Hello World!&quot;);
  &lt;/script&gt;
  &lt;noscript&gt;
    alert(&quot;Sorry, the JavaScript part of this page will not be executed because JavaScript is not running in your browser. Is JavaScript intentionally deactivated?&quot;);
  &lt;/noscript&gt;
&lt;/head&gt;
&lt;body&gt;
  &lt;!-- HTML code goes here --&gt;
&lt;/body&gt;
&lt;/html&gt;
&lt;/syntaxhighlight&gt;

== JavaScript in XHTML files ==
XHTML uses a stricter syntax than HTML. This leads to small differences.

First, for ''Internal JavaScript'' it's necessary that the scripts are introduced and finished with the two additional lines shown in the following example.

&lt;syntaxhighlight lang=&quot;html&quot;&gt;
&lt;script&gt;
  // &lt;![CDATA[
  alert(&quot;Hello World!&quot;);
  // ]]&gt;
&lt;/script&gt;
&lt;/syntaxhighlight&gt;

Second, for  ''External JavaScript'' the &lt;code&gt;type&lt;/code&gt; attribute is required.

== Reference ==
{{Reflist|40em}}


[[de:Websiteentwicklung: JavaScript: Der script-Tag]]
[[pl:JavaScript/Skrypt w przeglądarce]]</text>
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  </page>
  <page>
    <title>Cookbook:Cookie</title>
    <ns>102</ns>
    <id>27364</id>
    <revision>
      <id>4631219</id>
      <parentid>4611873</parentid>
      <timestamp>2026-04-18T12:48:11Z</timestamp>
      <contributor>
        <username>~2026-23858-31</username>
        <id>3577411</id>
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      <text bytes="25795" sha1="flsywm22zdgccnv97lwjml2elwewweo" xml:space="preserve">== Characteristics ==
It can be challenging to draw a precise line between all cookies and other sweet baked goods (e.g. cakes),&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Understanding the Science of Cookies {{!}} Institute of Culinary Education |url=https://www.ice.edu/blog/understanding-science-cookies |access-date=2025-01-01 |website=www.ice.edu}}&lt;/ref&gt; since the term is applied to a wide variety of small, sweet, baked morsels.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Rinsky |first=Glenn |url=https://books.google.com/books?id=mZhyDwAAQBAJ&amp;newbks=0&amp;hl=en |title=The Pastry Chef's Companion: A Comprehensive Resource Guide for the Baking and Pastry Professional |last2=Rinsky |first2=Laura Halpin |date=2008-02-28 |publisher=John Wiley &amp; Sons |isbn=978-0-470-00955-0 |language=en}}&lt;/ref&gt; In general, cookies are small, flattened, and individually shaped,&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Davidson |first=Alan |url=http://www.oxfordreference.com/view/10.1093/acref/9780199677337.001.0001/acref-9780199677337 |title=The Oxford Companion to Food |date=2014-01-01 |publisher=Oxford University Press |isbn=978-0-19-967733-7 |editor-last=Jaine |editor-first=Tom |doi=10.1093/acref/9780199677337.001.0001}}&lt;/ref&gt; baked free-form on a sheet and finishing with low moisture content.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Manley |first=Duncan |url=https://books.google.com/books?id=v5NwAgAAQBAJ&amp;newbks=0&amp;hl=en |title=Manley’s Technology of Biscuits, Crackers and Cookies |date=2011-09-28 |publisher=Elsevier |isbn=978-0-85709-364-6 |language=en}}&lt;/ref&gt; They are usually eaten with the hands.

Beyond these similarities, cookies come in a wide range of sizes, textures, shapes, flavors, and more.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The ideal characteristics will depend both on the specific type of cookie and the preferences of the individual&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Amendola |first=Joseph |url=https://books.google.com/books?id=bQcxoepvxOwC&amp;newbks=0&amp;hl=en |title=Understanding Baking: The Art and Science of Baking |last2=Rees |first2=Nicole |date=2003-01-03 |publisher=Wiley |isbn=978-0-471-44418-3 |language=en}}&lt;/ref&gt;—for example, some prefer chewy chocolate chip cookies while other prefer theirs crisp.

=== Composition ===
The primary components in cookie dough are typically some kind of sugar, fat, and flour, with the option of eggs and additional flavorings depending on the variety of cookie. All sugars, of course, contribute sweetness and browning to the cookie.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Ruhlman |first=Michael |url=https://books.google.com/books?id=cO6-wjf_XdgC&amp;newbks=0&amp;hl=en |title=The Elements of Cooking: Translating the Chef's Craft for Every Kitchen |date=2008 |publisher=Black Incorporated |isbn=978-1-86395-143-2 |language=en}}&lt;/ref&gt; Granulated sugar helps incorporate air into cookie dough,&lt;ref name=&quot;:2&quot; /&gt; especially when creamed well with a solid fat or egg.&lt;ref name=&quot;:10&quot; /&gt; Granulated sugar also melts in the oven before recrystallizing upon cooling, thereby contributing a crispness to the finished cookie.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:2&quot; /&gt; Liquid sugars like glucose syrup, honey, and molasses will not cause crystallization in this way, instead attracting water and causing softness.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;

Flour in cookies is primarily structural,&lt;ref name=&quot;:12&quot; /&gt; though it can contribute to color—bleached, low-protein wheat flour brown less, for example.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; High-protein and high-starch wheat flours (e.g. bread and cake flours, respectively) absorb more water and cause less spread than cookies made with pastry or all-purpose flours.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Pastry and all-purpose flours, however, are very commonly used.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Having a high ratio of flour to water yields dry, crumbly cookies without either gluten development or significant starch gelation.&lt;ref name=&quot;:12&quot; /&gt; A high ratio of water to flour allows for starch gelation but not gluten development, and the texture will be either cake-like or crisp.&lt;ref name=&quot;:12&quot; /&gt;

Fats contribute flavor, a feeling of moistness, and tenderness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; As lubricants, they disrupt the solid particles and contribute fluidity overall.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; The higher the melting point and the less liquid the fat, the less spread will occur in the oven;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; additionally, lower water content in the fat will also reduce spread compared to a pure fat.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Butter is more flavorful than shortening but will cause more spread.&lt;ref name=&quot;:15&quot; /&gt;

Chemical leaveners may be used in cookies, both for their leavening power and secondary effects. Baking soda, for example, neutralizes some of the acidity in the cookie dough, resulting in more spread and more browning.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Baking powder, on the other hand, will not significantly affect the acidity of the dough.&lt;ref name=&quot;:15&quot; /&gt;

The bulk of the water in cookie dough tends to come from egg, which also contributes protein for binding and a little fat from the yolk.&lt;ref name=&quot;:10&quot; /&gt; As a result of the moisture, eggs often contribute a puffiness and cakiness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; 

Additional flavorings and mix-ins may be incorporated into cookies, such as dried fruits, nuts, seeds, chocolate, candies, and more.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt;

=== Spread ===
The spread of a cookie refers to how much the dough spreads out horizontally during baking, and a number of factors can contribute to increased spread. Spread occurs because as the dough heats up in the oven, it becomes more thin and fluid, coheres less well, and spreads out.&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=Figoni |first=Paula |url=https://books.google.com/books?id=8ZpUDwAAQBAJ&amp;newbks=0&amp;hl=en |title=How Baking Works: Exploring the Fundamentals of Baking Science |date=2010-11-09 |publisher=John Wiley &amp; Sons |isbn=978-0-470-39267-6 |language=en}}&lt;/ref&gt; Eventually, with enough heat and time, the starches in the dough gelatinize and the proteins coagulate, which prevents any further spread of the dough.&lt;ref name=&quot;:3&quot; /&gt; A higher proportion of liquid in the dough evidently increases fluidity and spread.&lt;ref name=&quot;:11&quot; /&gt; A higher overall sugar content also has the effect of drawing more water into the fluid phase of the dough, making the dough more fluid and increasing spread.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Sugar also interferes with the gelatinization and coagulation, so the spread can proceed for longer before setting.&lt;ref name=&quot;:3&quot; /&gt; High-protein flour binds more water than does low-protein flour,&lt;ref name=&quot;:11&quot; /&gt; decreasing dough fluidity and therefore decreasing spread.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Adding starch to the dough achieves a similar effect.&lt;ref name=&quot;:6&quot; /&gt; Using fats with a lower melting point increases the dough fluidity and spread&lt;ref name=&quot;:11&quot; /&gt;—oil causes more spread than does butter,&lt;ref name=&quot;:3&quot; /&gt; which in turn causes more spread than does vegetable shortening.&lt;ref name=&quot;:11&quot; /&gt; Increasing the amount of air in the dough lowers its cohesive strength, thereby increasing spread; as a result, incorporating more air through extended creaming and additional leaveners will increase spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Leaveners that reduce the acidity of the dough (e.g. baking soda instead of baking powder) delay protein coagulation and setting of the dough.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Oat pulp cookies on tray.jpg|thumb|Cookies with relatively little spread]]
The baking environment will also influence spread. Greasing the baking surface decreases the friction between the dough and the surface, causing greater spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; A lower oven temperature increases spread because it takes longer for the proteins and starches to heat up and set the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Perhaps counterintuitively, however, chilling the dough actually reduces spread by making it take longer for the entire dough mass to warm through and become fluid; the crust can therefore form and set before much spread has a chance to occur.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Stemple Creek Ranch - August 2024 - Sarah Stierch 34.jpg|thumb|Cookies with a high amount of spread]]
The following table summarizes factors that affect cookie spread.
{| class=&quot;wikitable&quot;
!Decreased spread
!Increased spread
|-
|
* Cold dough
* Hotter oven
* Less liquid
* High melting point fat
* High-protein flour
* Ungreased pan
* Add starch/flour
* Less air
* Less sugar
* Coarser sugar
* Powdered sugar (contains starch) instead of granulated
* Increase acidity
|
* Warm dough
* Cooler oven
* More liquid
* Low melting point fat
* Low-protein flour
* Greased pan
* Reduce starch/flour
* More air
* More sugar
* Finer sugar
* Granulated sugar instead of powdered
* Decrease acidity
|}

=== Texture ===
A variety of factors can contribute to a cookie's chewiness, including an overall high moisture content, a high ratio of sugar and liquid to fat, a high egg content, high-protein flour, and increased stirring to develop any gluten in the flour.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Soft cookies also require a high moisture content, but they have less sugar, and the sugars used may be particularly water-attracting (e.g. honey, glucose/corn syrup).&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Conversely, the main cause of crispness in a cookie is very low moisture.&lt;ref name=&quot;:8&quot; /&gt; This can be achieved by simply including little moisture in the initial dough, using high sugar and fat proportions for pliability with the low moisture, baking for long enough to evaporate the moisture, and shaping the cookies thinly in order to more quickly achieve the latter.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Note that cookies will not be crispy while still warm; they must first cool down to achieve their snap.&lt;ref name=&quot;:2&quot;&gt;{{Cite web |title=Cookie science: How to achieve your perfect chocolate chip cookie {{!}} King Arthur Baking |url=https://www.kingarthurbaking.com/blog/2016/12/21/cookie-science |access-date=2025-07-25 |website=www.kingarthurbaking.com |language=en}}&lt;/ref&gt;

The following table links cookie texture with a variety of parameters involved in their creation:&lt;ref name=&quot;:6&quot; /&gt;
{| class=&quot;wikitable&quot;
!Desired texture
!Fat content
!Sugar content
!Sugar type
!Liquid content
!Four type
!Shape
|-
|Crispy
|High
|High
|Granulated
|Low
|Strong
|Thin dough
|-
|Soft
|Low
|Low
|Hygroscopic
|High
|Weak
|Thick dough
|-
|Chewy
|High
|High
|Hygroscopic
|High
|Strong
|
|-
|High-spread
|High
|High
|Coarse granulated
|High
|Weak
|
|}
== Types ==
As mentioned, a huge variety of cookies exists, and these can be classified in a few ways.

=== By mixing method ===
Cookie dough mixing methods are very similar to those for cakes.&lt;ref name=&quot;:8&quot; /&gt; In the one-stage method, all ingredients are mixed together at once&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Goldstein |first=Darra |url=http://www.oxfordreference.com/view/10.1093/acref/9780199313396.001.0001/acref-9780199313396 |title=The Oxford Companion to Sugar and Sweets |date=2015-01-01 |publisher=Oxford University Press |isbn=978-0-19-931339-6 |doi=10.1093/acref/9780199313396.001.0001}}&lt;/ref&gt;—this works adequately if there is not a lot of risk of gluten development.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; In creaming, the solid fat and sugar are beaten together before incorporating the flour and any liquids;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the degree of creaming will impact the aeration and spread as described above. With sanding, the flour and fat are rubbed together until crumbly or sandy before incorporating the moist ingredients.&lt;ref name=&quot;:8&quot; /&gt; Sponge/wafer/tuile cookie batters are made as with foaming, and the batter is delicate.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== By shaping ===
'''Drop''' cookies are formed from a soft dough portioned out in dollops onto the baking pan.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; This is often done with portion scoops, spoons, or the hands;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the dough may also be rolled between the hands into a more compact ball before panning and baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Some doughs may need to be flattened slightly for optimal spread and shaping.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=Jl0EEAAAQBAJ&amp;newbks=0 |title=Baking and Pastry: Mastering the Art and Craft |date=2015-02-25 |publisher=John Wiley &amp; Sons |isbn=978-0-470-92865-3 |language=en}}&lt;/ref&gt; Classic chocolate chip cookies and oatmeal cookies are examples of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Doughs for drop cookies are often made by the creaming method.&lt;ref name=&quot;:15&quot; /&gt; These doughs can often be made into icebox cookies (see below).&lt;ref name=&quot;:6&quot; /&gt;

A similar variety of cookie is the '''piped''', '''bagged''', '''pressed''', or '''spritz''' cookie,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; which also uses a soft dough that is stiff enough to hold a shape.&lt;ref name=&quot;:8&quot; /&gt; Here, the dough is forced through a nozzle, either using a pastry bag or a specialty [[Cookbook:Cookie press|cookie press]],&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; onto the prepared baking sheet. In general, this is done to create distinctive shapes and designs,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; so the dough should not be one that will spread significantly.&lt;ref name=&quot;:15&quot; /&gt; '''Molded''' cookies are also formed into specific and sometimes intricate designs, using hands, stamps, or molds&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;—the dough here may need to be chilled or otherwise stiffened for ease of working.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;

With so-called '''icebox''' cookies, the soft dough is shaped into a log and chilled in the refrigerator or freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Once solidified, cookies are evenly sliced off the log for baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; These cookies can be convenient, as the dough can be made and stored for some time before you are ready to bake the cookies.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Breaking with the above styles, '''rolled-and-cut''' or '''cutout''' cookies are made by rolling a relatively stiff dough out in large sheets on a floured surface before cutting out individual cookies using knives or specialized cutters.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=iGiM0YpaoDQC&amp;newbks=0 |title=The Professional Chef |date=2011-09-13 |publisher=John Wiley &amp; Sons |isbn=978-0-470-42135-2 |language=en}}&lt;/ref&gt; It's important to chill the dough sufficiently before rolling it out in order to avoid difficult handling.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Cookies should also be cut starting from the outside of the sheet and working towards the inside.&lt;ref name=&quot;:6&quot; /&gt; One downside to this style of cookie is that it tends to produce a lot of scraps—these can be pressed together and re-rolled, but the dough tends to get tougher as this continues, especially as more flour is incorporated.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If needed, the dough can be rolled out between sheets of parchment, plastic, or silicone.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

'''Bar''' cookies—not to be confused with sheet cookies (below), which are sometimes erroneously called bar cookies&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;—are made by baking a stiff dough in a large mass and then cutting into individual cookies while still warm.&lt;ref name=&quot;:8&quot; /&gt; Biscotti are one well-known example of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Like bar cookies, '''sheet/pan cookies''' or '''traybakes''' are portioned after baking in sheets.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; For this type, the dough or batter is pressed or spread in a shallow layer in the baking pan;&lt;ref name=&quot;:6&quot; /&gt; once baked, it is unmolded and cut into individual portion sizes.&lt;ref name=&quot;:6&quot; /&gt; Clean and even cuts can be more easily achieved by first chilling the baked mass by either refrigerating or freezing,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; depending on the product. Some consider brownies as sheet cookies,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; while others consider them cakes. Lemon bars, nut bars are other examples of preparations sometimes classified as sheet cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

'''Wafer''', '''tuile''', '''tulipe''', or '''stencil''' cookies are made with a thin dough or batter.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This batter is spread on silicone or parchment-lined pans, sometimes using a stencil cutout (hence the name),&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; before baking. Then, while still warm, the cookies can be trimmed or formed into a variety of shapes such as cups, rolls, cones, and more before cooling and setting.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This produces a thin, delicate, crisp cookie.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Due to their quick baking and complex handling, they should be prepared in small batches.&lt;ref name=&quot;:14&quot; /&gt;

=== Other ===
'''Sandwich cookies''' are, as the name implies, made by sandwiching a filling between two cookies. These include macarons, oreos, alfajores, linzer cookies, and more.

== Selection and storage ==
Due to their high sugar content and low moisture content, cookies generally will not spoil at room temperature.&lt;ref name=&quot;:10&quot; /&gt; Depending on their storage conditions, however, they may go stale; crispy cookies will absorb moisture and become chewy,&lt;ref name=&quot;:7&quot; /&gt; while soft cookies will lose moisture and become hard.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The best way to store cookies is in an airtight container,&lt;ref name=&quot;:5&quot; /&gt; with a desiccant for crisp cookies and a moist item like an apple slice for soft cookies.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Unfrosted cookie dough and baked cookies also freeze well when protected from air.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

== Use ==

Cookies are commonly served on their own as a snack or dessert.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; However, they can also be used as ingredients in their own right, either as decor&lt;ref name=&quot;:6&quot; /&gt; or as an integral component of another dish. An example of the latter is the use of crisp cookies to make crumb crusts.

== Techniques ==

=== Mixing ===
The mixing technique used when preparing the cookie dough will largely affect how the dough bakes and therefore the final texture of the cookie.&lt;ref name=&quot;:6&quot; /&gt; Creaming the butter before adding granulated sugar and not much afterwards will reduce the amount of incorporated air and reduce the puff and spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; When making dense cookies that will hold their shape, you should avoid excessive or vigorous creaming and incorporation of air.&lt;ref name=&quot;:6&quot; /&gt;''&lt;ref name=&quot;:3&quot; /&gt;'' If a gluten-containing flour is used in a dough with a moderate amount of liquid, mixing should be kept to a minimum after the addition of the flour to prevent toughness.&lt;ref name=&quot;:6&quot; /&gt; It is also important to scrape down the mixing bowl when preparing the dough to ensure that all ingredients are properly distributed.&lt;ref name=&quot;:6&quot; /&gt;

=== Forming ===
Cookies should be formed according to their type and placed on a baking sheet prepared according to the recipe specifications.&lt;ref name=&quot;:14&quot; /&gt; The most important consideration when forming any variety of cookie is consistency.&lt;ref name=&quot;:8&quot; /&gt; Each cookie should be the same size, shape, and thickness to ensure proper and even baking across the entire sheet.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; It is also important to space the cookies out enough that they will not touch each other when spreading in the oven,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; and so that heat can circulate evenly.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; The amount of space necessary will depend on the size and the cookie type. If adding garnishes before baking, these should be applied to the surface immediately after forming and pressed in gently to ensure they adhere properly to the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== Baking and cooling ===
Most cookies are baked at a moderately high temperature for a short time to allow proper baking and coloration—somewhere from 8–14 minutes at 325–375°F (163–191°C) is common.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Doneness is often indicated by color,&lt;ref name=&quot;:8&quot; /&gt; though this can be difficult to ascertain in darker cookies such as those containing chocolate. If possible, take a peek at the undersides of the cookies, since these often cook faster than the top. If there is a risk of overcooking on the bottom relative to the rest of the cookies, double-layering the pans will help insulate the bottom from the heat.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; Note as well that some carryover cooking will occur even after the cookies are removed from the oven.&lt;ref name=&quot;:6&quot; /&gt;

Often, cookies need to rest for a moment on the pans before moving to cooling racks, since many cookies are too soft to move while warm.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; However, timely removal of cookies from unlined pans is often critical, since there is a risk of the cookies sticking to the pan once they set.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Excessive carryover cooking can also occur.&lt;ref name=&quot;:14&quot; /&gt; Cool cookies completely to room temperature before storage.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;

=== Finishing ===
After baking, the cookies can be finished in a variety of ways.&lt;ref name=&quot;:6&quot; /&gt; Some are decorated, sandwiched, or filled with frosting, icing, chocolate, jam, fondant, marshmallow, and more.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; In most cases, it's important to cool the cookies completely before decorating and finishing.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If storing the cookies by freezing, wait to finish them until you remove them from the freezer and return them to room temperature, since the decorations are often damaged in the freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

=== Troubleshooting ===
The following table outlines some possible issues and causes that can arise when making cookies:&lt;ref name=&quot;:8&quot; /&gt;

{| class=&quot;wikitable&quot;
!Problem
!Potential cause
|-
| rowspan=&quot;5&quot; |Too tough
|Flour too strong
|-
|Too much flour
|-
|Incorrect amount of sugar
|-
|Not enough shortening
|-
|Mixed too long or improper mixing
|-
| rowspan=&quot;5&quot; |Too crumbly
|Improper mixing
|-
|Too much sugar
|-
|Too much shortening
|-
|Too much leavening
|-
|Not enough eggs
|-
| rowspan=&quot;6&quot; |Too hard
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
|Flour too strong
|-
|Not enough shortening
|-
|Not enough liquid
|-
| rowspan=&quot;5&quot; |Too dry 
|Not enough liquid
|-
|Not enough shortening
|-
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
| rowspan=&quot;3&quot; |Not browned enough
|Baking temperature too low
|-
|Underbaked
|-
|Not enough sugar
|-
| rowspan=&quot;3&quot; |Too brown 
|Baking temperature too high
|-
|Baked too long
|-
|Too much sugar
|-
| rowspan=&quot;4&quot; |Poor flavor
|Poor-quality ingredients
|-
|Flavoring ingredients left out
|-
|Dirty baking pans
|-
|Ingredients improperly measured
|-
| rowspan=&quot;2&quot; |Sugary surface or crust
|Improper mixing
|-
|Too much sugar
|-
| rowspan=&quot;6&quot; |Too much spread
|Baking temperature too low
|-
|Not enough flour
|-
|Too much sugar
|-
|Too much leavening (chemical leaveners or creaming)
|-
|Too much liquid
|-
|Pans greased too heavily
|-
| rowspan=&quot;6&quot; |Not enough spread
|Baking temperature too high
|-
|Too much flour or flour too strong
|-
|Not enough sugar
|-
|Not enough leavening
|-
|Not enough liquid
|-
|Insufficient pan grease
|-
| rowspan=&quot;3&quot; |Stick to pans 
|Pans improperly greased
|-
|Too much sugar
|-
|Improper mixing
|}

== Recipes ==

=== For cookies ===
{{div col|3}}
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes for cookies&lt;/categorytree&gt;
{{div col end}}

=== Using cookies ===
&lt;div style=&quot;column-count:3&quot;&gt;
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes using cookies&lt;/categorytree&gt;
&lt;/div&gt;

== Gallery ==
&lt;gallery mode=&quot;packed&quot;&gt;
File:2ChocolateChipCookies.jpg|Chocolate chip drop cookies
File:Macaron trio.jpg|Macarons
File:Biscotti 1.jpg|Biscotti
File:Alfajores-de-maicena-biscuits-recipe.jpg|Alfajores
File:Diamond-cut pecan bars (5631275901).jpg|Pecan bars
File:Round gingersnaps.jpg|Gingersnaps
File:Fortune cookies.jpg|Fortune cookies
File:Cattongues.jpg|Langues de chat
File:Caramel-colored-wafer-sticks.jpg|Tuile cookies shaped into cigars
File:Sofra Coconut macaroon, October 2009.jpg|Macaroon
File:2020-07-23 20 05 30 Wholesale Pantry Organic Vanilla Animal Crackers in Ewing Township, Mercer County, New Jersey.jpg|Animal crackers
File:Vegan Black and White Cookies (8746952832).jpg|Black-and-white cookies
File:Hamantaschen1.jpg|Hamantaschen
File:Russian teacakes (8307560889).jpg|Russian teacakes
File:Stroopwafels 01.jpg|Stroopwafels
&lt;/gallery&gt;

== References ==</text>
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        <id>3097823</id>
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      <comment>Reverted edits by [[Special:Contribs/~2026-23858-31|~2026-23858-31]] ([[User talk:~2026-23858-31|talk]]) to last version by Codename Noreste: unexplained content removal</comment>
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{{Ingredient summary
| Image = [[File:Koekjestrommel open.jpg|300px]]
}}

{{ingredient}}

'''Cookies''' are a class of sweet baked good, overlapping somewhat with [[Cookbook:Cake|cakes]] and [[Cookbook:Pastry|pastries]]. The term &quot;cookie&quot; is reported to derive from the Dutch for &quot;little cake&quot;,&lt;ref name=&quot;:7&quot;&gt;{{Cite book |last=Friberg |first=Bo |url=https://books.google.com/books?id=gmpkzgAACAAJ&amp;newbks=0&amp;hl=en |title=The Professional Pastry Chef: Fundamentals of Baking and Pastry |date=2016-09-13 |publisher=Wiley |isbn=978-0-470-46629-2 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite book |last=Labensky |first=Sarah |url=https://books.google.com/books?id=k87uoQEACAAJ&amp;newbks=0&amp;hl=en |title=On Baking: A Textbook of Baking and Pastry Fundamentals, Updated Edition |last2=Martel |first2=Priscilla |last3=Damme |first3=Eddy Van |date=2015-01-06 |publisher=Pearson Education |isbn=978-0-13-388675-7 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:10&quot;&gt;{{Cite book |last=McGee |first=Harold |url=https://www.google.com/books/edition/On_Food_and_Cooking/bKVCtH4AjwgC?hl=en&amp;gbpv=0 |title=On Food and Cooking: The Science and Lore of the Kitchen |date=2007-03-20 |publisher=Simon and Schuster |isbn=978-1-4165-5637-4 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot;&gt;{{Cite book |last=Gisslen |first=Wayne |url=https://books.google.com/books?id=KGjpCgAAQBAJ&amp;newbks=0&amp;hl=en |title=Professional Baking |date=2016-09-21 |publisher=John Wiley &amp; Sons |isbn=978-1-119-14844-9 |language=en}}&lt;/ref&gt; and it encompasses a wide variety of baked goods, including sweet biscuits in British English.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Savory biscuits as designated by British English are classified in the Cookbook as [[Cookbook:Cracker|crackers]].

== Characteristics ==
It can be challenging to draw a precise line between all cookies and other sweet baked goods (e.g. cakes),&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Understanding the Science of Cookies {{!}} Institute of Culinary Education |url=https://www.ice.edu/blog/understanding-science-cookies |access-date=2025-01-01 |website=www.ice.edu}}&lt;/ref&gt; since the term is applied to a wide variety of small, sweet, baked morsels.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Rinsky |first=Glenn |url=https://books.google.com/books?id=mZhyDwAAQBAJ&amp;newbks=0&amp;hl=en |title=The Pastry Chef's Companion: A Comprehensive Resource Guide for the Baking and Pastry Professional |last2=Rinsky |first2=Laura Halpin |date=2008-02-28 |publisher=John Wiley &amp; Sons |isbn=978-0-470-00955-0 |language=en}}&lt;/ref&gt; In general, cookies are small, flattened, and individually shaped,&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Davidson |first=Alan |url=http://www.oxfordreference.com/view/10.1093/acref/9780199677337.001.0001/acref-9780199677337 |title=The Oxford Companion to Food |date=2014-01-01 |publisher=Oxford University Press |isbn=978-0-19-967733-7 |editor-last=Jaine |editor-first=Tom |doi=10.1093/acref/9780199677337.001.0001}}&lt;/ref&gt; baked free-form on a sheet and finishing with low moisture content.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Manley |first=Duncan |url=https://books.google.com/books?id=v5NwAgAAQBAJ&amp;newbks=0&amp;hl=en |title=Manley’s Technology of Biscuits, Crackers and Cookies |date=2011-09-28 |publisher=Elsevier |isbn=978-0-85709-364-6 |language=en}}&lt;/ref&gt; They are usually eaten with the hands.

Beyond these similarities, cookies come in a wide range of sizes, textures, shapes, flavors, and more.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The ideal characteristics will depend both on the specific type of cookie and the preferences of the individual&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Amendola |first=Joseph |url=https://books.google.com/books?id=bQcxoepvxOwC&amp;newbks=0&amp;hl=en |title=Understanding Baking: The Art and Science of Baking |last2=Rees |first2=Nicole |date=2003-01-03 |publisher=Wiley |isbn=978-0-471-44418-3 |language=en}}&lt;/ref&gt;—for example, some prefer chewy chocolate chip cookies while other prefer theirs crisp.

=== Composition ===
The primary components in cookie dough are typically some kind of sugar, fat, and flour, with the option of eggs and additional flavorings depending on the variety of cookie. All sugars, of course, contribute sweetness and browning to the cookie.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Ruhlman |first=Michael |url=https://books.google.com/books?id=cO6-wjf_XdgC&amp;newbks=0&amp;hl=en |title=The Elements of Cooking: Translating the Chef's Craft for Every Kitchen |date=2008 |publisher=Black Incorporated |isbn=978-1-86395-143-2 |language=en}}&lt;/ref&gt; Granulated sugar helps incorporate air into cookie dough,&lt;ref name=&quot;:2&quot; /&gt; especially when creamed well with a solid fat or egg.&lt;ref name=&quot;:10&quot; /&gt; Granulated sugar also melts in the oven before recrystallizing upon cooling, thereby contributing a crispness to the finished cookie.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:2&quot; /&gt; Liquid sugars like glucose syrup, honey, and molasses will not cause crystallization in this way, instead attracting water and causing softness.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;

Flour in cookies is primarily structural,&lt;ref name=&quot;:12&quot; /&gt; though it can contribute to color—bleached, low-protein wheat flour brown less, for example.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; High-protein and high-starch wheat flours (e.g. bread and cake flours, respectively) absorb more water and cause less spread than cookies made with pastry or all-purpose flours.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Pastry and all-purpose flours, however, are very commonly used.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Having a high ratio of flour to water yields dry, crumbly cookies without either gluten development or significant starch gelation.&lt;ref name=&quot;:12&quot; /&gt; A high ratio of water to flour allows for starch gelation but not gluten development, and the texture will be either cake-like or crisp.&lt;ref name=&quot;:12&quot; /&gt;

Fats contribute flavor, a feeling of moistness, and tenderness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; As lubricants, they disrupt the solid particles and contribute fluidity overall.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; The higher the melting point and the less liquid the fat, the less spread will occur in the oven;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; additionally, lower water content in the fat will also reduce spread compared to a pure fat.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Butter is more flavorful than shortening but will cause more spread.&lt;ref name=&quot;:15&quot; /&gt;

Chemical leaveners may be used in cookies, both for their leavening power and secondary effects. Baking soda, for example, neutralizes some of the acidity in the cookie dough, resulting in more spread and more browning.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Baking powder, on the other hand, will not significantly affect the acidity of the dough.&lt;ref name=&quot;:15&quot; /&gt;

The bulk of the water in cookie dough tends to come from egg, which also contributes protein for binding and a little fat from the yolk.&lt;ref name=&quot;:10&quot; /&gt; As a result of the moisture, eggs often contribute a puffiness and cakiness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; 

Additional flavorings and mix-ins may be incorporated into cookies, such as dried fruits, nuts, seeds, chocolate, candies, and more.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt;

=== Spread ===
The spread of a cookie refers to how much the dough spreads out horizontally during baking, and a number of factors can contribute to increased spread. Spread occurs because as the dough heats up in the oven, it becomes more thin and fluid, coheres less well, and spreads out.&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=Figoni |first=Paula |url=https://books.google.com/books?id=8ZpUDwAAQBAJ&amp;newbks=0&amp;hl=en |title=How Baking Works: Exploring the Fundamentals of Baking Science |date=2010-11-09 |publisher=John Wiley &amp; Sons |isbn=978-0-470-39267-6 |language=en}}&lt;/ref&gt; Eventually, with enough heat and time, the starches in the dough gelatinize and the proteins coagulate, which prevents any further spread of the dough.&lt;ref name=&quot;:3&quot; /&gt; A higher proportion of liquid in the dough evidently increases fluidity and spread.&lt;ref name=&quot;:11&quot; /&gt; A higher overall sugar content also has the effect of drawing more water into the fluid phase of the dough, making the dough more fluid and increasing spread.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Sugar also interferes with the gelatinization and coagulation, so the spread can proceed for longer before setting.&lt;ref name=&quot;:3&quot; /&gt; High-protein flour binds more water than does low-protein flour,&lt;ref name=&quot;:11&quot; /&gt; decreasing dough fluidity and therefore decreasing spread.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Adding starch to the dough achieves a similar effect.&lt;ref name=&quot;:6&quot; /&gt; Using fats with a lower melting point increases the dough fluidity and spread&lt;ref name=&quot;:11&quot; /&gt;—oil causes more spread than does butter,&lt;ref name=&quot;:3&quot; /&gt; which in turn causes more spread than does vegetable shortening.&lt;ref name=&quot;:11&quot; /&gt; Increasing the amount of air in the dough lowers its cohesive strength, thereby increasing spread; as a result, incorporating more air through extended creaming and additional leaveners will increase spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Leaveners that reduce the acidity of the dough (e.g. baking soda instead of baking powder) delay protein coagulation and setting of the dough.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Oat pulp cookies on tray.jpg|thumb|Cookies with relatively little spread]]
The baking environment will also influence spread. Greasing the baking surface decreases the friction between the dough and the surface, causing greater spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; A lower oven temperature increases spread because it takes longer for the proteins and starches to heat up and set the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Perhaps counterintuitively, however, chilling the dough actually reduces spread by making it take longer for the entire dough mass to warm through and become fluid; the crust can therefore form and set before much spread has a chance to occur.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Stemple Creek Ranch - August 2024 - Sarah Stierch 34.jpg|thumb|Cookies with a high amount of spread]]
The following table summarizes factors that affect cookie spread.
{| class=&quot;wikitable&quot;
!Decreased spread
!Increased spread
|-
|
* Cold dough
* Hotter oven
* Less liquid
* High melting point fat
* High-protein flour
* Ungreased pan
* Add starch/flour
* Less air
* Less sugar
* Coarser sugar
* Powdered sugar (contains starch) instead of granulated
* Increase acidity
|
* Warm dough
* Cooler oven
* More liquid
* Low melting point fat
* Low-protein flour
* Greased pan
* Reduce starch/flour
* More air
* More sugar
* Finer sugar
* Granulated sugar instead of powdered
* Decrease acidity
|}

=== Texture ===
A variety of factors can contribute to a cookie's chewiness, including an overall high moisture content, a high ratio of sugar and liquid to fat, a high egg content, high-protein flour, and increased stirring to develop any gluten in the flour.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Soft cookies also require a high moisture content, but they have less sugar, and the sugars used may be particularly water-attracting (e.g. honey, glucose/corn syrup).&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Conversely, the main cause of crispness in a cookie is very low moisture.&lt;ref name=&quot;:8&quot; /&gt; This can be achieved by simply including little moisture in the initial dough, using high sugar and fat proportions for pliability with the low moisture, baking for long enough to evaporate the moisture, and shaping the cookies thinly in order to more quickly achieve the latter.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Note that cookies will not be crispy while still warm; they must first cool down to achieve their snap.&lt;ref name=&quot;:2&quot;&gt;{{Cite web |title=Cookie science: How to achieve your perfect chocolate chip cookie {{!}} King Arthur Baking |url=https://www.kingarthurbaking.com/blog/2016/12/21/cookie-science |access-date=2025-07-25 |website=www.kingarthurbaking.com |language=en}}&lt;/ref&gt;

The following table links cookie texture with a variety of parameters involved in their creation:&lt;ref name=&quot;:6&quot; /&gt;
{| class=&quot;wikitable&quot;
!Desired texture
!Fat content
!Sugar content
!Sugar type
!Liquid content
!Four type
!Shape
|-
|Crispy
|High
|High
|Granulated
|Low
|Strong
|Thin dough
|-
|Soft
|Low
|Low
|Hygroscopic
|High
|Weak
|Thick dough
|-
|Chewy
|High
|High
|Hygroscopic
|High
|Strong
|
|-
|High-spread
|High
|High
|Coarse granulated
|High
|Weak
|
|}
== Types ==
As mentioned, a huge variety of cookies exists, and these can be classified in a few ways.

=== By mixing method ===
Cookie dough mixing methods are very similar to those for cakes.&lt;ref name=&quot;:8&quot; /&gt; In the one-stage method, all ingredients are mixed together at once&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Goldstein |first=Darra |url=http://www.oxfordreference.com/view/10.1093/acref/9780199313396.001.0001/acref-9780199313396 |title=The Oxford Companion to Sugar and Sweets |date=2015-01-01 |publisher=Oxford University Press |isbn=978-0-19-931339-6 |doi=10.1093/acref/9780199313396.001.0001}}&lt;/ref&gt;—this works adequately if there is not a lot of risk of gluten development.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; In creaming, the solid fat and sugar are beaten together before incorporating the flour and any liquids;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the degree of creaming will impact the aeration and spread as described above. With sanding, the flour and fat are rubbed together until crumbly or sandy before incorporating the moist ingredients.&lt;ref name=&quot;:8&quot; /&gt; Sponge/wafer/tuile cookie batters are made as with foaming, and the batter is delicate.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== By shaping ===
'''Drop''' cookies are formed from a soft dough portioned out in dollops onto the baking pan.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; This is often done with portion scoops, spoons, or the hands;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the dough may also be rolled between the hands into a more compact ball before panning and baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Some doughs may need to be flattened slightly for optimal spread and shaping.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=Jl0EEAAAQBAJ&amp;newbks=0 |title=Baking and Pastry: Mastering the Art and Craft |date=2015-02-25 |publisher=John Wiley &amp; Sons |isbn=978-0-470-92865-3 |language=en}}&lt;/ref&gt; Classic chocolate chip cookies and oatmeal cookies are examples of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Doughs for drop cookies are often made by the creaming method.&lt;ref name=&quot;:15&quot; /&gt; These doughs can often be made into icebox cookies (see below).&lt;ref name=&quot;:6&quot; /&gt;

A similar variety of cookie is the '''piped''', '''bagged''', '''pressed''', or '''spritz''' cookie,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; which also uses a soft dough that is stiff enough to hold a shape.&lt;ref name=&quot;:8&quot; /&gt; Here, the dough is forced through a nozzle, either using a pastry bag or a specialty [[Cookbook:Cookie press|cookie press]],&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; onto the prepared baking sheet. In general, this is done to create distinctive shapes and designs,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; so the dough should not be one that will spread significantly.&lt;ref name=&quot;:15&quot; /&gt; '''Molded''' cookies are also formed into specific and sometimes intricate designs, using hands, stamps, or molds&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;—the dough here may need to be chilled or otherwise stiffened for ease of working.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;

With so-called '''icebox''' cookies, the soft dough is shaped into a log and chilled in the refrigerator or freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Once solidified, cookies are evenly sliced off the log for baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; These cookies can be convenient, as the dough can be made and stored for some time before you are ready to bake the cookies.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Breaking with the above styles, '''rolled-and-cut''' or '''cutout''' cookies are made by rolling a relatively stiff dough out in large sheets on a floured surface before cutting out individual cookies using knives or specialized cutters.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=iGiM0YpaoDQC&amp;newbks=0 |title=The Professional Chef |date=2011-09-13 |publisher=John Wiley &amp; Sons |isbn=978-0-470-42135-2 |language=en}}&lt;/ref&gt; It's important to chill the dough sufficiently before rolling it out in order to avoid difficult handling.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Cookies should also be cut starting from the outside of the sheet and working towards the inside.&lt;ref name=&quot;:6&quot; /&gt; One downside to this style of cookie is that it tends to produce a lot of scraps—these can be pressed together and re-rolled, but the dough tends to get tougher as this continues, especially as more flour is incorporated.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If needed, the dough can be rolled out between sheets of parchment, plastic, or silicone.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

'''Bar''' cookies—not to be confused with sheet cookies (below), which are sometimes erroneously called bar cookies&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;—are made by baking a stiff dough in a large mass and then cutting into individual cookies while still warm.&lt;ref name=&quot;:8&quot; /&gt; Biscotti are one well-known example of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Like bar cookies, '''sheet/pan cookies''' or '''traybakes''' are portioned after baking in sheets.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; For this type, the dough or batter is pressed or spread in a shallow layer in the baking pan;&lt;ref name=&quot;:6&quot; /&gt; once baked, it is unmolded and cut into individual portion sizes.&lt;ref name=&quot;:6&quot; /&gt; Clean and even cuts can be more easily achieved by first chilling the baked mass by either refrigerating or freezing,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; depending on the product. Some consider brownies as sheet cookies,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; while others consider them cakes. Lemon bars, nut bars are other examples of preparations sometimes classified as sheet cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

'''Wafer''', '''tuile''', '''tulipe''', or '''stencil''' cookies are made with a thin dough or batter.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This batter is spread on silicone or parchment-lined pans, sometimes using a stencil cutout (hence the name),&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; before baking. Then, while still warm, the cookies can be trimmed or formed into a variety of shapes such as cups, rolls, cones, and more before cooling and setting.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This produces a thin, delicate, crisp cookie.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Due to their quick baking and complex handling, they should be prepared in small batches.&lt;ref name=&quot;:14&quot; /&gt;

=== Other ===
'''Sandwich cookies''' are, as the name implies, made by sandwiching a filling between two cookies. These include macarons, oreos, alfajores, linzer cookies, and more.

== Selection and storage ==
Due to their high sugar content and low moisture content, cookies generally will not spoil at room temperature.&lt;ref name=&quot;:10&quot; /&gt; Depending on their storage conditions, however, they may go stale; crispy cookies will absorb moisture and become chewy,&lt;ref name=&quot;:7&quot; /&gt; while soft cookies will lose moisture and become hard.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The best way to store cookies is in an airtight container,&lt;ref name=&quot;:5&quot; /&gt; with a desiccant for crisp cookies and a moist item like an apple slice for soft cookies.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Unfrosted cookie dough and baked cookies also freeze well when protected from air.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

== Use ==

Cookies are commonly served on their own as a snack or dessert.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; However, they can also be used as ingredients in their own right, either as decor&lt;ref name=&quot;:6&quot; /&gt; or as an integral component of another dish. An example of the latter is the use of crisp cookies to make crumb crusts.

== Techniques ==

=== Mixing ===
The mixing technique used when preparing the cookie dough will largely affect how the dough bakes and therefore the final texture of the cookie.&lt;ref name=&quot;:6&quot; /&gt; Creaming the butter before adding granulated sugar and not much afterwards will reduce the amount of incorporated air and reduce the puff and spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; When making dense cookies that will hold their shape, you should avoid excessive or vigorous creaming and incorporation of air.&lt;ref name=&quot;:6&quot; /&gt;''&lt;ref name=&quot;:3&quot; /&gt;'' If a gluten-containing flour is used in a dough with a moderate amount of liquid, mixing should be kept to a minimum after the addition of the flour to prevent toughness.&lt;ref name=&quot;:6&quot; /&gt; It is also important to scrape down the mixing bowl when preparing the dough to ensure that all ingredients are properly distributed.&lt;ref name=&quot;:6&quot; /&gt;

=== Forming ===
Cookies should be formed according to their type and placed on a baking sheet prepared according to the recipe specifications.&lt;ref name=&quot;:14&quot; /&gt; The most important consideration when forming any variety of cookie is consistency.&lt;ref name=&quot;:8&quot; /&gt; Each cookie should be the same size, shape, and thickness to ensure proper and even baking across the entire sheet.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; It is also important to space the cookies out enough that they will not touch each other when spreading in the oven,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; and so that heat can circulate evenly.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; The amount of space necessary will depend on the size and the cookie type. If adding garnishes before baking, these should be applied to the surface immediately after forming and pressed in gently to ensure they adhere properly to the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== Baking and cooling ===
Most cookies are baked at a moderately high temperature for a short time to allow proper baking and coloration—somewhere from 8–14 minutes at 325–375°F (163–191°C) is common.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Doneness is often indicated by color,&lt;ref name=&quot;:8&quot; /&gt; though this can be difficult to ascertain in darker cookies such as those containing chocolate. If possible, take a peek at the undersides of the cookies, since these often cook faster than the top. If there is a risk of overcooking on the bottom relative to the rest of the cookies, double-layering the pans will help insulate the bottom from the heat.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; Note as well that some carryover cooking will occur even after the cookies are removed from the oven.&lt;ref name=&quot;:6&quot; /&gt;

Often, cookies need to rest for a moment on the pans before moving to cooling racks, since many cookies are too soft to move while warm.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; However, timely removal of cookies from unlined pans is often critical, since there is a risk of the cookies sticking to the pan once they set.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Excessive carryover cooking can also occur.&lt;ref name=&quot;:14&quot; /&gt; Cool cookies completely to room temperature before storage.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;

=== Finishing ===
After baking, the cookies can be finished in a variety of ways.&lt;ref name=&quot;:6&quot; /&gt; Some are decorated, sandwiched, or filled with frosting, icing, chocolate, jam, fondant, marshmallow, and more.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; In most cases, it's important to cool the cookies completely before decorating and finishing.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If storing the cookies by freezing, wait to finish them until you remove them from the freezer and return them to room temperature, since the decorations are often damaged in the freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

=== Troubleshooting ===
The following table outlines some possible issues and causes that can arise when making cookies:&lt;ref name=&quot;:8&quot; /&gt;

{| class=&quot;wikitable&quot;
!Problem
!Potential cause
|-
| rowspan=&quot;5&quot; |Too tough
|Flour too strong
|-
|Too much flour
|-
|Incorrect amount of sugar
|-
|Not enough shortening
|-
|Mixed too long or improper mixing
|-
| rowspan=&quot;5&quot; |Too crumbly
|Improper mixing
|-
|Too much sugar
|-
|Too much shortening
|-
|Too much leavening
|-
|Not enough eggs
|-
| rowspan=&quot;6&quot; |Too hard
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
|Flour too strong
|-
|Not enough shortening
|-
|Not enough liquid
|-
| rowspan=&quot;5&quot; |Too dry 
|Not enough liquid
|-
|Not enough shortening
|-
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
| rowspan=&quot;3&quot; |Not browned enough
|Baking temperature too low
|-
|Underbaked
|-
|Not enough sugar
|-
| rowspan=&quot;3&quot; |Too brown 
|Baking temperature too high
|-
|Baked too long
|-
|Too much sugar
|-
| rowspan=&quot;4&quot; |Poor flavor
|Poor-quality ingredients
|-
|Flavoring ingredients left out
|-
|Dirty baking pans
|-
|Ingredients improperly measured
|-
| rowspan=&quot;2&quot; |Sugary surface or crust
|Improper mixing
|-
|Too much sugar
|-
| rowspan=&quot;6&quot; |Too much spread
|Baking temperature too low
|-
|Not enough flour
|-
|Too much sugar
|-
|Too much leavening (chemical leaveners or creaming)
|-
|Too much liquid
|-
|Pans greased too heavily
|-
| rowspan=&quot;6&quot; |Not enough spread
|Baking temperature too high
|-
|Too much flour or flour too strong
|-
|Not enough sugar
|-
|Not enough leavening
|-
|Not enough liquid
|-
|Insufficient pan grease
|-
| rowspan=&quot;3&quot; |Stick to pans 
|Pans improperly greased
|-
|Too much sugar
|-
|Improper mixing
|}

== Recipes ==

=== For cookies ===
{{div col|3}}
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes for cookies&lt;/categorytree&gt;
{{div col end}}

=== Using cookies ===
&lt;div style=&quot;column-count:3&quot;&gt;
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes using cookies&lt;/categorytree&gt;
&lt;/div&gt;

== Gallery ==
&lt;gallery mode=&quot;packed&quot;&gt;
File:2ChocolateChipCookies.jpg|Chocolate chip drop cookies
File:Macaron trio.jpg|Macarons
File:Biscotti 1.jpg|Biscotti
File:Alfajores-de-maicena-biscuits-recipe.jpg|Alfajores
File:Diamond-cut pecan bars (5631275901).jpg|Pecan bars
File:Round gingersnaps.jpg|Gingersnaps
File:Fortune cookies.jpg|Fortune cookies
File:Cattongues.jpg|Langues de chat
File:Caramel-colored-wafer-sticks.jpg|Tuile cookies shaped into cigars
File:Sofra Coconut macaroon, October 2009.jpg|Macaroon
File:2020-07-23 20 05 30 Wholesale Pantry Organic Vanilla Animal Crackers in Ewing Township, Mercer County, New Jersey.jpg|Animal crackers
File:Vegan Black and White Cookies (8746952832).jpg|Black-and-white cookies
File:Hamantaschen1.jpg|Hamantaschen
File:Russian teacakes (8307560889).jpg|Russian teacakes
File:Stroopwafels 01.jpg|Stroopwafels
&lt;/gallery&gt;

== References ==</text>
      <sha1>3vjxn7knsrt23qbxmour41bzafky2vt</sha1>
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      <text bytes="25795" sha1="flsywm22zdgccnv97lwjml2elwewweo" xml:space="preserve">== Characteristics ==
It can be challenging to draw a precise line between all cookies and other sweet baked goods (e.g. cakes),&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Understanding the Science of Cookies {{!}} Institute of Culinary Education |url=https://www.ice.edu/blog/understanding-science-cookies |access-date=2025-01-01 |website=www.ice.edu}}&lt;/ref&gt; since the term is applied to a wide variety of small, sweet, baked morsels.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Rinsky |first=Glenn |url=https://books.google.com/books?id=mZhyDwAAQBAJ&amp;newbks=0&amp;hl=en |title=The Pastry Chef's Companion: A Comprehensive Resource Guide for the Baking and Pastry Professional |last2=Rinsky |first2=Laura Halpin |date=2008-02-28 |publisher=John Wiley &amp; Sons |isbn=978-0-470-00955-0 |language=en}}&lt;/ref&gt; In general, cookies are small, flattened, and individually shaped,&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Davidson |first=Alan |url=http://www.oxfordreference.com/view/10.1093/acref/9780199677337.001.0001/acref-9780199677337 |title=The Oxford Companion to Food |date=2014-01-01 |publisher=Oxford University Press |isbn=978-0-19-967733-7 |editor-last=Jaine |editor-first=Tom |doi=10.1093/acref/9780199677337.001.0001}}&lt;/ref&gt; baked free-form on a sheet and finishing with low moisture content.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Manley |first=Duncan |url=https://books.google.com/books?id=v5NwAgAAQBAJ&amp;newbks=0&amp;hl=en |title=Manley’s Technology of Biscuits, Crackers and Cookies |date=2011-09-28 |publisher=Elsevier |isbn=978-0-85709-364-6 |language=en}}&lt;/ref&gt; They are usually eaten with the hands.

Beyond these similarities, cookies come in a wide range of sizes, textures, shapes, flavors, and more.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The ideal characteristics will depend both on the specific type of cookie and the preferences of the individual&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Amendola |first=Joseph |url=https://books.google.com/books?id=bQcxoepvxOwC&amp;newbks=0&amp;hl=en |title=Understanding Baking: The Art and Science of Baking |last2=Rees |first2=Nicole |date=2003-01-03 |publisher=Wiley |isbn=978-0-471-44418-3 |language=en}}&lt;/ref&gt;—for example, some prefer chewy chocolate chip cookies while other prefer theirs crisp.

=== Composition ===
The primary components in cookie dough are typically some kind of sugar, fat, and flour, with the option of eggs and additional flavorings depending on the variety of cookie. All sugars, of course, contribute sweetness and browning to the cookie.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Ruhlman |first=Michael |url=https://books.google.com/books?id=cO6-wjf_XdgC&amp;newbks=0&amp;hl=en |title=The Elements of Cooking: Translating the Chef's Craft for Every Kitchen |date=2008 |publisher=Black Incorporated |isbn=978-1-86395-143-2 |language=en}}&lt;/ref&gt; Granulated sugar helps incorporate air into cookie dough,&lt;ref name=&quot;:2&quot; /&gt; especially when creamed well with a solid fat or egg.&lt;ref name=&quot;:10&quot; /&gt; Granulated sugar also melts in the oven before recrystallizing upon cooling, thereby contributing a crispness to the finished cookie.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:2&quot; /&gt; Liquid sugars like glucose syrup, honey, and molasses will not cause crystallization in this way, instead attracting water and causing softness.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;

Flour in cookies is primarily structural,&lt;ref name=&quot;:12&quot; /&gt; though it can contribute to color—bleached, low-protein wheat flour brown less, for example.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; High-protein and high-starch wheat flours (e.g. bread and cake flours, respectively) absorb more water and cause less spread than cookies made with pastry or all-purpose flours.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Pastry and all-purpose flours, however, are very commonly used.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Having a high ratio of flour to water yields dry, crumbly cookies without either gluten development or significant starch gelation.&lt;ref name=&quot;:12&quot; /&gt; A high ratio of water to flour allows for starch gelation but not gluten development, and the texture will be either cake-like or crisp.&lt;ref name=&quot;:12&quot; /&gt;

Fats contribute flavor, a feeling of moistness, and tenderness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; As lubricants, they disrupt the solid particles and contribute fluidity overall.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; The higher the melting point and the less liquid the fat, the less spread will occur in the oven;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; additionally, lower water content in the fat will also reduce spread compared to a pure fat.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Butter is more flavorful than shortening but will cause more spread.&lt;ref name=&quot;:15&quot; /&gt;

Chemical leaveners may be used in cookies, both for their leavening power and secondary effects. Baking soda, for example, neutralizes some of the acidity in the cookie dough, resulting in more spread and more browning.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Baking powder, on the other hand, will not significantly affect the acidity of the dough.&lt;ref name=&quot;:15&quot; /&gt;

The bulk of the water in cookie dough tends to come from egg, which also contributes protein for binding and a little fat from the yolk.&lt;ref name=&quot;:10&quot; /&gt; As a result of the moisture, eggs often contribute a puffiness and cakiness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; 

Additional flavorings and mix-ins may be incorporated into cookies, such as dried fruits, nuts, seeds, chocolate, candies, and more.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt;

=== Spread ===
The spread of a cookie refers to how much the dough spreads out horizontally during baking, and a number of factors can contribute to increased spread. Spread occurs because as the dough heats up in the oven, it becomes more thin and fluid, coheres less well, and spreads out.&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=Figoni |first=Paula |url=https://books.google.com/books?id=8ZpUDwAAQBAJ&amp;newbks=0&amp;hl=en |title=How Baking Works: Exploring the Fundamentals of Baking Science |date=2010-11-09 |publisher=John Wiley &amp; Sons |isbn=978-0-470-39267-6 |language=en}}&lt;/ref&gt; Eventually, with enough heat and time, the starches in the dough gelatinize and the proteins coagulate, which prevents any further spread of the dough.&lt;ref name=&quot;:3&quot; /&gt; A higher proportion of liquid in the dough evidently increases fluidity and spread.&lt;ref name=&quot;:11&quot; /&gt; A higher overall sugar content also has the effect of drawing more water into the fluid phase of the dough, making the dough more fluid and increasing spread.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Sugar also interferes with the gelatinization and coagulation, so the spread can proceed for longer before setting.&lt;ref name=&quot;:3&quot; /&gt; High-protein flour binds more water than does low-protein flour,&lt;ref name=&quot;:11&quot; /&gt; decreasing dough fluidity and therefore decreasing spread.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Adding starch to the dough achieves a similar effect.&lt;ref name=&quot;:6&quot; /&gt; Using fats with a lower melting point increases the dough fluidity and spread&lt;ref name=&quot;:11&quot; /&gt;—oil causes more spread than does butter,&lt;ref name=&quot;:3&quot; /&gt; which in turn causes more spread than does vegetable shortening.&lt;ref name=&quot;:11&quot; /&gt; Increasing the amount of air in the dough lowers its cohesive strength, thereby increasing spread; as a result, incorporating more air through extended creaming and additional leaveners will increase spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Leaveners that reduce the acidity of the dough (e.g. baking soda instead of baking powder) delay protein coagulation and setting of the dough.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Oat pulp cookies on tray.jpg|thumb|Cookies with relatively little spread]]
The baking environment will also influence spread. Greasing the baking surface decreases the friction between the dough and the surface, causing greater spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; A lower oven temperature increases spread because it takes longer for the proteins and starches to heat up and set the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Perhaps counterintuitively, however, chilling the dough actually reduces spread by making it take longer for the entire dough mass to warm through and become fluid; the crust can therefore form and set before much spread has a chance to occur.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Stemple Creek Ranch - August 2024 - Sarah Stierch 34.jpg|thumb|Cookies with a high amount of spread]]
The following table summarizes factors that affect cookie spread.
{| class=&quot;wikitable&quot;
!Decreased spread
!Increased spread
|-
|
* Cold dough
* Hotter oven
* Less liquid
* High melting point fat
* High-protein flour
* Ungreased pan
* Add starch/flour
* Less air
* Less sugar
* Coarser sugar
* Powdered sugar (contains starch) instead of granulated
* Increase acidity
|
* Warm dough
* Cooler oven
* More liquid
* Low melting point fat
* Low-protein flour
* Greased pan
* Reduce starch/flour
* More air
* More sugar
* Finer sugar
* Granulated sugar instead of powdered
* Decrease acidity
|}

=== Texture ===
A variety of factors can contribute to a cookie's chewiness, including an overall high moisture content, a high ratio of sugar and liquid to fat, a high egg content, high-protein flour, and increased stirring to develop any gluten in the flour.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Soft cookies also require a high moisture content, but they have less sugar, and the sugars used may be particularly water-attracting (e.g. honey, glucose/corn syrup).&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Conversely, the main cause of crispness in a cookie is very low moisture.&lt;ref name=&quot;:8&quot; /&gt; This can be achieved by simply including little moisture in the initial dough, using high sugar and fat proportions for pliability with the low moisture, baking for long enough to evaporate the moisture, and shaping the cookies thinly in order to more quickly achieve the latter.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Note that cookies will not be crispy while still warm; they must first cool down to achieve their snap.&lt;ref name=&quot;:2&quot;&gt;{{Cite web |title=Cookie science: How to achieve your perfect chocolate chip cookie {{!}} King Arthur Baking |url=https://www.kingarthurbaking.com/blog/2016/12/21/cookie-science |access-date=2025-07-25 |website=www.kingarthurbaking.com |language=en}}&lt;/ref&gt;

The following table links cookie texture with a variety of parameters involved in their creation:&lt;ref name=&quot;:6&quot; /&gt;
{| class=&quot;wikitable&quot;
!Desired texture
!Fat content
!Sugar content
!Sugar type
!Liquid content
!Four type
!Shape
|-
|Crispy
|High
|High
|Granulated
|Low
|Strong
|Thin dough
|-
|Soft
|Low
|Low
|Hygroscopic
|High
|Weak
|Thick dough
|-
|Chewy
|High
|High
|Hygroscopic
|High
|Strong
|
|-
|High-spread
|High
|High
|Coarse granulated
|High
|Weak
|
|}
== Types ==
As mentioned, a huge variety of cookies exists, and these can be classified in a few ways.

=== By mixing method ===
Cookie dough mixing methods are very similar to those for cakes.&lt;ref name=&quot;:8&quot; /&gt; In the one-stage method, all ingredients are mixed together at once&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Goldstein |first=Darra |url=http://www.oxfordreference.com/view/10.1093/acref/9780199313396.001.0001/acref-9780199313396 |title=The Oxford Companion to Sugar and Sweets |date=2015-01-01 |publisher=Oxford University Press |isbn=978-0-19-931339-6 |doi=10.1093/acref/9780199313396.001.0001}}&lt;/ref&gt;—this works adequately if there is not a lot of risk of gluten development.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; In creaming, the solid fat and sugar are beaten together before incorporating the flour and any liquids;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the degree of creaming will impact the aeration and spread as described above. With sanding, the flour and fat are rubbed together until crumbly or sandy before incorporating the moist ingredients.&lt;ref name=&quot;:8&quot; /&gt; Sponge/wafer/tuile cookie batters are made as with foaming, and the batter is delicate.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== By shaping ===
'''Drop''' cookies are formed from a soft dough portioned out in dollops onto the baking pan.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; This is often done with portion scoops, spoons, or the hands;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the dough may also be rolled between the hands into a more compact ball before panning and baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Some doughs may need to be flattened slightly for optimal spread and shaping.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=Jl0EEAAAQBAJ&amp;newbks=0 |title=Baking and Pastry: Mastering the Art and Craft |date=2015-02-25 |publisher=John Wiley &amp; Sons |isbn=978-0-470-92865-3 |language=en}}&lt;/ref&gt; Classic chocolate chip cookies and oatmeal cookies are examples of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Doughs for drop cookies are often made by the creaming method.&lt;ref name=&quot;:15&quot; /&gt; These doughs can often be made into icebox cookies (see below).&lt;ref name=&quot;:6&quot; /&gt;

A similar variety of cookie is the '''piped''', '''bagged''', '''pressed''', or '''spritz''' cookie,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; which also uses a soft dough that is stiff enough to hold a shape.&lt;ref name=&quot;:8&quot; /&gt; Here, the dough is forced through a nozzle, either using a pastry bag or a specialty [[Cookbook:Cookie press|cookie press]],&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; onto the prepared baking sheet. In general, this is done to create distinctive shapes and designs,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; so the dough should not be one that will spread significantly.&lt;ref name=&quot;:15&quot; /&gt; '''Molded''' cookies are also formed into specific and sometimes intricate designs, using hands, stamps, or molds&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;—the dough here may need to be chilled or otherwise stiffened for ease of working.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;

With so-called '''icebox''' cookies, the soft dough is shaped into a log and chilled in the refrigerator or freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Once solidified, cookies are evenly sliced off the log for baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; These cookies can be convenient, as the dough can be made and stored for some time before you are ready to bake the cookies.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Breaking with the above styles, '''rolled-and-cut''' or '''cutout''' cookies are made by rolling a relatively stiff dough out in large sheets on a floured surface before cutting out individual cookies using knives or specialized cutters.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=iGiM0YpaoDQC&amp;newbks=0 |title=The Professional Chef |date=2011-09-13 |publisher=John Wiley &amp; Sons |isbn=978-0-470-42135-2 |language=en}}&lt;/ref&gt; It's important to chill the dough sufficiently before rolling it out in order to avoid difficult handling.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Cookies should also be cut starting from the outside of the sheet and working towards the inside.&lt;ref name=&quot;:6&quot; /&gt; One downside to this style of cookie is that it tends to produce a lot of scraps—these can be pressed together and re-rolled, but the dough tends to get tougher as this continues, especially as more flour is incorporated.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If needed, the dough can be rolled out between sheets of parchment, plastic, or silicone.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

'''Bar''' cookies—not to be confused with sheet cookies (below), which are sometimes erroneously called bar cookies&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;—are made by baking a stiff dough in a large mass and then cutting into individual cookies while still warm.&lt;ref name=&quot;:8&quot; /&gt; Biscotti are one well-known example of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Like bar cookies, '''sheet/pan cookies''' or '''traybakes''' are portioned after baking in sheets.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; For this type, the dough or batter is pressed or spread in a shallow layer in the baking pan;&lt;ref name=&quot;:6&quot; /&gt; once baked, it is unmolded and cut into individual portion sizes.&lt;ref name=&quot;:6&quot; /&gt; Clean and even cuts can be more easily achieved by first chilling the baked mass by either refrigerating or freezing,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; depending on the product. Some consider brownies as sheet cookies,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; while others consider them cakes. Lemon bars, nut bars are other examples of preparations sometimes classified as sheet cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

'''Wafer''', '''tuile''', '''tulipe''', or '''stencil''' cookies are made with a thin dough or batter.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This batter is spread on silicone or parchment-lined pans, sometimes using a stencil cutout (hence the name),&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; before baking. Then, while still warm, the cookies can be trimmed or formed into a variety of shapes such as cups, rolls, cones, and more before cooling and setting.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This produces a thin, delicate, crisp cookie.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Due to their quick baking and complex handling, they should be prepared in small batches.&lt;ref name=&quot;:14&quot; /&gt;

=== Other ===
'''Sandwich cookies''' are, as the name implies, made by sandwiching a filling between two cookies. These include macarons, oreos, alfajores, linzer cookies, and more.

== Selection and storage ==
Due to their high sugar content and low moisture content, cookies generally will not spoil at room temperature.&lt;ref name=&quot;:10&quot; /&gt; Depending on their storage conditions, however, they may go stale; crispy cookies will absorb moisture and become chewy,&lt;ref name=&quot;:7&quot; /&gt; while soft cookies will lose moisture and become hard.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The best way to store cookies is in an airtight container,&lt;ref name=&quot;:5&quot; /&gt; with a desiccant for crisp cookies and a moist item like an apple slice for soft cookies.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Unfrosted cookie dough and baked cookies also freeze well when protected from air.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

== Use ==

Cookies are commonly served on their own as a snack or dessert.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; However, they can also be used as ingredients in their own right, either as decor&lt;ref name=&quot;:6&quot; /&gt; or as an integral component of another dish. An example of the latter is the use of crisp cookies to make crumb crusts.

== Techniques ==

=== Mixing ===
The mixing technique used when preparing the cookie dough will largely affect how the dough bakes and therefore the final texture of the cookie.&lt;ref name=&quot;:6&quot; /&gt; Creaming the butter before adding granulated sugar and not much afterwards will reduce the amount of incorporated air and reduce the puff and spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; When making dense cookies that will hold their shape, you should avoid excessive or vigorous creaming and incorporation of air.&lt;ref name=&quot;:6&quot; /&gt;''&lt;ref name=&quot;:3&quot; /&gt;'' If a gluten-containing flour is used in a dough with a moderate amount of liquid, mixing should be kept to a minimum after the addition of the flour to prevent toughness.&lt;ref name=&quot;:6&quot; /&gt; It is also important to scrape down the mixing bowl when preparing the dough to ensure that all ingredients are properly distributed.&lt;ref name=&quot;:6&quot; /&gt;

=== Forming ===
Cookies should be formed according to their type and placed on a baking sheet prepared according to the recipe specifications.&lt;ref name=&quot;:14&quot; /&gt; The most important consideration when forming any variety of cookie is consistency.&lt;ref name=&quot;:8&quot; /&gt; Each cookie should be the same size, shape, and thickness to ensure proper and even baking across the entire sheet.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; It is also important to space the cookies out enough that they will not touch each other when spreading in the oven,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; and so that heat can circulate evenly.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; The amount of space necessary will depend on the size and the cookie type. If adding garnishes before baking, these should be applied to the surface immediately after forming and pressed in gently to ensure they adhere properly to the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== Baking and cooling ===
Most cookies are baked at a moderately high temperature for a short time to allow proper baking and coloration—somewhere from 8–14 minutes at 325–375°F (163–191°C) is common.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Doneness is often indicated by color,&lt;ref name=&quot;:8&quot; /&gt; though this can be difficult to ascertain in darker cookies such as those containing chocolate. If possible, take a peek at the undersides of the cookies, since these often cook faster than the top. If there is a risk of overcooking on the bottom relative to the rest of the cookies, double-layering the pans will help insulate the bottom from the heat.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; Note as well that some carryover cooking will occur even after the cookies are removed from the oven.&lt;ref name=&quot;:6&quot; /&gt;

Often, cookies need to rest for a moment on the pans before moving to cooling racks, since many cookies are too soft to move while warm.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; However, timely removal of cookies from unlined pans is often critical, since there is a risk of the cookies sticking to the pan once they set.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Excessive carryover cooking can also occur.&lt;ref name=&quot;:14&quot; /&gt; Cool cookies completely to room temperature before storage.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;

=== Finishing ===
After baking, the cookies can be finished in a variety of ways.&lt;ref name=&quot;:6&quot; /&gt; Some are decorated, sandwiched, or filled with frosting, icing, chocolate, jam, fondant, marshmallow, and more.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; In most cases, it's important to cool the cookies completely before decorating and finishing.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If storing the cookies by freezing, wait to finish them until you remove them from the freezer and return them to room temperature, since the decorations are often damaged in the freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

=== Troubleshooting ===
The following table outlines some possible issues and causes that can arise when making cookies:&lt;ref name=&quot;:8&quot; /&gt;

{| class=&quot;wikitable&quot;
!Problem
!Potential cause
|-
| rowspan=&quot;5&quot; |Too tough
|Flour too strong
|-
|Too much flour
|-
|Incorrect amount of sugar
|-
|Not enough shortening
|-
|Mixed too long or improper mixing
|-
| rowspan=&quot;5&quot; |Too crumbly
|Improper mixing
|-
|Too much sugar
|-
|Too much shortening
|-
|Too much leavening
|-
|Not enough eggs
|-
| rowspan=&quot;6&quot; |Too hard
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
|Flour too strong
|-
|Not enough shortening
|-
|Not enough liquid
|-
| rowspan=&quot;5&quot; |Too dry 
|Not enough liquid
|-
|Not enough shortening
|-
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
| rowspan=&quot;3&quot; |Not browned enough
|Baking temperature too low
|-
|Underbaked
|-
|Not enough sugar
|-
| rowspan=&quot;3&quot; |Too brown 
|Baking temperature too high
|-
|Baked too long
|-
|Too much sugar
|-
| rowspan=&quot;4&quot; |Poor flavor
|Poor-quality ingredients
|-
|Flavoring ingredients left out
|-
|Dirty baking pans
|-
|Ingredients improperly measured
|-
| rowspan=&quot;2&quot; |Sugary surface or crust
|Improper mixing
|-
|Too much sugar
|-
| rowspan=&quot;6&quot; |Too much spread
|Baking temperature too low
|-
|Not enough flour
|-
|Too much sugar
|-
|Too much leavening (chemical leaveners or creaming)
|-
|Too much liquid
|-
|Pans greased too heavily
|-
| rowspan=&quot;6&quot; |Not enough spread
|Baking temperature too high
|-
|Too much flour or flour too strong
|-
|Not enough sugar
|-
|Not enough leavening
|-
|Not enough liquid
|-
|Insufficient pan grease
|-
| rowspan=&quot;3&quot; |Stick to pans 
|Pans improperly greased
|-
|Too much sugar
|-
|Improper mixing
|}

== Recipes ==

=== For cookies ===
{{div col|3}}
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes for cookies&lt;/categorytree&gt;
{{div col end}}

=== Using cookies ===
&lt;div style=&quot;column-count:3&quot;&gt;
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes using cookies&lt;/categorytree&gt;
&lt;/div&gt;

== Gallery ==
&lt;gallery mode=&quot;packed&quot;&gt;
File:2ChocolateChipCookies.jpg|Chocolate chip drop cookies
File:Macaron trio.jpg|Macarons
File:Biscotti 1.jpg|Biscotti
File:Alfajores-de-maicena-biscuits-recipe.jpg|Alfajores
File:Diamond-cut pecan bars (5631275901).jpg|Pecan bars
File:Round gingersnaps.jpg|Gingersnaps
File:Fortune cookies.jpg|Fortune cookies
File:Cattongues.jpg|Langues de chat
File:Caramel-colored-wafer-sticks.jpg|Tuile cookies shaped into cigars
File:Sofra Coconut macaroon, October 2009.jpg|Macaroon
File:2020-07-23 20 05 30 Wholesale Pantry Organic Vanilla Animal Crackers in Ewing Township, Mercer County, New Jersey.jpg|Animal crackers
File:Vegan Black and White Cookies (8746952832).jpg|Black-and-white cookies
File:Hamantaschen1.jpg|Hamantaschen
File:Russian teacakes (8307560889).jpg|Russian teacakes
File:Stroopwafels 01.jpg|Stroopwafels
&lt;/gallery&gt;

== References ==</text>
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      <comment>Reverted edits by [[Special:Contribs/~2026-23858-31|~2026-23858-31]] ([[User talk:~2026-23858-31|talk]]) to last version by MathXplore: unexplained content removal</comment>
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{{Ingredient summary
| Image = [[File:Koekjestrommel open.jpg|300px]]
}}

{{ingredient}}

'''Cookies''' are a class of sweet baked good, overlapping somewhat with [[Cookbook:Cake|cakes]] and [[Cookbook:Pastry|pastries]]. The term &quot;cookie&quot; is reported to derive from the Dutch for &quot;little cake&quot;,&lt;ref name=&quot;:7&quot;&gt;{{Cite book |last=Friberg |first=Bo |url=https://books.google.com/books?id=gmpkzgAACAAJ&amp;newbks=0&amp;hl=en |title=The Professional Pastry Chef: Fundamentals of Baking and Pastry |date=2016-09-13 |publisher=Wiley |isbn=978-0-470-46629-2 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite book |last=Labensky |first=Sarah |url=https://books.google.com/books?id=k87uoQEACAAJ&amp;newbks=0&amp;hl=en |title=On Baking: A Textbook of Baking and Pastry Fundamentals, Updated Edition |last2=Martel |first2=Priscilla |last3=Damme |first3=Eddy Van |date=2015-01-06 |publisher=Pearson Education |isbn=978-0-13-388675-7 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:10&quot;&gt;{{Cite book |last=McGee |first=Harold |url=https://www.google.com/books/edition/On_Food_and_Cooking/bKVCtH4AjwgC?hl=en&amp;gbpv=0 |title=On Food and Cooking: The Science and Lore of the Kitchen |date=2007-03-20 |publisher=Simon and Schuster |isbn=978-1-4165-5637-4 |language=en}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot;&gt;{{Cite book |last=Gisslen |first=Wayne |url=https://books.google.com/books?id=KGjpCgAAQBAJ&amp;newbks=0&amp;hl=en |title=Professional Baking |date=2016-09-21 |publisher=John Wiley &amp; Sons |isbn=978-1-119-14844-9 |language=en}}&lt;/ref&gt; and it encompasses a wide variety of baked goods, including sweet biscuits in British English.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Savory biscuits as designated by British English are classified in the Cookbook as [[Cookbook:Cracker|crackers]].

== Characteristics ==
It can be challenging to draw a precise line between all cookies and other sweet baked goods (e.g. cakes),&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Understanding the Science of Cookies {{!}} Institute of Culinary Education |url=https://www.ice.edu/blog/understanding-science-cookies |access-date=2025-01-01 |website=www.ice.edu}}&lt;/ref&gt; since the term is applied to a wide variety of small, sweet, baked morsels.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot;&gt;{{Cite book |last=Rinsky |first=Glenn |url=https://books.google.com/books?id=mZhyDwAAQBAJ&amp;newbks=0&amp;hl=en |title=The Pastry Chef's Companion: A Comprehensive Resource Guide for the Baking and Pastry Professional |last2=Rinsky |first2=Laura Halpin |date=2008-02-28 |publisher=John Wiley &amp; Sons |isbn=978-0-470-00955-0 |language=en}}&lt;/ref&gt; In general, cookies are small, flattened, and individually shaped,&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Davidson |first=Alan |url=http://www.oxfordreference.com/view/10.1093/acref/9780199677337.001.0001/acref-9780199677337 |title=The Oxford Companion to Food |date=2014-01-01 |publisher=Oxford University Press |isbn=978-0-19-967733-7 |editor-last=Jaine |editor-first=Tom |doi=10.1093/acref/9780199677337.001.0001}}&lt;/ref&gt; baked free-form on a sheet and finishing with low moisture content.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite book |last=Manley |first=Duncan |url=https://books.google.com/books?id=v5NwAgAAQBAJ&amp;newbks=0&amp;hl=en |title=Manley’s Technology of Biscuits, Crackers and Cookies |date=2011-09-28 |publisher=Elsevier |isbn=978-0-85709-364-6 |language=en}}&lt;/ref&gt; They are usually eaten with the hands.

Beyond these similarities, cookies come in a wide range of sizes, textures, shapes, flavors, and more.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The ideal characteristics will depend both on the specific type of cookie and the preferences of the individual&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book |last=Amendola |first=Joseph |url=https://books.google.com/books?id=bQcxoepvxOwC&amp;newbks=0&amp;hl=en |title=Understanding Baking: The Art and Science of Baking |last2=Rees |first2=Nicole |date=2003-01-03 |publisher=Wiley |isbn=978-0-471-44418-3 |language=en}}&lt;/ref&gt;—for example, some prefer chewy chocolate chip cookies while other prefer theirs crisp.

=== Composition ===
The primary components in cookie dough are typically some kind of sugar, fat, and flour, with the option of eggs and additional flavorings depending on the variety of cookie. All sugars, of course, contribute sweetness and browning to the cookie.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:12&quot;&gt;{{Cite book |last=Ruhlman |first=Michael |url=https://books.google.com/books?id=cO6-wjf_XdgC&amp;newbks=0&amp;hl=en |title=The Elements of Cooking: Translating the Chef's Craft for Every Kitchen |date=2008 |publisher=Black Incorporated |isbn=978-1-86395-143-2 |language=en}}&lt;/ref&gt; Granulated sugar helps incorporate air into cookie dough,&lt;ref name=&quot;:2&quot; /&gt; especially when creamed well with a solid fat or egg.&lt;ref name=&quot;:10&quot; /&gt; Granulated sugar also melts in the oven before recrystallizing upon cooling, thereby contributing a crispness to the finished cookie.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:2&quot; /&gt; Liquid sugars like glucose syrup, honey, and molasses will not cause crystallization in this way, instead attracting water and causing softness.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;

Flour in cookies is primarily structural,&lt;ref name=&quot;:12&quot; /&gt; though it can contribute to color—bleached, low-protein wheat flour brown less, for example.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; High-protein and high-starch wheat flours (e.g. bread and cake flours, respectively) absorb more water and cause less spread than cookies made with pastry or all-purpose flours.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Pastry and all-purpose flours, however, are very commonly used.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; Having a high ratio of flour to water yields dry, crumbly cookies without either gluten development or significant starch gelation.&lt;ref name=&quot;:12&quot; /&gt; A high ratio of water to flour allows for starch gelation but not gluten development, and the texture will be either cake-like or crisp.&lt;ref name=&quot;:12&quot; /&gt;

Fats contribute flavor, a feeling of moistness, and tenderness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; As lubricants, they disrupt the solid particles and contribute fluidity overall.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; The higher the melting point and the less liquid the fat, the less spread will occur in the oven;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; additionally, lower water content in the fat will also reduce spread compared to a pure fat.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Butter is more flavorful than shortening but will cause more spread.&lt;ref name=&quot;:15&quot; /&gt;

Chemical leaveners may be used in cookies, both for their leavening power and secondary effects. Baking soda, for example, neutralizes some of the acidity in the cookie dough, resulting in more spread and more browning.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Baking powder, on the other hand, will not significantly affect the acidity of the dough.&lt;ref name=&quot;:15&quot; /&gt;

The bulk of the water in cookie dough tends to come from egg, which also contributes protein for binding and a little fat from the yolk.&lt;ref name=&quot;:10&quot; /&gt; As a result of the moisture, eggs often contribute a puffiness and cakiness to cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; 

Additional flavorings and mix-ins may be incorporated into cookies, such as dried fruits, nuts, seeds, chocolate, candies, and more.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt;

=== Spread ===
The spread of a cookie refers to how much the dough spreads out horizontally during baking, and a number of factors can contribute to increased spread. Spread occurs because as the dough heats up in the oven, it becomes more thin and fluid, coheres less well, and spreads out.&lt;ref name=&quot;:3&quot;&gt;{{Cite book |last=Figoni |first=Paula |url=https://books.google.com/books?id=8ZpUDwAAQBAJ&amp;newbks=0&amp;hl=en |title=How Baking Works: Exploring the Fundamentals of Baking Science |date=2010-11-09 |publisher=John Wiley &amp; Sons |isbn=978-0-470-39267-6 |language=en}}&lt;/ref&gt; Eventually, with enough heat and time, the starches in the dough gelatinize and the proteins coagulate, which prevents any further spread of the dough.&lt;ref name=&quot;:3&quot; /&gt; A higher proportion of liquid in the dough evidently increases fluidity and spread.&lt;ref name=&quot;:11&quot; /&gt; A higher overall sugar content also has the effect of drawing more water into the fluid phase of the dough, making the dough more fluid and increasing spread.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Sugar also interferes with the gelatinization and coagulation, so the spread can proceed for longer before setting.&lt;ref name=&quot;:3&quot; /&gt; High-protein flour binds more water than does low-protein flour,&lt;ref name=&quot;:11&quot; /&gt; decreasing dough fluidity and therefore decreasing spread.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Adding starch to the dough achieves a similar effect.&lt;ref name=&quot;:6&quot; /&gt; Using fats with a lower melting point increases the dough fluidity and spread&lt;ref name=&quot;:11&quot; /&gt;—oil causes more spread than does butter,&lt;ref name=&quot;:3&quot; /&gt; which in turn causes more spread than does vegetable shortening.&lt;ref name=&quot;:11&quot; /&gt; Increasing the amount of air in the dough lowers its cohesive strength, thereby increasing spread; as a result, incorporating more air through extended creaming and additional leaveners will increase spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Leaveners that reduce the acidity of the dough (e.g. baking soda instead of baking powder) delay protein coagulation and setting of the dough.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Oat pulp cookies on tray.jpg|thumb|Cookies with relatively little spread]]
The baking environment will also influence spread. Greasing the baking surface decreases the friction between the dough and the surface, causing greater spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; A lower oven temperature increases spread because it takes longer for the proteins and starches to heat up and set the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Perhaps counterintuitively, however, chilling the dough actually reduces spread by making it take longer for the entire dough mass to warm through and become fluid; the crust can therefore form and set before much spread has a chance to occur.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;
[[File:Stemple Creek Ranch - August 2024 - Sarah Stierch 34.jpg|thumb|Cookies with a high amount of spread]]
The following table summarizes factors that affect cookie spread.
{| class=&quot;wikitable&quot;
!Decreased spread
!Increased spread
|-
|
* Cold dough
* Hotter oven
* Less liquid
* High melting point fat
* High-protein flour
* Ungreased pan
* Add starch/flour
* Less air
* Less sugar
* Coarser sugar
* Powdered sugar (contains starch) instead of granulated
* Increase acidity
|
* Warm dough
* Cooler oven
* More liquid
* Low melting point fat
* Low-protein flour
* Greased pan
* Reduce starch/flour
* More air
* More sugar
* Finer sugar
* Granulated sugar instead of powdered
* Decrease acidity
|}

=== Texture ===
A variety of factors can contribute to a cookie's chewiness, including an overall high moisture content, a high ratio of sugar and liquid to fat, a high egg content, high-protein flour, and increased stirring to develop any gluten in the flour.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Soft cookies also require a high moisture content, but they have less sugar, and the sugars used may be particularly water-attracting (e.g. honey, glucose/corn syrup).&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Conversely, the main cause of crispness in a cookie is very low moisture.&lt;ref name=&quot;:8&quot; /&gt; This can be achieved by simply including little moisture in the initial dough, using high sugar and fat proportions for pliability with the low moisture, baking for long enough to evaporate the moisture, and shaping the cookies thinly in order to more quickly achieve the latter.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Note that cookies will not be crispy while still warm; they must first cool down to achieve their snap.&lt;ref name=&quot;:2&quot;&gt;{{Cite web |title=Cookie science: How to achieve your perfect chocolate chip cookie {{!}} King Arthur Baking |url=https://www.kingarthurbaking.com/blog/2016/12/21/cookie-science |access-date=2025-07-25 |website=www.kingarthurbaking.com |language=en}}&lt;/ref&gt;

The following table links cookie texture with a variety of parameters involved in their creation:&lt;ref name=&quot;:6&quot; /&gt;
{| class=&quot;wikitable&quot;
!Desired texture
!Fat content
!Sugar content
!Sugar type
!Liquid content
!Four type
!Shape
|-
|Crispy
|High
|High
|Granulated
|Low
|Strong
|Thin dough
|-
|Soft
|Low
|Low
|Hygroscopic
|High
|Weak
|Thick dough
|-
|Chewy
|High
|High
|Hygroscopic
|High
|Strong
|
|-
|High-spread
|High
|High
|Coarse granulated
|High
|Weak
|
|}
== Types ==
As mentioned, a huge variety of cookies exists, and these can be classified in a few ways.

=== By mixing method ===
Cookie dough mixing methods are very similar to those for cakes.&lt;ref name=&quot;:8&quot; /&gt; In the one-stage method, all ingredients are mixed together at once&lt;ref name=&quot;:5&quot;&gt;{{Cite book |last=Goldstein |first=Darra |url=http://www.oxfordreference.com/view/10.1093/acref/9780199313396.001.0001/acref-9780199313396 |title=The Oxford Companion to Sugar and Sweets |date=2015-01-01 |publisher=Oxford University Press |isbn=978-0-19-931339-6 |doi=10.1093/acref/9780199313396.001.0001}}&lt;/ref&gt;—this works adequately if there is not a lot of risk of gluten development.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; In creaming, the solid fat and sugar are beaten together before incorporating the flour and any liquids;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the degree of creaming will impact the aeration and spread as described above. With sanding, the flour and fat are rubbed together until crumbly or sandy before incorporating the moist ingredients.&lt;ref name=&quot;:8&quot; /&gt; Sponge/wafer/tuile cookie batters are made as with foaming, and the batter is delicate.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== By shaping ===
'''Drop''' cookies are formed from a soft dough portioned out in dollops onto the baking pan.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; This is often done with portion scoops, spoons, or the hands;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; the dough may also be rolled between the hands into a more compact ball before panning and baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Some doughs may need to be flattened slightly for optimal spread and shaping.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=Jl0EEAAAQBAJ&amp;newbks=0 |title=Baking and Pastry: Mastering the Art and Craft |date=2015-02-25 |publisher=John Wiley &amp; Sons |isbn=978-0-470-92865-3 |language=en}}&lt;/ref&gt; Classic chocolate chip cookies and oatmeal cookies are examples of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; Doughs for drop cookies are often made by the creaming method.&lt;ref name=&quot;:15&quot; /&gt; These doughs can often be made into icebox cookies (see below).&lt;ref name=&quot;:6&quot; /&gt;

A similar variety of cookie is the '''piped''', '''bagged''', '''pressed''', or '''spritz''' cookie,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; which also uses a soft dough that is stiff enough to hold a shape.&lt;ref name=&quot;:8&quot; /&gt; Here, the dough is forced through a nozzle, either using a pastry bag or a specialty [[Cookbook:Cookie press|cookie press]],&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; onto the prepared baking sheet. In general, this is done to create distinctive shapes and designs,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; so the dough should not be one that will spread significantly.&lt;ref name=&quot;:15&quot; /&gt; '''Molded''' cookies are also formed into specific and sometimes intricate designs, using hands, stamps, or molds&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;—the dough here may need to be chilled or otherwise stiffened for ease of working.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;

With so-called '''icebox''' cookies, the soft dough is shaped into a log and chilled in the refrigerator or freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt; Once solidified, cookies are evenly sliced off the log for baking.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; These cookies can be convenient, as the dough can be made and stored for some time before you are ready to bake the cookies.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Breaking with the above styles, '''rolled-and-cut''' or '''cutout''' cookies are made by rolling a relatively stiff dough out in large sheets on a floured surface before cutting out individual cookies using knives or specialized cutters.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot;&gt;{{Cite book |last=The Culinary Institute of America (CIA) |first= |url=https://books.google.com/books?id=iGiM0YpaoDQC&amp;newbks=0 |title=The Professional Chef |date=2011-09-13 |publisher=John Wiley &amp; Sons |isbn=978-0-470-42135-2 |language=en}}&lt;/ref&gt; It's important to chill the dough sufficiently before rolling it out in order to avoid difficult handling.&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Cookies should also be cut starting from the outside of the sheet and working towards the inside.&lt;ref name=&quot;:6&quot; /&gt; One downside to this style of cookie is that it tends to produce a lot of scraps—these can be pressed together and re-rolled, but the dough tends to get tougher as this continues, especially as more flour is incorporated.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If needed, the dough can be rolled out between sheets of parchment, plastic, or silicone.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

'''Bar''' cookies—not to be confused with sheet cookies (below), which are sometimes erroneously called bar cookies&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;—are made by baking a stiff dough in a large mass and then cutting into individual cookies while still warm.&lt;ref name=&quot;:8&quot; /&gt; Biscotti are one well-known example of this variety.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

Like bar cookies, '''sheet/pan cookies''' or '''traybakes''' are portioned after baking in sheets.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; For this type, the dough or batter is pressed or spread in a shallow layer in the baking pan;&lt;ref name=&quot;:6&quot; /&gt; once baked, it is unmolded and cut into individual portion sizes.&lt;ref name=&quot;:6&quot; /&gt; Clean and even cuts can be more easily achieved by first chilling the baked mass by either refrigerating or freezing,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; depending on the product. Some consider brownies as sheet cookies,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; while others consider them cakes. Lemon bars, nut bars are other examples of preparations sometimes classified as sheet cookies.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

'''Wafer''', '''tuile''', '''tulipe''', or '''stencil''' cookies are made with a thin dough or batter.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This batter is spread on silicone or parchment-lined pans, sometimes using a stencil cutout (hence the name),&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; before baking. Then, while still warm, the cookies can be trimmed or formed into a variety of shapes such as cups, rolls, cones, and more before cooling and setting.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; This produces a thin, delicate, crisp cookie.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Due to their quick baking and complex handling, they should be prepared in small batches.&lt;ref name=&quot;:14&quot; /&gt;

=== Other ===
'''Sandwich cookies''' are, as the name implies, made by sandwiching a filling between two cookies. These include macarons, oreos, alfajores, linzer cookies, and more.

== Selection and storage ==
Due to their high sugar content and low moisture content, cookies generally will not spoil at room temperature.&lt;ref name=&quot;:10&quot; /&gt; Depending on their storage conditions, however, they may go stale; crispy cookies will absorb moisture and become chewy,&lt;ref name=&quot;:7&quot; /&gt; while soft cookies will lose moisture and become hard.&lt;ref name=&quot;:10&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; The best way to store cookies is in an airtight container,&lt;ref name=&quot;:5&quot; /&gt; with a desiccant for crisp cookies and a moist item like an apple slice for soft cookies.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Unfrosted cookie dough and baked cookies also freeze well when protected from air.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

== Use ==

Cookies are commonly served on their own as a snack or dessert.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; However, they can also be used as ingredients in their own right, either as decor&lt;ref name=&quot;:6&quot; /&gt; or as an integral component of another dish. An example of the latter is the use of crisp cookies to make crumb crusts.

== Techniques ==

=== Mixing ===
The mixing technique used when preparing the cookie dough will largely affect how the dough bakes and therefore the final texture of the cookie.&lt;ref name=&quot;:6&quot; /&gt; Creaming the butter before adding granulated sugar and not much afterwards will reduce the amount of incorporated air and reduce the puff and spread.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt; When making dense cookies that will hold their shape, you should avoid excessive or vigorous creaming and incorporation of air.&lt;ref name=&quot;:6&quot; /&gt;''&lt;ref name=&quot;:3&quot; /&gt;'' If a gluten-containing flour is used in a dough with a moderate amount of liquid, mixing should be kept to a minimum after the addition of the flour to prevent toughness.&lt;ref name=&quot;:6&quot; /&gt; It is also important to scrape down the mixing bowl when preparing the dough to ensure that all ingredients are properly distributed.&lt;ref name=&quot;:6&quot; /&gt;

=== Forming ===
Cookies should be formed according to their type and placed on a baking sheet prepared according to the recipe specifications.&lt;ref name=&quot;:14&quot; /&gt; The most important consideration when forming any variety of cookie is consistency.&lt;ref name=&quot;:8&quot; /&gt; Each cookie should be the same size, shape, and thickness to ensure proper and even baking across the entire sheet.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; It is also important to space the cookies out enough that they will not touch each other when spreading in the oven,&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; and so that heat can circulate evenly.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; The amount of space necessary will depend on the size and the cookie type. If adding garnishes before baking, these should be applied to the surface immediately after forming and pressed in gently to ensure they adhere properly to the dough.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== Baking and cooling ===
Most cookies are baked at a moderately high temperature for a short time to allow proper baking and coloration—somewhere from 8–14 minutes at 325–375°F (163–191°C) is common.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; Doneness is often indicated by color,&lt;ref name=&quot;:8&quot; /&gt; though this can be difficult to ascertain in darker cookies such as those containing chocolate. If possible, take a peek at the undersides of the cookies, since these often cook faster than the top. If there is a risk of overcooking on the bottom relative to the rest of the cookies, double-layering the pans will help insulate the bottom from the heat.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; Note as well that some carryover cooking will occur even after the cookies are removed from the oven.&lt;ref name=&quot;:6&quot; /&gt;

Often, cookies need to rest for a moment on the pans before moving to cooling racks, since many cookies are too soft to move while warm.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; However, timely removal of cookies from unlined pans is often critical, since there is a risk of the cookies sticking to the pan once they set.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; Excessive carryover cooking can also occur.&lt;ref name=&quot;:14&quot; /&gt; Cool cookies completely to room temperature before storage.&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;

=== Finishing ===
After baking, the cookies can be finished in a variety of ways.&lt;ref name=&quot;:6&quot; /&gt; Some are decorated, sandwiched, or filled with frosting, icing, chocolate, jam, fondant, marshmallow, and more.&lt;ref name=&quot;:7&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;&lt;ref name=&quot;:14&quot; /&gt; In most cases, it's important to cool the cookies completely before decorating and finishing.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt; If storing the cookies by freezing, wait to finish them until you remove them from the freezer and return them to room temperature, since the decorations are often damaged in the freezer.&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

=== Troubleshooting ===
The following table outlines some possible issues and causes that can arise when making cookies:&lt;ref name=&quot;:8&quot; /&gt;

{| class=&quot;wikitable&quot;
!Problem
!Potential cause
|-
| rowspan=&quot;5&quot; |Too tough
|Flour too strong
|-
|Too much flour
|-
|Incorrect amount of sugar
|-
|Not enough shortening
|-
|Mixed too long or improper mixing
|-
| rowspan=&quot;5&quot; |Too crumbly
|Improper mixing
|-
|Too much sugar
|-
|Too much shortening
|-
|Too much leavening
|-
|Not enough eggs
|-
| rowspan=&quot;6&quot; |Too hard
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
|Flour too strong
|-
|Not enough shortening
|-
|Not enough liquid
|-
| rowspan=&quot;5&quot; |Too dry 
|Not enough liquid
|-
|Not enough shortening
|-
|Baked too long
|-
|Baking temperature too low
|-
|Too much flour
|-
| rowspan=&quot;3&quot; |Not browned enough
|Baking temperature too low
|-
|Underbaked
|-
|Not enough sugar
|-
| rowspan=&quot;3&quot; |Too brown 
|Baking temperature too high
|-
|Baked too long
|-
|Too much sugar
|-
| rowspan=&quot;4&quot; |Poor flavor
|Poor-quality ingredients
|-
|Flavoring ingredients left out
|-
|Dirty baking pans
|-
|Ingredients improperly measured
|-
| rowspan=&quot;2&quot; |Sugary surface or crust
|Improper mixing
|-
|Too much sugar
|-
| rowspan=&quot;6&quot; |Too much spread
|Baking temperature too low
|-
|Not enough flour
|-
|Too much sugar
|-
|Too much leavening (chemical leaveners or creaming)
|-
|Too much liquid
|-
|Pans greased too heavily
|-
| rowspan=&quot;6&quot; |Not enough spread
|Baking temperature too high
|-
|Too much flour or flour too strong
|-
|Not enough sugar
|-
|Not enough leavening
|-
|Not enough liquid
|-
|Insufficient pan grease
|-
| rowspan=&quot;3&quot; |Stick to pans 
|Pans improperly greased
|-
|Too much sugar
|-
|Improper mixing
|}

== Recipes ==

=== For cookies ===
{{div col|3}}
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes for cookies&lt;/categorytree&gt;
{{div col end}}

=== Using cookies ===
&lt;div style=&quot;column-count:3&quot;&gt;
&lt;categorytree mode=&quot;all&quot; hideprefix=&quot;always&quot;&gt;Recipes using cookies&lt;/categorytree&gt;
&lt;/div&gt;

== Gallery ==
&lt;gallery mode=&quot;packed&quot;&gt;
File:2ChocolateChipCookies.jpg|Chocolate chip drop cookies
File:Macaron trio.jpg|Macarons
File:Biscotti 1.jpg|Biscotti
File:Alfajores-de-maicena-biscuits-recipe.jpg|Alfajores
File:Diamond-cut pecan bars (5631275901).jpg|Pecan bars
File:Round gingersnaps.jpg|Gingersnaps
File:Fortune cookies.jpg|Fortune cookies
File:Cattongues.jpg|Langues de chat
File:Caramel-colored-wafer-sticks.jpg|Tuile cookies shaped into cigars
File:Sofra Coconut macaroon, October 2009.jpg|Macaroon
File:2020-07-23 20 05 30 Wholesale Pantry Organic Vanilla Animal Crackers in Ewing Township, Mercer County, New Jersey.jpg|Animal crackers
File:Vegan Black and White Cookies (8746952832).jpg|Black-and-white cookies
File:Hamantaschen1.jpg|Hamantaschen
File:Russian teacakes (8307560889).jpg|Russian teacakes
File:Stroopwafels 01.jpg|Stroopwafels
&lt;/gallery&gt;

== References ==</text>
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    <title>Arabic/Arabic numbers</title>
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      <contributor>
        <username>~2026-24002-16</username>
        <id>3577656</id>
      </contributor>
      <comment>/* ١٠-٠ */Added Arabic digit two</comment>
      <origin>4631305</origin>
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      <text bytes="8189" sha1="cw7wbn7t4h0b6rmmtivuuy3fbszxltw" xml:space="preserve">== The Arabic Numbers ==
Arabic Numerals (how the numbers look) consist of two types: the numerals used in most of the world, which originate from Arabic, and those used in the Arabic language today. Interestingly, while the Arabic language is written from right to left, Arabic Numerals are written from left to right, for example:

“I am 42 years old today!” is translated into Arabic as “!اليوم عندي ٤٢ سنة.”

A pattern you might notice is that words containing a ١ or ٢ have word-forms that differ from any patterns
== Comparison of Arabic numerals ==
Sometimes, the numbers themselves are written in different forms, depending on the country.
{|class=&quot;wikitable nounderlines&quot; style=&quot;text-align:center;line-height:normal&quot;
|-
|Western Arabic 
|0 (number)|0 || 1 (number)|1 || 2 (number)|2 || 3 (number)|3 || 4 (number)|4 || 5 (number)|5 || 6 (number)|6 || 7 (number)|7 || 8 (number)|8 || 9 (number)|9 || 10 (number)|10
|-
|Eastern Arabic numerals{{efn|Arabic (Unicode block) U+0660 through U+0669}}
| {{lang|ar|٠}} || {{lang|ar|١}} || {{lang|ar|٢}} || {{lang|ar|٣}} || {{lang|ar|٤}} || {{lang|ar|٥}} || {{lang|ar|٦}} || {{lang|ar|٧}} || {{lang|ar|٨}} || {{lang|ar|٩}} || {{lang|ar|١٠}}
|- 
|Persian{{efn|Arabic (Unicode block) U+06F0 through U+06F9; The numbers 4, 5, and 6 are different from Eastern Arabic}}
| {{lang|fa|۰}} || {{lang|fa|۱}} || {{lang|fa|۲}} || {{lang|fa|۳}} || {{lang|fa|۴}} || {{lang|fa|۵}} || {{lang|fa|۶}} || {{lang|fa|۷}} || {{lang|fa|۸}} || {{lang|fa|۹}} || {{lang|fa|۱۰}}
|-
|Urdu{{efn|Same Unicode characters as the Persian, but language is set to Urdu. The numerals 4, 6 and 7 are different from Persian; on some devices, this row may appear identical to Persian}}
| {{lang|ur|۰}} || {{lang|ur|۱}} || {{lang|ur|۲}} || {{lang|ur|۳}} || {{lang|ur|۴}} || {{lang|ur|۵}} || {{lang|ur|۶}} || {{lang|ur|۷}} || {{lang|ur|۸}} || {{lang|ur|۹}} || {{lang|ur|۱۰}}
|-
|colspan=12; style=&quot;text-align:left|{{Notelist}}
|}
Though the Persian &amp; Urdu variants are not often found outside of Iran &amp; Pakistan respectively, it is not impossible to run into them. It is also accepted to use western Arabic numerals, especially in typing where they are heavily favored. In this course, we will only use the eastern Arabic numerals in order to help better recognize them

==١٠-٠==
Numbers ١٠-١ have no rhyme or reason, so the best thing to do is memorize them

*&lt;font size=&quot;+2&quot;&gt;'''٠ - صفر'''&lt;/font&gt;
:ṣifr
:''0 - Zero''

*&lt;font size=&quot;+2&quot;&gt;'''١ - واحد'''&lt;/font&gt;
:wāḥid
:''1 - One''

* &lt;font size=&quot;+2&quot;&gt;'''٢- اثنان'''&lt;/font&gt;
:iṯnān
:''2 - Two''

*&lt;font size=&quot;+2&quot;&gt;'''٣ - ثلاثة'''&lt;/font&gt;
:ṯalāṯa
:''3 - Three''

*&lt;font size=&quot;+2&quot;&gt;'''٤ - أربعة'''&lt;/font&gt;
:ʾarbaʿa
:''4 - Four''

*&lt;font size=&quot;+2&quot;&gt;'''٥ - خمسة'''&lt;/font&gt;
:ḵamsa
:''5 - Five''

*&lt;font size=&quot;+2&quot;&gt;'''٦ - ستة'''&lt;/font&gt;
:sitta
:''6 - Six''

*&lt;font size=&quot;+2&quot;&gt;'''٧ - سبعة'''&lt;/font&gt;
:sabʿa
:''7 - Seven''

*&lt;font size=&quot;+2&quot;&gt;'''٨ - ثمانية'''&lt;/font&gt;
:ṯamāniya
:''8 - Eight''

*&lt;font size=&quot;+2&quot;&gt;'''٩ - تسعة'''&lt;/font&gt;
:tisʿa
:''9 - Nine''

*&lt;font size=&quot;+2&quot;&gt;'''١٠ - عشرة'''&lt;/font&gt;
:ʿašara
:''10 - Ten''

==١٩-١١==
Numbers ١٩-١١ are constructed by placing &quot;عشر&quot; after the number, with the exception of ١١ and ١٢ which have special word-forms. Even though these numbers are written with two separate words, they are spoken as if they are one, so ة is pronounced as ت
*&lt;font size=&quot;+2&quot;&gt;'''١١ - أحد عشر'''&lt;/font&gt;
:ʾāḥida ʿašr
:''11 - Eleven''

*&lt;font size=&quot;+2&quot;&gt;'''١٢ - اثنا عشر'''&lt;/font&gt;
:iṯnā ʿašr
:''12 - Twelve''

*&lt;font size=&quot;+2&quot;&gt;'''١٣ - ثلاثة عشر'''&lt;/font&gt;
:ṯalāṯata ʿašr
:''13 - Thirteen''

*&lt;font size=&quot;+2&quot;&gt;'''١٤ - أربعة عشر'''&lt;/font&gt;
:ʾarbaʿata ʿašr
:''14 - Fourteen''

*&lt;font size=&quot;+2&quot;&gt;'''١٥ - خمسة عشر'''&lt;/font&gt;
:ḵamsata ʿašr
:''15 - Fifteen''

*&lt;font size=&quot;+2&quot;&gt;'''١٦ - ستة عشر'''&lt;/font&gt;
:sittata ʿašr
:''16 - Sixteen''

*&lt;font size=&quot;+2&quot;&gt;'''١٧ - سبعة عشر'''&lt;/font&gt;
:sabʿata ʿašr
:''17 - Seventeen''

*&lt;font size=&quot;+2&quot;&gt;'''١٨ - ثمانية عشر'''&lt;/font&gt;
:ṯamāniyata ʿašr
:''18 - Eighteen''

*&lt;font size=&quot;+2&quot;&gt;'''١٩ - تسعة عشر'''&lt;/font&gt;
:tisʿata ʿašr
:''19 - Nineteen''

==٩٠-١٠==
Numbers ending with ٠, also known as tens in English, are formed methodically by replacing ـة- with ـون-; the numbers ١٠ and ٢٠ have special forms

*&lt;font size=&quot;+2&quot;&gt;'''١٠ - عشرة'''&lt;/font&gt;
:ʿašara
:''10 - Ten''

*&lt;font size=&quot;+2&quot;&gt;'''٢٠ - عشرون'''&lt;/font&gt;
:ʾišrūn
:''20 - Twenty''

*&lt;font size=&quot;+2&quot;&gt;'''٣٠ - ثلاثون'''&lt;/font&gt;
:ṯalāṯūn
:''30 - Thirty''

*&lt;font size=&quot;+2&quot;&gt;'''٤٠ - أربعون'''&lt;/font&gt;
:ʾarbaʿūn
:''40 - Forty''

*&lt;font size=&quot;+2&quot;&gt;'''٥٠ - خمسون'''&lt;/font&gt;
:ḵamsūn
:''50 - Fifty''

*&lt;font size=&quot;+2&quot;&gt;'''٦٠ - ستون'''&lt;/font&gt;
:sittūn
:''60 - Sixty''

*&lt;font size=&quot;+2&quot;&gt;'''٧٠ - سبعون'''&lt;/font&gt;
:sabʿūn
:''70 - Seventy''

*&lt;font size=&quot;+2&quot;&gt;'''٨٠ - ثمانون'''&lt;/font&gt;
:ṯamānūn
:''80 - Eighty''

*&lt;font size=&quot;+2&quot;&gt;'''٩٠ - تسعون'''&lt;/font&gt;
:tisʿūn
:''90 - Ninety''

==٩٠٠-١٠٠==
Hundreds also have their own forms, made by replacing ـة- and attaching the word for hundred to the base number, ـمائة-; the numbers ١٠٠ and ٢٠٠ have special forms
*&lt;font size=&quot;+2&quot;&gt;'''١٠٠ - مائة'''&lt;/font&gt;
:māʾa
:''100 - One hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٢٠٠ - مائتين'''&lt;/font&gt;
:māʾatayn
:''200 - Two hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٣٠٠ - ثلاثمائة'''&lt;/font&gt;
:ṯalāṯumāʾa
:''300 - Three hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٤٠٠ - أربعمائة'''&lt;/font&gt;
:ʾarbaʿumāʾa
:''400 - Four hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٥٠٠ - خمسمائة'''&lt;/font&gt;
:ḵamsumāʾa
:''500 - Five hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٦٠٠ - ستمائة'''&lt;/font&gt;
:sittumāʾa
:''600 - Six hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٧٠٠ - سبعمائة'''&lt;/font&gt;
:sabʿumāʾa
:''700 - Seven hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٨٠٠ - ثمانيمائة'''&lt;/font&gt;
:ṯamāniyumāʾa
:''800 - Eight hundred''

*&lt;font size=&quot;+2&quot;&gt;'''٩٠٠ - تسعمائة'''&lt;/font&gt;
:tisʿumāʾa
:''900 - Nine hundred''

==٩٩٩-٠==
Other complex numbers under a thousand can be made placing the two words next to each other and adding the word &quot;and&quot;, &quot;-و&quot;, to the beginning of the next word; keep in mind that when reading numbers, the order is hundreds, ''ones,'' and ''then'' tens. Here are a few examples:

*&lt;font size=&quot;+2&quot;&gt;'''٢٩٥ - مائتين وخمسة وتسعون'''&lt;/font&gt;
:māʾatayn wa-ḵamsa wa-tisʿūn 
:''295 - Two hundred ninety-five (lit. two-hundred and-five and-ninety)''

*&lt;font size=&quot;+2&quot;&gt;'''٣٦٣ - ثلاثمائة وثلاثة وستون'''&lt;/font&gt;
:ṯalāṯumāʾa wa-ṯalāṯa wa-sittūn
:''363 - Three hundred sixty-three (lit. three-hundred and-three and-sixty)''

*&lt;font size=&quot;+2&quot;&gt;'''٤٧ - سبعة وأربعون'''&lt;/font&gt;
:sabʿa wa-ʾarbaʿūn
:''47 - Forty-seven (lit. seven and-forty)''

*&lt;font size=&quot;+2&quot;&gt;'''٥٠٢ - خمسمائة واثنان'''&lt;/font&gt;
:ḵamsumāʾa wa-iṯnān
:''502 - Five hundred two (lit. five-hundred and-two)''

*&lt;font size=&quot;+2&quot;&gt;'''٧٣٠ - سبعمائة وثلاثون'''&lt;/font&gt;
:sabʿumāʾa wa-ṯalāṯūn
:''730 Seven hundred thirty (lit. seven-hundred and-thirty)''

*&lt;font size=&quot;+2&quot;&gt;'''٦١ - واحد وستون'''&lt;/font&gt;
:wāḥid wa-sittūn
:''61 - Sixty-one (lit. one and-sixty)''

*&lt;font size=&quot;+2&quot;&gt;'''٧١٨ - سبعمائة وثمانية عشر'''&lt;/font&gt;
:sabʿumāʾa wa-ṯamāniyata ʿašr
:''718 - Seven hundred eighteen (lit. seven-hundred and-eighteen)''

*&lt;font size=&quot;+2&quot;&gt;'''٩٩٩ - تسعمائة وتسعة وتسعون'''&lt;/font&gt;
:tisʿumāʾa wa-tisʿa wa-tisʿūn
:''999 - Nine hundred ninety-nine (lit. nine-hundred and-nine and-ninety)''

*&lt;font size=&quot;+2&quot;&gt;'''٥٥ - خمسة وخمسون'''&lt;/font&gt;
:ḵamsa wa-ḵamsūn
:''55 - Fifty-five (lit. five and-fifty)''

*&lt;font size=&quot;+2&quot;&gt;'''٦٤٢ - ستمائة واثنان وأربعون'''&lt;/font&gt;
:sittumāʾa wa-iṯnān wa-ʾarbaʿūn
:''642 - Six hundred forty-two (lit. six-hundred and-two and-forty)''
----

&lt;center&gt;Back to [[Arabic]]&lt;/center&gt;

{{BookCat}}</text>
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  <page>
    <title>Cantonese</title>
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      <timestamp>2026-04-19T03:42:24Z</timestamp>
      <contributor>
        <username>JKuroha</username>
        <id>3516066</id>
      </contributor>
      <comment>Added details</comment>
      <origin>4631288</origin>
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      <text bytes="4522" sha1="5v4m06j2cd8ezpcntgvq61wb0nqg1wm" xml:space="preserve">{{status|50%}}
[[Image:Hongkong_victoria_peak.jpg|thumb|right|200px|&lt;div style=&quot;text-align: center;&quot;&gt;香港&lt;/div&gt;&lt;br/&gt;Victoria Harbour as seen from Victoria Peak 由山頂望向維多利亞港]]

Welcome to the wikibook course on '''Cantonese'''. To use this book, your web browser must first be configured to [[Chinese/Displaying Chinese Characters|display Chinese characters]] {{stage short|50%|Jan 24, 2005}}. If the characters in the box below appear as blank boxes or garbage such as �?�?, your browser is not properly configured.

{| border=&quot;1&quot; cellspacing=&quot;0&quot; cellpadding=&quot;6&quot; align=&quot;center&quot;
 | style=&quot;background-color:#eeeeee;&quot; |如果你識廣東話嘅話，唔該幫下手！
 |}

== Introduction / 序 ==
*[[Cantonese/About Cantonese|About Cantonese&lt;br/&gt; 廣東話係乜]] {{stage short|100%|May 21, 2005}}
*[[Cantonese/How To Use This Textbook|How To Use This Textbook&lt;br/&gt;點用呢本教科書]] {{stage short|100%|May 24, 2005}}
*[[Cantonese/How To Study Cantonese|How To Study Cantonese&lt;br/&gt;點學廣東話]] {{stage short|100%|May 24, 2005}}
*[[Cantonese/Pronunciation|Pronunciation&lt;br/&gt; 廣東話發音]] {{stage short|75%|May 21, 2005}}

== Lessons / 課程 ==
*[[Cantonese/Lesson 1|Lesson 1: Hello!&lt;br/&gt; 第一課：你好！]] {{stage short|75%|May 22, 2005}}
*[[Cantonese/Lesson 2|Lesson 2: Introductions II&lt;br/&gt; 第二課：介紹 2]] {{stage short|00%|May 21, 2005}}
**[[Cantonese/Lesson 2/Conversation|Conversation&lt;br/&gt; 傾偈]]
**[[Cantonese/Lesson 2/Notes|Grammar Notes&lt;br/&gt; 文法]]
**[[Cantonese/Lesson 2/Drills|Practice Drills&lt;br/&gt; 練習]]
*[[Cantonese/Lesson 3|Lesson 3: An introduction to particles&lt;br/&gt; 第三課：助語詞]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 4|Lesson 4: Word order and Verbs&lt;br/&gt; 第四課：詞序同動詞]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 5|Lesson 5: Measure words&lt;br/&gt; 第五課：量詞]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 6|Lesson 6: More on interrogatives&lt;br/&gt; 第六課：疑問助詞]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 7|Lesson 7: What's this?&lt;br/&gt; 第七課：呢啲係咩黎架？]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 8|Lesson 8: Introduce yourself to strangers&lt;br/&gt; 第八課：向陌生人介紹自己]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 9|Lesson 9: It's Monday today.&lt;br/&gt; 第九課：今日係(星期/禮拜)一。]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/Lesson 10|Lesson 10: Ancient Origins of Cantonese Words.&lt;br/&gt; 第十課：粵語嘅本字。]]

== Appendices / 附錄 ==
*[[Cantonese/Cantonese-English Glossary|Cantonese-English Glossary&lt;br/&gt;粤英詞彙]]{{stage short|00%|May 21, 2005}}
*[[Cantonese/Solutions to Exercises|Solutions to Exercises&lt;br/&gt;題解]] {{stage short|00%|May 21, 2005}}
*[[Cantonese/English-Cantonese Glossary|English-Cantonese Glossary&lt;br/&gt;英粤詞彙]] {{stage short|00%|May 21, 2005}}
&lt;!--*[[Cantonese/Greetings|Greetings&lt;br/&gt;問候]] {{stage short|00%|May 21, 2005}}--&gt;
*[[Cantonese/Numbers|Numbers&lt;br/&gt;數字/數目/數目字]] {{stage short|25%|May 21, 2005}}
*[[Cantonese/Romanization Systems|Cantonese Romanization Schemes&lt;br/&gt;粤語注音]] {{stage short|75%|May 21, 2005}}
&lt;!--*[[Cantonese/Slang|Slang&lt;br/&gt;俚語]] {{stage short|00%|May 21, 2005}}--&gt;
*[[Cantonese/Web Resources|Web Resources&lt;br/&gt;網上資源]] {{stage short|50%|May 22, 2005}}
*[[Cantonese/Common phrases|Common phrases]]
&lt;!-- Some appendicies are commented out for now, because they may be  a reduplication of the work on cantonese.sheik.co.uk or www.cantonese.ca.  Would it be better to just link to these sites? --&gt;

== References / 參考 ==
*[http://humanum.arts.cuhk.edu.hk/Lexis/lexi-can/ Cantonese Syllabary] - Look up the Cantonese pronunciations of a character or look for all the characters of a particular pronunciation.  Includes sound files, simple definitions, variant pronunciations, and more.  Can switch to virtually any Cantonese romanization system.

== Related Books / 相關維基教科書 ==
*[[Chinese|Standard Mandarin&lt;br/&gt; 普通话]]
*[[East Asian Calligraphy|How to write Chinese in the right way &lt;br/&gt; 如何正確書寫中文]]

== Contributors / 編者 ==
*[[Cantonese/Contributor's Guide|Contributor's Guide]] {{stage short|100%|Dec 21, 2006}}
*[[Cantonese/Planning|Textbook Planning&lt;br/&gt;課文安排]] {{stage short|25%|May 21, 2005}}
*Contributors: [[User:xingmu|xingmu]]

== See also / 睇埋 ==
*[[Cangjie input method]]

{{shelves|chinese language}}
{{alphabetical|C}}
{{Category 3}}</text>
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  <page>
    <title>C Programming</title>
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      <timestamp>2026-04-19T08:48:21Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>49843</id>
      </contributor>
      <comment>Replacing The_C_Programming_Language_logo.svg with [[File:C1stEdition.svg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:Duplicate|Duplicate]]: Exact or scaled-down duplicate: [[:c::File:C1stEdition.svg|]]).</comment>
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      <text bytes="4694" sha1="c3goumsaqikuebx0ghynnhoy1ek019w" xml:space="preserve">{{Featured book}}

&lt;div style=&quot;text-align: center;&quot;&gt;
''Wikibooks Contributors Present:''

[[Image:C1stEdition.svg|100px]]

&lt;big&gt;&lt;big&gt;&lt;big&gt;'''C Programming'''&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;

&lt;div style=&quot;text-align: center;&quot;&gt;''A comprehensive look at the C programming language and its features.''&lt;/div&gt;
&lt;/div&gt;

__NOTOC__

==Table of contents ==
&lt;noinclude&gt;{{Book Search}}
{{reading level|intermediate}}
{{PDF version}}
&lt;!--{{Collection}}--&gt;
{{Print version}}
{{E-Book Reader PDF version|C_Programming_eBook_Reader.pdf}}
&lt;/noinclude&gt;

=== Introduction ===
:{{stage short|100%}} [[/Why learn C?/|Why learn C?]]
:{{stage short|100%}} [[/History/|History]]
:{{stage short|75%}} [[/What you need before you can learn/|What you need before you can learn]]
:{{stage short|50%}} [[/Obtaining a compiler/|Obtaining a compiler]]

=== Beginning C ===
:{{stage short|100%}} [[/Intro exercise/|Intro exercise]]
:{{stage short|100%}} [[/Preliminaries/|Preliminaries]]
:{{stage short|100%}} [[/Basics of compilation/|Basics of compilation]]
:{{stage short|100%}} [[/Simple output/|Simple output]]
:{{stage short|100%}} [[/Variables/|Variables]]
:{{stage short|100%}} [[/Simple input/|Simple input]]
:{{stage short|50%}} [[/Operators and type casting/|Operators and type casting]]
:{{stage short|75%}} [[/Arrays and strings/|Arrays and strings]]
:{{stage short|75%}} [[/Program flow control/|Program flow control]]
:{{stage short|50%}} [[/Procedures and functions/|Procedures and functions]]
:{{stage short|100%}} [[/Headers and libraries/|Headers and libraries]]
:{{stage short|75%}} [[/Beginning exercises/|Beginning exercises]]

=== Intermediate C ===
:{{stage short|100%}} [[/Advanced data types|Advanced data types]]
:{{stage short|50%}} [[/Pointers and arrays/|Pointers and arrays]]
:{{stage short|100%}} [[/Side effects and sequence points/|Side effects and sequence points]]
:{{stage short|50%}} [[/Memory management/|Memory management]]
:{{stage short|50%}} [[/Error handling/|Error handling]]
:{{stage short|75%}} [[/Stream IO/|Stream I/O]]
:{{stage short|75%}} [[/String manipulation/|String manipulation]]
:{{stage short|75%}} [[/Further math/|Further math]]
:{{stage short|50%}} [[/Standard libraries/|Standard libraries]]

=== Advanced C ===
:{{stage short|50%}} [[/Common practices/|Common practices]]
:{{stage short|50%}} [[/Preprocessor directives and macros/|Preprocessor directives and macros]]
:{{stage short|50%}} [[/Networking in UNIX/|Networking in UNIX]]
:{{stage short|100%}} [[/X macros and serialization/|X macros and serialization]]
:{{stage short|25%}} [[/Coroutines/|Coroutines]]

=== C and beyond ===
:{{stage short|50%}} [[/Particularities of C/|Particularities of C]]
:{{stage short|50%}} [[/Low-level IO/|Low-level I/O]]
:{{stage short|25%}} [[/Mixing languages/|Mixing languages]]
:{{stage short|75%}} [[/GObject/|GObject]]

=== Reference tables ===
This section has some tables and lists of C entities.
: {{stage short|25%}} [[/Standard library reference/|Standard library reference]]
: {{stage short|100%}} [[/Statements/|Statements]]
: {{stage short|25%}} [[/Preprocessor reference/|Preprocessor reference]]
: {{stage short|75%}} [[/Language Reference/|Language Reference]]
::* [[/Language Reference/#Table of Keywords|Table of Keywords]]
::* [[/Language Reference/#Table of Operators|Table of Operators]]
::* [[/Language Reference/#Table of Data Types|Table of Data Types]]
==== Platform reference ====
: {{stage short|25%}} [[/POSIX Reference/|POSIX]]
: {{stage short|25%}} [[/GNU C Library Reference/|GNU C Library]]
: {{stage short|25%}} [[/MS Windows Reference/|MS Windows]]

===Appendices===
: {{stage short|100%}} [[/Alternative tokens/|Alternative tokens]]
: {{stage short|25%}} [[/C Compilers Reference List/|C Compilers Reference List]]
: {{stage short|100%}} [[/Code style/|Code style]]
: {{stage short|25%}} [[/Exercise solutions/|Exercise solutions]]
: {{stage short|75%}} [[/Index/|Index]]
: {{stage short|50%}} [[/Links/|Links]]&lt;noinclude&gt;

==Related Wikibooks==
* [[A Little C Primer]]
* [[Data Structures]]
* [[GCC Debugging]]
* [[GNU C Compiler Internals]]

&lt;/noinclude&gt;{{Shelves|C programming language}}&lt;noinclude&gt;
{{alphabetical|C}}
{{status|100%}}


[[bn:সি প্রোগ্রামিং]]
[[ca:Programació en C]]
[[cs:Kurz programování v C]]
[[de:C-Programmierung]]
[[el:Προγραμματισμός στην C]]
[[et:Programmeerimiskeel C]]
[[es:Programación en C]]
[[fr:Programmation C]]
[[gl:C]]
[[it:C]]
[[he:שפת C]]
[[ka:C (პროგრამირება)]]
[[nl:Programmeren in C]]
[[ja:C言語]]
[[pl:C]]
[[pt:Programar em C]]
[[ru:Язык Си в примерах]]
[[fi:C]]
[[sv:Programmering i ANSI-C]]
[[ta:சி]]

&lt;/noinclude&gt;</text>
      <sha1>c3goumsaqikuebx0ghynnhoy1ek019w</sha1>
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  </page>
  <page>
    <title>Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6/3. d4</title>
    <ns>0</ns>
    <id>42319</id>
    <revision>
      <id>4631297</id>
      <parentid>4529306</parentid>
      <timestamp>2026-04-19T09:40:26Z</timestamp>
      <contributor>
        <username>JCrue</username>
        <id>2226064</id>
      </contributor>
      <origin>4631297</origin>
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      <text bytes="1588" sha1="5ceuz6oss1hy0bazigxhu6g2hk0s6yc" xml:space="preserve">{{Chess Opening Theory/Position
|name=Open Sicilian with ...Nc6
|eco=[[Chess/ECOB|B32]]
|parent=[[Chess Opening Theory/1. e4/1...c5|Sicilian defence]] → [[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6|2...Nc6]]
}}
== 3. d4 · Open Sicilian with ...Nc6 ==
White takes space in the centre. White invites Black to take the d-pawn. Black can hardly allow White to advance to e5, taking more space and displacing their knight. 

[[/3...cxd4|'''3...cxd4''']] eliminates the pawn and White can recapture 4. Nxd4. White's advantage is that they now have open lines for all their pieces to develop. Black's advantage is that they have both central pawns to White's one. 

'''3...Nxd4?!''' is legal but barely warrants mentioning. 4. Nxd4 cxd4 5. Qxd4 only gives White a helping hand with their attack.  

'''3...e6?''' to increase control of the e5 square does not prevent White from playing e5.  

==Theory table==

1.e4 c5 2.Nf3 Nc6 3.d4

&lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;4&quot;&gt;
        &lt;tr&gt;
                &lt;th&gt;&lt;/th&gt;
                &lt;th align=&quot;left&quot;&gt;3&lt;/th&gt;
        &lt;/tr&gt;
        &lt;tr&gt;
                &lt;th align=&quot;right&quot;&gt;Main Line&lt;/th&gt;
                &lt;td&gt;...&lt;br&gt;[[/3...cxd4|cxd4]]&lt;/td&gt;
                &lt;td&gt;+=&lt;/td&gt;
        &lt;/tr&gt;
        &lt;tr&gt;
                &lt;th align=&quot;right&quot;&gt;&lt;/th&gt;
                &lt;td&gt;...&lt;br&gt;[[/3...e6|e6]]&lt;/td&gt;
                &lt;td&gt;Nc3&lt;br&gt;&amp;nbsp;&lt;/td&gt;
        &lt;/tr&gt;
&lt;/table&gt;
{{ChessMid}}

==References==
{{reflist}}
=== See also ===
{{Wikipedia|Sicilian Defence}}
{{BCO2}}
{{Chess Opening Theory/Footer}}

[[fi:Shakkiaapinen/Peli/1. e4/1...c5/2. Rf3/2...Rc6/3. d4]]</text>
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  <page>
    <title>Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6/3. d4/3...cxd4</title>
    <ns>0</ns>
    <id>53611</id>
    <revision>
      <id>4631298</id>
      <parentid>4529305</parentid>
      <timestamp>2026-04-19T09:47:23Z</timestamp>
      <contributor>
        <username>JCrue</username>
        <id>2226064</id>
      </contributor>
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      <text bytes="1335" sha1="0u22hhjdnj10upwszqbety6861c2s4v" xml:space="preserve">{{Chess Opening Theory/Position
|Open Sicilian with ...Nc6
|eco=[[Chess/ECOB|B26]]
|parent=[[Chess Opening Theory/1. e4/1...c5|Sicilian defence]] → [[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6|2...Nc6]] → [[../|3. d4]]
}}
== 3...cxd4 ==
Black captures White's d-pawn and reduces White's control of the centre. They are now playing with two centre pawns to White's one. 

[[/4. Nxd4|'''4. Nxd4''']] recaptures the pawn. White now has opened lines for their bishops to develop and join the attack. Their knight on d4 may subsequently move to b3, where it assists in the defence of their king following queen's side castling and is out of the way of a king's side pawn storm. 

'''4. c3''' is also in keeping with the position. This is a gambit that transposes into the Smith-Morra gambit if accepted. White sacrifices a pawn or two to extend their development lead. 

==Theory table==
'''1.e4 c5 2.Nf3 Nc6 3.d4 cxd4'''
&lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;4&quot;&gt;
	&lt;tr&gt;
		&lt;th&gt;&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;4&lt;/th&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Main Line&lt;/th&gt;
		&lt;td&gt;[[/4. Nxd4/|Nxd4]]&lt;br&gt;-&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
&lt;/table&gt;
{{ChessMid}}

==References==
{{reflist}}
=== See also ===

{{Wikipedia|Sicilian Defence}}
{{BCO2}}
{{Chess Opening Theory/Footer}}

[[fi:Shakkiaapinen/Peli/1. e4/1...c5/2. Rf3/2...Rc6/3. d4/3...cxd4]]</text>
      <sha1>0u22hhjdnj10upwszqbety6861c2s4v</sha1>
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  <page>
    <title>Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6/3. d4/3...cxd4/4. Nxd4</title>
    <ns>0</ns>
    <id>53612</id>
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        <username>JCrue</username>
        <id>2226064</id>
      </contributor>
      <comment>summary of continuations</comment>
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      <format>text/x-wiki</format>
      <text bytes="4486" sha1="7r10xwh8sb9vuvjq9nmreweqlsutaxw" xml:space="preserve">{{Chess Opening Theory/Position
|name=Open Sicilian with ...Nc6
|eco=[[Chess/ECOB|B32]]
|parent=[[Chess Opening Theory/1. e4/1...c5|Sicilian defence]] → [[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6|2...Nc6]] → [[../|3...cxd4]]
}}
== 4. Nxd4 ==
White recaptures the pawn. They have open lines for their pieces to develop. Black has the positional advantage of two centre pawns to White's one.  

The two knights are at a standoff. Neither wants to take first: if Black trades knights, '''4...Nxd4!?''', after 5. Qxd4 White will have better control of the centre and Black cannot easily chase away the White queen. If White plays volunteers to trade, say 4...Nf6 5. Nxc6!? bxc6, Black has brought in another pawn to help control the centre and support playing d5.

The main moves here are 4...Nf6, 4...e5, and 4...g6. 

[[/4...Nf6|'''4...Nf6''']] is the main line. This develops a piece while attack the undefended e4-pawn. This elicits White to defend it with 5. Nc3 which prevents White from playing c4 and clamping down on the d5 square, where the major lines are either 5...d6, the classical Sicilian, or 5...e5, the Sveshnikov or Lasker-Pelikan variation, both of which prevent White from trading knights and playing e5 with tempo. A sideline is 5. f3, to retain the option of c4. 

[[/4...e5|'''4...e5''']] immediately is the '''Löwenthal variation'''. The main line is 5. Nb5, threatening Nd6+, where 5...d6 is the Kalashnikov variation.  

[[/4...g6|'''4...g6''']] is the '''accelerated dragon variation'''. This prepares to fianchetto the king's bishop, playing a dragon variation-style set-up without spending a move on ...d6 yet. The most critical approach is 5. c4, the Maróczy bind. 

=== Other moves ===

[[/4...d5|'''4...d5''']] is the '''Nimzo-American variation'''. 5. exd5 Qxd5 6. Be3 is most critical; 5. Bb5 is common and leads to a queenless middle game, e.g. 5. Bb5 dxe4 6. Nxc6 Qxd1+ 7. Kxd1. 

[[/4...Qb6|'''4...Qb6''']] is the '''Godiva variation'''. This attacks White's knight and 5. Be3? to defend it drops the b2 pawn. 5. Nb3 is called for. 

=== Transpositions ===

[[/4...e6|'''4...e6''']] is a transposition into the '''Taimanov variation'''. It is more common nowadays to play the Taimanov with 2...e6 and 4...Nc6 instead, as the move order with 2...Nc6 allows the possibility of the sideline 3. Bb5, the Rossolimo. 

[[/4...Qc7|'''4...Qc7''']] is the '''Flohr variation''', which usually transposes to the Bastrikov line in the Taimanov after 5. Nc3 e6.

==Theory table==
'''1.e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4'''
&lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;4&quot;&gt;
	&lt;tr&gt;
		&lt;th&gt;&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;4&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;5&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;6&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;7&lt;/th&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Main Line&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Nf6|Nf6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Taimanov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[../../../../2...e6/3. d4/3...cxd4/4. Nxd4/4...Nc6|e6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Bxc3+&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Kalashnikov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...e5|e5]]&lt;/td&gt;
		&lt;td&gt;Nb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;N1c3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;a6&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;[[Chess/Accelerated Dragon|Accelerated Dragon]]&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...g6|g6]]&lt;/td&gt;
		&lt;td&gt;c4&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Ng4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Nimzovich Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...d5|d5]]&lt;/td&gt;
		&lt;td&gt;exd5&lt;br&gt;Qxd5&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Flohr Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Qc7|Qc7]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;f4

&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Godiva Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Nb3&lt;br&gt;Qb4+&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bd3&lt;br&gt;Ne5&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Qd8&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;Neg4&lt;/td&gt;
		&lt;td&gt;Bd4&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;fxe5&lt;br&gt;dxe5&lt;/td&gt;
		&lt;td&gt;Bb5+&lt;br&gt;Bd7&lt;/td&gt;
		&lt;td&gt;Bxd7+&lt;br&gt;Qxd7&lt;/td&gt;
		&lt;td&gt;Bc5&lt;br&gt;0-0-0&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;a5&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;0-0&lt;br&gt;0-0&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;b6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
&lt;/table&gt;
{{ChessMid}}

{{Wikipedia|Sicilian Defence}}

==References==
{{reflist}}
{{BCO2}}
{{Chess Opening Theory/Footer}}</text>
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    </revision>
    <revision>
      <id>4631300</id>
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      <timestamp>2026-04-19T10:39:37Z</timestamp>
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        <username>JCrue</username>
        <id>2226064</id>
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      <text bytes="4486" sha1="36zfpdw1tjykpqehdragaaow7m8wwls" xml:space="preserve">{{Chess Opening Theory/Position
|name=Open Sicilian with ...Nc6
|eco=[[Chess/ECOB|B32]]
|parent=[[Chess Opening Theory/1. e4/1...c5|Sicilian defence]] → [[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6|2...Nc6]] → [[../|3...cxd4]]
}}
== 4. Nxd4 ==
White recaptures the pawn. They have open lines for their pieces to develop. Black has the positional advantage of two centre pawns to White's one.  

The two knights are at a standoff. Neither wants to take first: if Black trades knights, '''4...Nxd4!?''', after 5. Qxd4 White will have better control of the centre and Black cannot easily chase away the white queen. If White plays volunteers to trade, say 4...Nf6 5. Nxc6!? bxc6, Black has brought in another pawn to help control the centre and support playing d5.

The main moves here are 4...Nf6, 4...e5, and 4...g6. 

[[/4...Nf6|'''4...Nf6''']] is the main line. This develops a piece while attack the undefended e4-pawn. This elicits White to defend it with 5. Nc3 which prevents White from playing c4 and clamping down on the d5 square, where the major lines are either 5...d6, the classical Sicilian, or 5...e5, the Sveshnikov or Lasker-Pelikan variation, both of which prevent White from trading knights and playing e5 with tempo. A sideline is 5. f3, to retain the option of c4. 

[[/4...e5|'''4...e5''']] immediately is the '''Löwenthal variation'''. The main line is 5. Nb5, threatening Nd6+, where 5...d6 is the Kalashnikov variation.  

[[/4...g6|'''4...g6''']] is the '''accelerated dragon variation'''. This prepares to fianchetto the king's bishop, playing a dragon variation-style set-up without spending a move on ...d6 yet. The most critical approach is 5. c4, the Maróczy bind. 

=== Other moves ===

[[/4...d5|'''4...d5''']] is the '''Nimzo-American variation'''. 5. exd5 Qxd5 6. Be3 is most critical; 5. Bb5 is common and leads to a queenless middle game, e.g. 5. Bb5 dxe4 6. Nxc6 Qxd1+ 7. Kxd1. 

[[/4...Qb6|'''4...Qb6''']] is the '''Godiva variation'''. This attacks White's knight and 5. Be3? to defend it drops the b2 pawn. 5. Nb3 is called for. 

=== Transpositions ===

[[/4...e6|'''4...e6''']] is a transposition into the '''Taimanov variation'''. It is more common nowadays to play the Taimanov with 2...e6 and 4...Nc6 instead, as the move order with 2...Nc6 allows the possibility of the sideline 3. Bb5, the Rossolimo. 

[[/4...Qc7|'''4...Qc7''']] is the '''Flohr variation''', which usually transposes to the Bastrikov line in the Taimanov after 5. Nc3 e6.

==Theory table==
'''1.e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4'''
&lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;4&quot;&gt;
	&lt;tr&gt;
		&lt;th&gt;&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;4&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;5&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;6&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;7&lt;/th&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Main Line&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Nf6|Nf6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Taimanov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[../../../../2...e6/3. d4/3...cxd4/4. Nxd4/4...Nc6|e6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Bxc3+&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Kalashnikov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...e5|e5]]&lt;/td&gt;
		&lt;td&gt;Nb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;N1c3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;a6&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;[[Chess/Accelerated Dragon|Accelerated Dragon]]&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...g6|g6]]&lt;/td&gt;
		&lt;td&gt;c4&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Ng4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Nimzovich Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...d5|d5]]&lt;/td&gt;
		&lt;td&gt;exd5&lt;br&gt;Qxd5&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Flohr Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Qc7|Qc7]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;f4

&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Godiva Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Nb3&lt;br&gt;Qb4+&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bd3&lt;br&gt;Ne5&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Qd8&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;Neg4&lt;/td&gt;
		&lt;td&gt;Bd4&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;fxe5&lt;br&gt;dxe5&lt;/td&gt;
		&lt;td&gt;Bb5+&lt;br&gt;Bd7&lt;/td&gt;
		&lt;td&gt;Bxd7+&lt;br&gt;Qxd7&lt;/td&gt;
		&lt;td&gt;Bc5&lt;br&gt;0-0-0&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;a5&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;0-0&lt;br&gt;0-0&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;b6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
&lt;/table&gt;
{{ChessMid}}

{{Wikipedia|Sicilian Defence}}

==References==
{{reflist}}
{{BCO2}}
{{Chess Opening Theory/Footer}}</text>
      <sha1>36zfpdw1tjykpqehdragaaow7m8wwls</sha1>
    </revision>
    <revision>
      <id>4631301</id>
      <parentid>4631300</parentid>
      <timestamp>2026-04-19T10:39:59Z</timestamp>
      <contributor>
        <username>JCrue</username>
        <id>2226064</id>
      </contributor>
      <minor />
      <origin>4631301</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4476" sha1="9ntz4b8ofwtal6ggy20w6modj80aw7b" xml:space="preserve">{{Chess Opening Theory/Position
|name=Open Sicilian with ...Nc6
|eco=[[Chess/ECOB|B32]]
|parent=[[Chess Opening Theory/1. e4/1...c5|Sicilian defence]] → [[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...Nc6|2...Nc6]] → [[../|3...cxd4]]
}}
== 4. Nxd4 ==
White recaptures the pawn. They have open lines for their pieces to develop. Black has the positional advantage of two centre pawns to White's one.  

The two knights are at a standoff. Neither wants to take first: if Black trades knights, '''4...Nxd4!?''', after 5. Qxd4 White will have better control of the centre and Black cannot easily chase away the white queen. If White elects to trade, say 4...Nf6 5. Nxc6!? bxc6, Black has brought in another pawn to help control the centre and support playing d5.

The main moves here are 4...Nf6, 4...e5, and 4...g6. 

[[/4...Nf6|'''4...Nf6''']] is the main line. This develops a piece while attack the undefended e4-pawn. This elicits White to defend it with 5. Nc3 which prevents White from playing c4 and clamping down on the d5 square, where the major lines are either 5...d6, the classical Sicilian, or 5...e5, the Sveshnikov or Lasker-Pelikan variation, both of which prevent White from trading knights and playing e5 with tempo. A sideline is 5. f3, to retain the option of c4. 

[[/4...e5|'''4...e5''']] immediately is the '''Löwenthal variation'''. The main line is 5. Nb5, threatening Nd6+, where 5...d6 is the Kalashnikov variation.  

[[/4...g6|'''4...g6''']] is the '''accelerated dragon variation'''. This prepares to fianchetto the king's bishop, playing a dragon variation-style set-up without spending a move on ...d6 yet. The most critical approach is 5. c4, the Maróczy bind. 

=== Other moves ===

[[/4...d5|'''4...d5''']] is the '''Nimzo-American variation'''. 5. exd5 Qxd5 6. Be3 is most critical; 5. Bb5 is common and leads to a queenless middle game, e.g. 5. Bb5 dxe4 6. Nxc6 Qxd1+ 7. Kxd1. 

[[/4...Qb6|'''4...Qb6''']] is the '''Godiva variation'''. This attacks White's knight and 5. Be3? to defend it drops the b2 pawn. 5. Nb3 is called for. 

=== Transpositions ===

[[/4...e6|'''4...e6''']] is a transposition into the '''Taimanov variation'''. It is more common nowadays to play the Taimanov with 2...e6 and 4...Nc6 instead, as the move order with 2...Nc6 allows the possibility of the sideline 3. Bb5, the Rossolimo. 

[[/4...Qc7|'''4...Qc7''']] is the '''Flohr variation''', which usually transposes to the Bastrikov line in the Taimanov after 5. Nc3 e6.

==Theory table==
'''1.e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4'''
&lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;4&quot;&gt;
	&lt;tr&gt;
		&lt;th&gt;&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;4&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;5&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;6&lt;/th&gt;
		&lt;th align=&quot;left&quot;&gt;7&lt;/th&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Main Line&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Nf6|Nf6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Taimanov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[../../../../2...e6/3. d4/3...cxd4/4. Nxd4/4...Nc6|e6]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Bxc3+&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Kalashnikov Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...e5|e5]]&lt;/td&gt;
		&lt;td&gt;Nb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;N1c3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bg5&lt;br&gt;a6&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;[[Chess/Accelerated Dragon|Accelerated Dragon]]&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...g6|g6]]&lt;/td&gt;
		&lt;td&gt;c4&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Ng4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Nimzovich Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...d5|d5]]&lt;/td&gt;
		&lt;td&gt;exd5&lt;br&gt;Qxd5&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Bb4&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Flohr Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;[[/4...Qc7|Qc7]]&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e6&lt;/td&gt;
		&lt;td&gt;f4

&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;Godiva Variation&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Nb3&lt;br&gt;Qb4+&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Bd3&lt;br&gt;Ne5&lt;/td&gt;
		&lt;td&gt;a3&lt;br&gt;Qb6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Qd8&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;Neg4&lt;/td&gt;
		&lt;td&gt;Bd4&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;fxe5&lt;br&gt;dxe5&lt;/td&gt;
		&lt;td&gt;Bb5+&lt;br&gt;Bd7&lt;/td&gt;
		&lt;td&gt;Bxd7+&lt;br&gt;Qxd7&lt;/td&gt;
		&lt;td&gt;Bc5&lt;br&gt;0-0-0&lt;/td&gt;
		&lt;td&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
	&lt;tr&gt;
		&lt;th align=&quot;right&quot;&gt;&lt;/th&gt;
		&lt;td&gt;...&lt;br&gt;a5&lt;/td&gt;
		&lt;td&gt;Nc3&lt;br&gt;e5&lt;/td&gt;
		&lt;td&gt;Ndb5&lt;br&gt;d6&lt;/td&gt;
		&lt;td&gt;Be3&lt;br&gt;Nf6&lt;/td&gt;
		&lt;td&gt;Be2&lt;br&gt;Be7&lt;/td&gt;
		&lt;td&gt;0-0&lt;br&gt;0-0&lt;/td&gt;
		&lt;td&gt;f4&lt;br&gt;b6&lt;/td&gt;
		&lt;td&gt;Qd2&lt;br&gt;+=&lt;/td&gt;
	&lt;/tr&gt;
&lt;/table&gt;
{{ChessMid}}

{{Wikipedia|Sicilian Defence}}

==References==
{{reflist}}
{{BCO2}}
{{Chess Opening Theory/Footer}}</text>
      <sha1>9ntz4b8ofwtal6ggy20w6modj80aw7b</sha1>
    </revision>
  </page>
  <page>
    <title>User:TakuyaMurata/Calculus</title>
    <ns>2</ns>
    <id>70616</id>
    <revision>
      <id>4631235</id>
      <parentid>4624340</parentid>
      <timestamp>2026-04-18T15:35:30Z</timestamp>
      <contributor>
        <username>ShakespeareFan00</username>
        <id>46022</id>
      </contributor>
      <minor />
      <origin>4631235</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="39955" sha1="2khwb0ttbscsswektc4v323tkc900we" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm. &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;. If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space. Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space. Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39957" sha1="gf9e79k7e1y90m6ynm9wz5e26h6uxc1" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;. If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space. Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space. Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39959" sha1="qwf7szfb2nd18er4a9d0s39l6ie77x7" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;. If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space. Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space. Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39961" sha1="mrkl4xu7k1ec8gj5g6k9ns3xzmmu7o7" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space. Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space. Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39963" sha1="eib3zvzzwutpsxhgpvwxbz85ee3c9zi" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space. Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39969" sha1="5rsflax0v7rzmugaaatxmv17w7t6ii7" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39971" sha1="s06ez2axdzns8i4r2lctsh7xumg1176" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39973" sha1="st19fitlfygaya1r8mk8wdvjywj2epk" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39975" sha1="llojcuo6509f2l8cu1mkr66fkpqefxv" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39977" sha1="hd9my5mm64tkcm1l5l388d3wf9ps14r" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39981" sha1="1loxqmkq1jvq3xxin9fqkcbkwjrvx7u" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.''&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39979" sha1="etgak8h9hjxfdfmanyqplppohvy7cot" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then
:&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39983" sha1="jxwf3w8htsigv4erx6ztq0la77485gx" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then''
:''&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that
:&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39987" sha1="cf296jqqmativ41fb4petsiri6xvy11" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then''
:''&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that''
:''&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="39989" sha1="2x445l89alz8ag9b73f8wq1bke798bl" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then''
:''&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that''
:''&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.''

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;
# ''The set''
#:&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.

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      <text bytes="39995" sha1="q7jr75giumb3tbscs7p7d4qsrvlotoq" xml:space="preserve">== Module and linear space ==
An additive group &lt;math&gt;G&lt;/math&gt; is said to be a ''module'' over &lt;math&gt;R&lt;/math&gt;, or ''R-module'' for short, if the ''scalars'', the members of a ring &lt;math&gt;R&lt;/math&gt;, satisfy the following properties: if &lt;math&gt;x, y \in G&lt;/math&gt; and &lt;math&gt;\alpha, \beta \in R&lt;/math&gt;
* (i) Both &lt;math&gt;\alpha x&lt;/math&gt; and &lt;math&gt;x + y&lt;/math&gt; are in &lt;math&gt;G&lt;/math&gt;
* (ii) &lt;math&gt;(\alpha \beta) x = \alpha (\beta x)&lt;/math&gt; (associativity)
* (iii) &lt;math&gt;\alpha (x + y) = \alpha x + \alpha y&lt;/math&gt; and &lt;math&gt;(\alpha + \beta) x = \alpha x + \beta x&lt;/math&gt; (distribution law)
* (iv) &lt;math&gt;1_R x = x&lt;/math&gt;

By definition, every abelian group itself is a module over &lt;math&gt;\mathbb{Z}&lt;/math&gt;, since &lt;math&gt;x + x + x + ... = nx&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; is a scalar. Finally, a ''linear space'' is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space is ''finite-dimensional'' if it has a finite basis; that is, we can find linear independent vectors &lt;math&gt;e_1, e_2, ..., e_n&lt;/math&gt; so that &lt;math&gt;\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}&lt;/math&gt;. Such a basis need not be unique.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{V}&lt;/math&gt; be a finite-dimensional &lt;math&gt;\mathcal{K}&lt;/math&gt;-vector space. Then &lt;math&gt;\mathcal{V}^*&lt;/math&gt; has the same dimension as &lt;math&gt;\mathcal{V}&lt;/math&gt; does; that is, every basis for &lt;math&gt;\mathcal{V}&lt;/math&gt; has the same cardinality as every basis for &lt;math&gt;\mathcal{V}^*&lt;/math&gt; does.''

It can be shown that the map &lt;math&gt;\mathcal{V} \to \mathcal{V}^*&lt;/math&gt; cannot be defined constructively.[http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html] (TODO: need to detail this matter)

'''1 Theorem''' ''If &lt;math&gt;\mathcal{X}&lt;/math&gt; is a TVS and every finite subset of &lt;math&gt;\mathcal{X}&lt;/math&gt; is closed, it then follows that &lt;math&gt;\mathcal{X}&lt;/math&gt; is a Hausdorff space.''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y \in X&lt;/math&gt; with &lt;math&gt;x \ne y&lt;/math&gt; be given. Moreover, let &lt;math&gt;\Omega&lt;/math&gt; be the complement of the singleton &lt;math&gt;\{y\}&lt;/math&gt;, which is open by hypothesis. Since the function &lt;math&gt;f(z) = x + z&lt;/math&gt; is continuous at &lt;math&gt;0&lt;/math&gt; and &lt;math&gt;f(0) = x&lt;/math&gt; is in &lt;math&gt;\Omega&lt;/math&gt;, we can find an &lt;math&gt;\omega&lt;/math&gt; open and such that &lt;math&gt;\{x\} + \omega \subset \Omega&lt;/math&gt;. Here, we used, and would do so henceforward, the notation &lt;math&gt;A + B = &lt;/math&gt; the union of &lt;math&gt;\{ x + y \}&lt;/math&gt; taken ''all'' over &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;y \in B&lt;/math&gt;. Furthermore, since the function &lt;math&gt;g(x) = -x&lt;/math&gt; is continuous and so is its inverse, namely &lt;math&gt;g&lt;/math&gt;, we may assume that &lt;math&gt;\omega = -\omega&lt;/math&gt; by replacing &lt;math&gt;\omega&lt;/math&gt; by the intersection of &lt;math&gt;\omega&lt;/math&gt; and &lt;math&gt;-\omega&lt;/math&gt;. By repeating the same construction for each &lt;math&gt;x + z&lt;/math&gt; where &lt;math&gt;z \in \omega&lt;/math&gt;, we find &lt;math&gt;\omega&lt;/math&gt; so that &lt;math&gt;\{x\} + \omega + \omega \subset \Omega&lt;/math&gt;. It then follows that &lt;math&gt;\{x\} + \omega&lt;/math&gt; and &lt;math&gt;\{y\} + \omega&lt;/math&gt; are disjoint. Indeed, if we write &lt;math&gt;x + z = y + w&lt;/math&gt; for some &lt;math&gt;z, w \in \omega&lt;/math&gt;, then &lt;math&gt;y = x + z - w \in \Omega&lt;/math&gt;, a contradiction. &lt;math&gt;\square&lt;/math&gt;

== Normed spaces ==

A vector space is said to be ''normed'' if it is a metric space and its metric &lt;math&gt;d&lt;/math&gt; has the form:
:&lt;math&gt;d(x, y) = \|x - y\|&lt;/math&gt;
Here, the function &lt;math&gt;\|\cdot\|&lt;/math&gt;, called a ''norm'', has the property (in addition to that it induces the metric) that &lt;math&gt;\| \lambda x \| = |\lambda| \| x \|&lt;/math&gt; for any scalar &lt;math&gt;\lambda&lt;/math&gt;. We note that:
:&lt;math&gt;\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|&lt;/math&gt;
and
:&lt;math&gt;d(x+z, y+z) = \|x - y\| = d(x,y)&lt;/math&gt; for any &lt;math&gt;x, y, z&lt;/math&gt;.

It may go without saying but a vector space is ''infinite-dimensional'' if it is not finite-dimensional.

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt;, &lt;math&gt;\mathcal{Y}&lt;/math&gt; be normed spaces. If &lt;math&gt;\mathcal{X}&lt;/math&gt; is an infinite-dimensional and if &lt;math&gt;\mathcal{Y}&lt;/math&gt; is nonzero, there exists a linear operator &lt;math&gt;f:\mathcal{X} \to \mathcal{Y}&lt;/math&gt; that is not continuous.''&lt;br /&gt;

== Baire's theorem ==

A normed space is said to be ''complete'' when every Cauchy sequence in it converges in it.

'''3 Theorem''' ''Let &lt;math&gt;E&lt;/math&gt; is a subspace of a Banach space &lt;math&gt;G&lt;/math&gt; carrying the same norm. Then the following are equivalent:''
:''(a) &lt;math&gt;E&lt;/math&gt; is complete.''
:''(b) &lt;math&gt;E&lt;/math&gt; is closed in &lt;math&gt;G&lt;/math&gt;.''
:''(c) &lt;math&gt;\sum \|x_k\| &lt; \infty&lt;/math&gt; implies &lt;math&gt;\sum x_k&lt;/math&gt;.''
Proof: (i) Show (a) &lt;math&gt;\iff&lt;/math&gt; (b). If &lt;math&gt;E&lt;/math&gt; is complete, then every Cauchy sequence in &lt;math&gt;E&lt;/math&gt; has the limit in &lt;math&gt;E&lt;/math&gt;; thus, &lt;math&gt;E&lt;/math&gt; is closed. Conversely, if &lt;math&gt;E&lt;/math&gt; is closed, then every Cauchy sequence converges in &lt;math&gt;E&lt;/math&gt; since &lt;math&gt;G&lt;/math&gt; is complete. Hence, &lt;math&gt;E&lt;/math&gt; is complete. (ii) Show (a) &lt;math&gt;\iff&lt;/math&gt; (c). Let &lt;math&gt;x_j \in G&lt;/math&gt; be a Cauchy sequence. Then
:&lt;math&gt;\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Thus, &lt;math&gt;\sum x_k&lt;/math&gt; is Cauchy, and converges in &lt;math&gt;G&lt;/math&gt; since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence &lt;math&gt;x_k&lt;/math&gt; such that &lt;math&gt;\| x_{k+1} - x_k \| &lt; 2^{-k}&lt;/math&gt;. Then
:&lt;math&gt;\sum \| x_{k+1} - x_k \| &lt; \infty&lt;/math&gt;.
If the summation condition holds, then it follows that &lt;math&gt;\sum x_{k + 1} - x_k&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt;. Hence, &lt;math&gt;x_j&lt;/math&gt; converges in &lt;math&gt;G&lt;/math&gt; as well. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''&lt;math&gt;Q&lt;/math&gt; is incomplete but dense in &lt;math&gt;\mathbb{R}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;\mathbb{Q}&lt;/math&gt; is not closed in &lt;math&gt;\mathbb{R}&lt;/math&gt;. Since &lt;math&gt;\mathbb{R} \backslash \mathbb{Q}&lt;/math&gt; has empty interior, &lt;math&gt;\overline{\mathbb{Q}} = \overline{\mathbb{R}}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a set has ''dense complement'' if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

'''3 Theorem''' ''A complete normed space &lt;math&gt;G&lt;/math&gt; which is nonempty is never the union of a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement.''&lt;br /&gt;
Proof: Let &lt;math&gt;E_n \subset G&lt;/math&gt; be a sequence of subsets of &lt;math&gt;G&lt;/math&gt; with dense complement. Since &lt;math&gt;\overline{E_1}&lt;/math&gt; has empty interior and &lt;math&gt;G&lt;/math&gt; has nonempty interior, there exists an nonempty open ball &lt;math&gt;S_1 \subset (G \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-1}&lt;/math&gt;. Since &lt;math&gt;\overline{E^2}&lt;/math&gt; has empty interior and &lt;math&gt;S_1&lt;/math&gt; has empty interior, again there exists an nonempty open ball &lt;math&gt;S_2 \subset (S_1 \backslash \overline{E_1})&lt;/math&gt; with the radius &lt;math&gt;\le 2^{-2}&lt;/math&gt;. Iterating the construction ''ad infinitum'' we get the decreasing sequence &lt;math&gt;S_n&lt;/math&gt;. Now let &lt;math&gt;x_n&lt;/math&gt; be the sequence of the centers of &lt;math&gt;S_n&lt;/math&gt;. Then &lt;math&gt;x_n&lt;/math&gt; is Cauchy since: for some &lt;math&gt;N \le n, m&lt;/math&gt;
:&lt;math&gt;\|x_n - x_m\| &lt; 2^{-N} + 2^{-N} \to 0&lt;/math&gt; as &lt;math&gt;N \to \infty&lt;/math&gt;.
It then follows &lt;math&gt;x_n&lt;/math&gt; converges in &lt;math&gt;G \backslash \bigcup^{\infty} E_n&lt;/math&gt; from the compleness of &lt;math&gt;G&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary (open mapping theorem)''' ''If &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are Banach spaces, then a continuous linear surjection &lt;math&gt;f: A \to B&lt;/math&gt; maps an open set in &lt;math&gt;A&lt;/math&gt; to an open set in &lt;math&gt;B&lt;/math&gt;.''&lt;br /&gt;
Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

'''3 Corollary (Lipschitz continuity)''' ''Let &lt;math&gt;S_n&lt;/math&gt; = the set of functions &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; such that there exists some &lt;math&gt;x \in [0, 1]&lt;/math&gt; such that:''
:&lt;math&gt;|u(x + h) - u(x)| \le n |h|&lt;/math&gt; for all &lt;math&gt;x + h \in [0, 1]&lt;/math&gt;.
''Then (i) &lt;math&gt;\mathcal{C}^0 ([0, 1])&lt;/math&gt; is complete, (ii) &lt;math&gt;S_n&lt;/math&gt; is closed and has dense complement, and (iii) there exists a &lt;math&gt;u \in \mathcal{C}^0 ([0, 1])&lt;/math&gt; that is not in any &lt;math&gt;S_n&lt;/math&gt;; i.e., one that is differentiable nowhere.''&lt;br /&gt;
Proof: (i) &lt;math&gt;[0, 1]&lt;/math&gt; is complete; thus, &lt;math&gt;\mathcal{C}^0&lt;/math&gt; is a Banach space by some early theorem. (ii) Let &lt;math&gt;u_j \in S_n&lt;/math&gt; be a sequence, and suppose &lt;math&gt;u_j \to u&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;|u(x + h) - u(x)|&lt;/math&gt;
|&lt;math&gt;\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|&lt;/math&gt;
|-
|
|&lt;math&gt;\to n|h|&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
Thus, &lt;math&gt;u \in S_n&lt;/math&gt;; i.e., &lt;math&gt;S_n&lt;/math&gt; is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a &lt;math&gt;g \in \mathcal{C}^{\infty}([0, 1])&lt;/math&gt; such that:
:&lt;math&gt;\| u - g \|&lt;/math&gt;.
If we let &lt;math&gt;v = g + {\epsilon \over 2} \sin Nx&lt;/math&gt;, then
:&lt;math&gt;v \in \mathcal{C}^0 ([0, 1]) \backslash S_n&lt;/math&gt;
Hence, &lt;math&gt;S_n&lt;/math&gt; has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). &lt;math&gt;\square&lt;/math&gt;

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

'''3 Lemma''' ''In a topological space &lt;math&gt;X&lt;/math&gt;, the following are equivalent:''
* ''(i) Every countable union of closed sets with empty interior has empty interior.''
* ''(ii) Every countable intersection of open dense sets is dense.''
Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. &lt;math&gt;\square&lt;/math&gt;

When the above equivalent conditions are true, we say &lt;math&gt;X&lt;/math&gt; is a ''Baire space''.

'''3 Theorem''' If a Banach space &lt;math&gt;G&lt;/math&gt; has a ''Schauder basis'', a unique sequence of scalars such that
:&lt;math&gt;\| x - \sum_1^n \alpha_k x_k \| \to 0&lt;/math&gt; as &lt;math&gt;n \to \infty&lt;/math&gt;,
then &lt;math&gt;G&lt;/math&gt; is separable.&lt;br /&gt;
Proof:

The validity of the converse had been known as a ''Basis Problem'' for long time. It was, however, proven to be false in 19-something by someone.

== Duality ==

The kernel of a linear operator &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\ker(f)&lt;/math&gt;, is the set of all zero divisors for &lt;math&gt;f&lt;/math&gt;. A kernel of a linear operator is a linear space since &lt;math&gt;f(x) = 0 implies that f(\alpha x) = 0&lt;/math&gt; and &lt;math&gt;f(x) = 0 = f(y)&lt;/math&gt; implies &lt;math&gt;0 = f(x + y)&lt;/math&gt;. Moreover, a linear operator has zero kernel if and only it is injective.

'''3 Theorem''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear functional. Then &lt;math&gt;f&lt;/math&gt; is continuous if and only if &lt;math&gt;\ker(f)&lt;/math&gt; is closed.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is continuous, then &lt;math&gt;\ker(f) = f^{-1} \mid_{\{0\}}&lt;/math&gt; is closed since a finite set is closed. Conversely, suppose &lt;math&gt;f&lt;/math&gt; is not continuous. Then there exists a sequence &lt;math&gt;x_j \to x&lt;/math&gt; such that
:&lt;math&gt;\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0&lt;/math&gt;
In other words, &lt;math&gt;\ker(f)&lt;/math&gt; is not closed. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem''' ''If &lt;math&gt;f&lt;/math&gt; is a linear functional on &lt;math&gt;l^p&lt;/math&gt;, then''
:''&lt;math&gt;f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k&lt;/math&gt;''
Proof: Let &lt;math&gt;y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)&lt;/math&gt; where &lt;math&gt;\delta(j, k) = 1&lt;/math&gt; if &lt;math&gt;j = k&lt;/math&gt; else &lt;math&gt;0&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

The ''dual'' of a linear space &lt;math&gt;G&lt;/math&gt;, denoted by &lt;math&gt;G^*&lt;/math&gt;, is the set of all of linear operators from &lt;math&gt;G&lt;/math&gt; to &lt;math&gt;\mathbb{F}&lt;/math&gt; (i.e., either &lt;math&gt;\mathbb{C}&lt;/math&gt; or &lt;math&gt;\mathbb{R}&lt;/math&gt;). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

'''Theorem''' ''Let G be a normed linear space. Then''
:''&lt;math&gt;\|x\| = \sup_{\|y\| = 1} |&lt;x, y&gt;|&lt;/math&gt; and &lt;math&gt;\|y\| = \sup_{\|x\| = 1} |&lt;x, y&gt;|&lt;/math&gt;''.

The duality between a Banach space and its dual gives rise to.

Example: For &lt;math&gt;p&lt;/math&gt; finite, the dual of &lt;math&gt;l^p&lt;/math&gt; is &lt;math&gt;l^q&lt;/math&gt; where &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;.

'''3 Theorem (Krein-Milman)''' ''The unit ball of the dual of a real normed linear space has an extreme point.''&lt;br /&gt;
Proof: (TODO: to be written)

The theorem is equivalent to the AC. [http://publish.uwo.ca/~jbell/Axiom%20of%20Choice%20and%20Zorn.pdf]

== The Hahn-Banach theorem ==

'''3 Theorem (Hahn-Banach)''' ''Let &lt;math&gt;\mathcal{X}, \mathcal{Y}&lt;/math&gt; be normed vector spaces over real numbers. Then the following are equivalent.''
* (i) ''Every collection of mutually intersecting closed balls of &lt;math&gt;\mathcal{Y}&lt;/math&gt; has nonempty intersection. (binary intersection property)''
* (ii) ''If &lt;math&gt;\mathcal{M} \subset \mathcal{X}&lt;/math&gt; is a subspace and &lt;math&gt;f: \mathcal{M} \to \mathcal{Y}&lt;/math&gt; is a continuous linear operator, then &lt;math&gt;f&lt;/math&gt; can be extend to a &lt;math&gt;F&lt;/math&gt; on &lt;math&gt;\mathcal{X}&lt;/math&gt; such that &lt;math&gt;\|f\| = \|F\|&lt;/math&gt;. (dominated version)''
* (iii) ''If the linear variety &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; does not meet a non-empty open convex subset &lt;math&gt;G&lt;/math&gt; of &lt;math&gt;\mathcal{X}&lt;/math&gt;, then there exists a closed hyper-plane &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;{x} + \mathcal{M}&lt;/math&gt; that does not meet &lt;math&gt;G&lt;/math&gt; either. (geometric form)''

'''3 Corollary''' ''If the equivalent conditions hold in the theorem, &lt;math&gt;\mathcal{Y}&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Consider the identity map extended to the completion of &lt;math&gt;\mathcal{Y}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''Let &lt;math&gt;f&lt;/math&gt; be a linear operator from a Banach space &lt;math&gt;\mathcal{X}&lt;/math&gt; to a Banach space &lt;math&gt;\mathcal{Y}&lt;/math&gt;. If there exists a set &lt;math&gt;\Gamma&lt;/math&gt; and operators &lt;math&gt;f_1:\mathcal{X} \to l^\infty (\Gamma)&lt;/math&gt; and &lt;math&gt;f_2:l^\infty (\Gamma) \to \mathcal{X}&lt;/math&gt; such that &lt;math&gt;f_2 \circ f_1&lt;/math&gt; and &lt;math&gt;\|f_2\| = \|f_1\|&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; can be extended to a Banach space containing &lt;math&gt;\mathcal{X}&lt;/math&gt; without increase in norm.'' &lt;br /&gt;

== Hilbert spaces ==

A linear space &lt;math&gt;\mathcal{X}&lt;/math&gt; is called a ''pre-Hilbert space'' if for each ordered pair of &lt;math&gt;(x, y)&lt;/math&gt; there is a unique complex number called ''an inner product'' of &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; and denoted by &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; satisfying the following properties:
* (i) &lt;math&gt;\langle x, y \rangle_\mathcal{X}&lt;/math&gt; is a linear operator of &lt;math&gt;x&lt;/math&gt; when &lt;math&gt;y&lt;/math&gt; is fixed.
* (ii) &lt;math&gt;\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}&lt;/math&gt; (where the bar means the complex conjugation).
* (iii) &lt;math&gt;\langle x, x \rangle \ge 0&lt;/math&gt; with equality only when &lt;math&gt;x = 0&lt;/math&gt;.

When only one pre-Hilbert space is being considered we usually omit the subscript &lt;math&gt;\mathcal{X}&lt;/math&gt;.

We define &lt;math&gt;\| x \| = \langle x, x \rangle^{1/2}&lt;/math&gt; and indeed this is a norm. Indeed, it is clear that &lt;math&gt;\| \alpha x \| = | \alpha | \| x \|&lt;/math&gt; and (iii) is the reason that &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;. Finally, the triangular inequality follows from the next lemma.

'''3 Lemma (Schwarz's inequality)''' ''&lt;math&gt;|\langle x, y \rangle| \le \|x\|\|y\|&lt;/math&gt; where the equality holds if and only if we can write &lt;math&gt;x = \lambda y&lt;/math&gt; for some scalar &lt;math&gt;\lambda&lt;/math&gt;.''

If we assume the lemma, then since &lt;math&gt;\operatorname{Re}(\alpha) \le | \alpha |&lt;/math&gt; for any complex number &lt;math&gt;\alpha&lt;/math&gt; it follows:
:{|
|-
|&lt;math&gt;\| x + y \|^2&lt;/math&gt;
|&lt;math&gt;= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2&lt;/math&gt;
|-
|
|&lt;math&gt;\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\le (\| x \| + \| y \|)^2&lt;/math&gt;
|}

Proof of Lemma: The lemma is just a special case of the next theorem:

'''3 Theorem''' ''Let &lt;math&gt;\mathcal{H}&lt;/math&gt; be a pre-Hilbert and &lt;math&gt;S \subset \mathcal{H}&lt;/math&gt; be an orthonormal set (i.e., for &lt;math&gt;u, v \in E&lt;/math&gt; &lt;math&gt;\langle u, v \rangle = 1&lt;/math&gt; iff &lt;math&gt;u = v&lt;/math&gt; iff &lt;math&gt;\langle u, v \rangle&lt;/math&gt; is nonzero.)''
* ''(i) &lt;math&gt;\sum_{u \in S} \langle x, u \rangle \le |x|&lt;/math&gt; for any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt;.''
* ''(ii) The equality holds in (i) if and only if &lt;math&gt;S&lt;/math&gt; is maximal in the collection of all orthonormal subsets of &lt;math&gt;\mathcal{H}&lt;/math&gt; ordered by &lt;math&gt;\subset&lt;/math&gt;.''
Proof: (TODO)

'''3 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be a sequence in a pre-Hilbert space with &lt;math&gt;\|u_j\| = 1&lt;/math&gt;.'' If &lt;math&gt;\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2&lt;/math&gt; for any sequence &lt;math&gt;\alpha_j&lt;/math&gt; of scalars.
Proof: Let &lt;math&gt;I&lt;/math&gt; be a set of all pairs &lt;math&gt;(i, j)&lt;/math&gt; such that &lt;math&gt;m \le i \le n&lt;/math&gt;, &lt;math&gt;m \le j \le n&lt;/math&gt; and &lt;math&gt;i \ne j&lt;/math&gt;. By Hölder's inequality we get:
:&lt;math&gt;\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma&lt;/math&gt;.
Since
:&lt;math&gt;\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;,
we get the second inequality. Moreover, 
:&lt;math&gt;\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle  + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |&lt;/math&gt;
and this gives the first inequality. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Bessel's inequality)''' ''Let &lt;math&gt;U&lt;/math&gt; be an orthonormal subset of a pre-Hilbert space.'' Then for each &lt;math&gt;x&lt;/math&gt; in the space,
:&lt;math&gt;\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2&lt;/math&gt;
where the sum can be obtained over some countable subset of &lt;math&gt;U&lt;/math&gt; and the equality holds if and only if &lt;math&gt;U&lt;/math&gt; is maximal; i.e., &lt;math&gt;U&lt;/math&gt; is contained in no other orthogonal sets. &lt;br /&gt;
Proof: First suppose &lt;math&gt;U&lt;/math&gt; is finite; i.e., &lt;math&gt;U = \{ u_1, u_2, ... u_n \}&lt;/math&gt;. Let &lt;math&gt;\alpha_j = \langle x, u_j \rangle&lt;/math&gt;. Since for each &lt;math&gt;k&lt;/math&gt;, &lt;math&gt;\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0&lt;/math&gt;, by the preceding theorem or by direct computation,
:{|
|-
|&lt;math&gt;\|x\|^2&lt;/math&gt;
|&lt;math&gt;=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 &lt;/math&gt;
|-
|
|&lt;math&gt;\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2&lt;/math&gt;
|}
Now suppose that &lt;math&gt;U&lt;/math&gt; is maximal. Let &lt;math&gt;y = \sum_{j=1}^n \langle x, u_j \rangle u_j&lt;/math&gt;. Then by the same reasoning above, &lt;math&gt;x - y&lt;/math&gt; is orthogonal to every &lt;math&gt;u_j&lt;/math&gt;. But since the assumed maximality &lt;math&gt;x = y&lt;/math&gt;. Hence,
:&lt;math&gt;\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2&lt;/math&gt;. Conversely, suppose that &lt;math&gt;U&lt;/math&gt; is not maximal. Then there exists some nonzero &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;\langle x, u \rangle = 0&lt;/math&gt; for every &lt;math&gt;u \in U&lt;/math&gt;. Thus,
:&lt;math&gt;\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 &lt; \|x\|^2&lt;/math&gt;.
The general case follows from the application of Egorov's theorem. &lt;math&gt;\square&lt;/math&gt;

'''3 Corollary''' ''In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists.'' (TODO: need elaboration)

'''3 Lemma''' ''The function &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; is continuous each time &lt;math&gt;y&lt;/math&gt; is fixed.''&lt;br /&gt;
Proof: If &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt;, from Schwarz's inequality it follows:
:&lt;math&gt;| f(z) - f(x) | =  | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0&lt;/math&gt; as &lt;math&gt;z \to x&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Given a linear subspace &lt;math&gt;\mathcal{M}&lt;/math&gt; of &lt;math&gt;\mathcal{H}&lt;/math&gt;, we define: &lt;math&gt;\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}&lt;/math&gt;. In other words, &lt;math&gt;\mathcal{M}^\bot&lt;/math&gt; is the intersection of the kernels of the continuous functionals &lt;math&gt;f(x) = \langle x, y&lt;/math&gt;, which are closed; hence, &lt;math&gt;M^\bot&lt;/math&gt; is closed. (TODO: we can also show that &lt;math&gt;\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot&lt;/math&gt;)

'''3 Lemma''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a linear subspace of a pre-Hilbert space.''
''Then  &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; if and only if &lt;math&gt;\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}&lt;/math&gt;.''&lt;br /&gt;
Proof: The Schwarz inequality says the inequality
:&lt;math&gt;| \langle z, z + w \rangle | \le \|z\| \|z + w \|&lt;/math&gt;
is actually equality if and only if &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;z + w&lt;/math&gt; are linear dependent. &lt;math&gt;\square&lt;/math&gt;

'''3 Theorem (Riesz)''' ''Let &lt;math&gt;\mathcal{X}&lt;/math&gt; be a pre-Hilbert space and &lt;math&gt;\mathcal{M}&lt;/math&gt; be its subspace. The following are equivalent:''
* (i) ''&lt;math&gt;\mathcal{X}&lt;/math&gt; is a complete.''
* (ii) ''&lt;math&gt;\mathcal{M}&lt;/math&gt; is dense if and only if &lt;math&gt;\mathcal{M}^\bot = \{ 0 \}&lt;/math&gt;.''
* (iii) ''Every continuous linear functional on &lt;math&gt;\mathcal{X}^*&lt;/math&gt; has the form &lt;math&gt;f(x) = \langle x, y \rangle&lt;/math&gt; where y is uniquely determined by &lt;math&gt;f&lt;/math&gt;.''
Proof: If &lt;math&gt;\overline{\mathcal{M}} = \mathcal{H}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}&lt;/math&gt;. (Note: completeness was not needed.) Conversely, if &lt;math&gt;\overline{\mathcal{M}}&lt;/math&gt; is not dense, then it can be shown (TODO: using completeness) that there is &lt;math&gt;y \in \overline{\mathcal{M}}&lt;/math&gt; such that
:&lt;math&gt;\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}&lt;/math&gt;.
That is, &lt;math&gt;0 \ne x - y \in \overline{M}^\bot&lt;/math&gt;. In sum, (i) implies (ii). To show (iii), we may suppose that &lt;math&gt;f&lt;/math&gt; is not identically zero, and in view of (ii), there exists a &lt;math&gt;z \in \ker(f)^\bot&lt;/math&gt; with &lt;math&gt;\|z\| = 1&lt;/math&gt;. Since &lt;math&gt;f(xf(z) - f(x)z) = 0&lt;/math&gt;,
:&lt;math&gt;0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)&lt;/math&gt;.
The uniqueness holds since &lt;math&gt;\langle x, y \rangle = \langle x, y_2 \rangle&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; implies that &lt;math&gt;y = y_2&lt;/math&gt;. Finally, (iii) implies reflexivility which implies (i). &lt;math&gt;\square&lt;/math&gt;

A complete pre-Hilbert space is called a ''Hilbert space''.

'''3 Corollary''' ''Let &lt;math&gt;\mathcal{M}&lt;/math&gt; be a a closed linear subspace of a Hilbert space&lt;/math&gt;''
* ''(i) For any &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; we can write &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;z \in \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;y, z&lt;/math&gt; are uniquely determined by &lt;math&gt;x&lt;/math&gt;.''
* ''(ii) then &lt;math&gt;\mathcal{M}^{\bot\bot} = \bar \mathcal{M}&lt;/math&gt;.''
Proof: (i) Let &lt;math&gt;x \in \mathcal{H}&lt;/math&gt; be given. Define &lt;math&gt;f(w) = \langle w, x \rangle&lt;/math&gt; for each &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;. Since &lt;math&gt;f&lt;/math&gt; is continuous and linear on &lt;math&gt;\mathcal{M}&lt;/math&gt;, which is a Hilbert space, there is &lt;math&gt;y \in \mathcal{M}&lt;/math&gt; such that &lt;math&gt;f(w) = \langle w, y \rangle&lt;/math&gt;. It follows that &lt;math&gt;\langle w, x - y \rangle = 0&lt;/math&gt; for any &lt;math&gt;w \in \mathcal{M}&lt;/math&gt;; that is, &lt;math&gt;x - y \in \mathcal{M}^\bot&lt;/math&gt;. The uniqueness holds since if &lt;math&gt;y_2 \in \mathcal{M}&lt;/math&gt; and &lt;math&gt;x - y_2 \in \mathcal{M}^\bot&lt;/math&gt;, then &lt;math&gt;f(w) = \langle x, y_2 \rangle&lt;/math&gt; and the representation is unique. (ii) If &lt;math&gt;x \in \mathcal{M}&lt;/math&gt;, then since &lt;math&gt;x&lt;/math&gt; is orthogonal to &lt;math&gt;\mathcal{M}^{\bot \bot}&lt;/math&gt;. Thus, &lt;math&gt;\mathcal{M} \subset \mathcal{M}^{\bot\bot}&lt;/math&gt; and taking closure on both sides we get: &lt;math&gt;\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}&lt;/math&gt;.
Also, if &lt;math&gt;x \in \overline{M}^{\bot\bot}&lt;/math&gt;, then we write: &lt;math&gt;x = y + z&lt;/math&gt; where &lt;math&gt;y \in \overline{M}&lt;/math&gt; and &lt;math&gt;z \in \overline{M}^\bot&lt;/math&gt; and &lt;math&gt;\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0&lt;/math&gt;. Thus, &lt;math&gt;x = y \in \overline{\mathcal{M}}&lt;/math&gt;.
Since &lt;math&gt;\mathcal{M} \subset \overline{\mathcal{M}}&lt;/math&gt; implies that &lt;math&gt;\overline{\mathcal{M}} \subset \mathcal{M}^\bot&lt;/math&gt; and &lt;math&gt;\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}&lt;/math&gt;, the corollary follows. &lt;math&gt;\square&lt;/math&gt;

== Integration ==

'''3 Theorem (Fundamental Theorem of Calculus)''' ''The following are equivalent.''
* (i) The derivative of &lt;math&gt;\int_a^x f(t) dt&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;f(x)&lt;/math&gt;.
* (ii) &lt;math&gt;f&lt;/math&gt; is absolutely continuous.
Proof: Suppose (ii). Since we have:
:&lt;math&gt;\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)&lt;/math&gt;,
for any &lt;math&gt;a&lt;/math&gt;,
:&lt;math&gt;\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

== Differentiation ==

''Differentiation'' of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;x&lt;/math&gt; is to take the limit of the quotient by letting &lt;math&gt;h \rightarrow 0&lt;/math&gt;:
:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt;.
When the limit of the quotient indeed exists, we say &lt;math&gt;f&lt;/math&gt; is differentiable at &lt;math&gt;x&lt;/math&gt;.
The derivative of &lt;math&gt;f&lt;/math&gt;, denoted by &lt;math&gt;\dot f&lt;/math&gt;, is defined by &lt;math&gt;f(x)&lt;/math&gt; = the limit of the quotient at &lt;math&gt;x&lt;/math&gt;.

'''3.8. Theorem''' ''The power series:''
:&lt;math&gt;u = \sum_0^{\infty} a_j z^j&lt;/math&gt;
''is analytic inside the radius of convergence.''&lt;br /&gt;
Proof: The normal convergence of &lt;math&gt;u&lt;/math&gt; implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, thereby inducing topology;
:&lt;math&gt;\| x \| = \| \sum_1^n x_j^2 \|^{1/2}&lt;/math&gt;.

The topology in this way is often called a ''natural topology'' of &lt;math&gt;\mathit{R}^2&lt;/math&gt;, since so to speak we don't artificially induce a topology by defining &lt;math&gt;\sigma&lt;/math&gt;.

'''3. Theorem (Euler's formula)'''
:&lt;math&gt;z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)&lt;/math&gt; If &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.
Proof:

'''3. Theorem (Cauchy-Riemann equations)''' ''Suppose &lt;math&gt;u \in \mathcal{C}^1(\Omega)&lt;/math&gt;. We have:''
:''&lt;math&gt;{\partial u \over \partial \bar z} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt; if and only if &lt;math&gt;{\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''3. Corollary''' ''Let &lt;math&gt;u, v&lt;/math&gt; are non-constant and analytic in &lt;math&gt;\Omega&lt;/math&gt;. If &lt;math&gt;\mbox{Re }u = \mbox{Re }v&lt;/math&gt;, then &lt;math&gt;u = v&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;g = u - v&lt;/math&gt;. Then &lt;math&gt;0 = \mbox{Re }g = g + \bar g&lt;/math&gt;. Thus, &lt;math&gt;\mbox{Im }g = 0&lt;/math&gt;, and hence g = 0&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

This furnishes examples of functions that are not analytic. For example, &lt;math&gt;u(x + iy) = x + iy&lt;/math&gt; is analytic everywhere and that means &lt;math&gt;v(x + iy) = x + icy&lt;/math&gt; cannot be analytic unless &lt;math&gt;c = 1&lt;/math&gt;.

A operator &lt;math&gt;f&lt;/math&gt; is ''bounded'' if there exists a constant &lt;math&gt;C &gt; 0&lt;/math&gt; such that for every &lt;math&gt;x&lt;/math&gt;:
:&lt;math&gt;\| f(x) \| \le C \| x \|&lt;/math&gt;.

'''3.1 Theorem''' Given a bounded operator &lt;math&gt;f&lt;/math&gt;, if
:&lt;math&gt;\alpha = \inf \{ C : \| f(x) \le C \| x \| \}&lt;/math&gt;, &lt;math&gt;\beta = \sup_{\| x \| \le 1} \| f(x) \|&lt;/math&gt; and &lt;math&gt;\gamma = \sup_{\| x \| = 1} \| f(x) \|&lt;/math&gt;,
then &lt;math&gt;\alpha = \beta = \gamma&lt;/math&gt;.&lt;br /&gt;
Proof: Since &lt;math&gt;\beta \le \gamma&lt;/math&gt; can be verified (FIXME) and &lt;math&gt;\gamma&lt;/math&gt; is inf,
:&lt;math&gt;\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|&lt;/math&gt;.
Thus,
:&lt;math&gt;\alpha \le \beta \le \gamma&lt;/math&gt;.
But if &lt;math&gt;\alpha \|x\| &lt; \gamma \|x\|&lt;/math&gt; in the above, then this is absurd since &lt;math&gt;\gamma&lt;/math&gt; is sup; hence the theorem is proven.

We denote by &lt;math&gt;\| f \|&lt;/math&gt; either of the above values, and call it the ''norm'' of &lt;math&gt;f&lt;/math&gt;

'''3.2 Corollary''' ''A operator &lt;math&gt;f&lt;/math&gt; is bounded if and only if is continuous.''&lt;br /&gt;
Proof: If &lt;math&gt;f&lt;/math&gt; is bounded, then we find &lt;math&gt;\| f \|&lt;/math&gt; and since the identity: for every &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt;
:&lt;math&gt;\| f(x + h) - f(x) \| \le \| f \| \| h \|&lt;/math&gt;,
&lt;math&gt;f&lt;/math&gt; is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of &lt;math&gt;f&lt;/math&gt;, &lt;math&gt;\| f \|&lt;/math&gt;, and the theorem follows after the preceding theorem. &lt;math&gt;\square&lt;/math&gt;

'''3. Theorem''' ''If F is a linear space of dimension &lt;math&gt;n&lt;/math&gt;, then it has exactly &lt;math&gt;n&lt;/math&gt; subspaces including F and excluding {0}.''&lt;br /&gt;
Proof: F has a basis of n elements.

'''Theorem''' ''If H is complete, then &lt;math&gt;H^n = \{ \sum_1^n x_j e_j : x_j \in E \}&lt;/math&gt; (i.e., a cartesian space of E) is complete&lt;br /&gt;''
Proof: Let &lt;math&gt;z_j \in H&lt;/math&gt; be a Cauchy sequence. Then we have:
:&lt;math&gt;|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.
Since orthogonality, we have:
:&lt;math&gt;|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|&lt;/math&gt;,
and both &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; are also Cauchy sequences. Since completeness, the respective limits &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are in &lt;math&gt;E&lt;/math&gt;; thus, the limit &lt;math&gt;z = x e_1 + y e_2&lt;/math&gt; is in E_2. &lt;math&gt;\square&lt;/math&gt;

The theorem shows in particular that &lt;math&gt;\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}&lt;/math&gt; are complete.

'''3. Theorem (Hamel basis)''' ''The Axiom of Choice implies that every linear space has a basis''&lt;br /&gt;
Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt [http://www.dpmms.cam.ac.uk/%7Ekf262/COBOR/L5.pdf]. '''3. Theorem (Fixed Point Theorem)''' ''Suppose a function f maps a closed subset &lt;math&gt;F&lt;/math&gt; of a Banach space to itself, and further suppose that there exists some &lt;math&gt;c &lt; 1&lt;/math&gt; such that &lt;math&gt;\|f(x) - f(y)\| \le c \| x - y \|&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; has a unique fixed point.''&lt;br /&gt;
Proof: Let &lt;math&gt;s_n&lt;/math&gt; be a sequence: &lt;math&gt;x, f(x), f(f(x)), f(f(f(x))), ... &lt;/math&gt;. For any &lt;math&gt;n&lt;/math&gt; for some &lt;math&gt;x \in F&lt;/math&gt;. Then we have:
:&lt;math&gt;\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|&lt;/math&gt;.
By induction it follows:
:&lt;math&gt;\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|&lt;/math&gt;.
Thus, &lt;math&gt;s_n&lt;/math&gt; is a Cauchy sequence since:
:{|
|-
|&lt;math&gt;\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j&lt;/math&gt;
|-
|&lt;math&gt;= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})&lt;/math&gt;.
|}
That &lt;math&gt;F&lt;/math&gt; is closed puts the limit of &lt;math&gt;s_n&lt;/math&gt; in &lt;math&gt;F&lt;/math&gt;. Finally, the uniqueness follows since if &lt;math&gt;f(x) = x&lt;/math&gt; and &lt;math&gt;f(y) = y&lt;/math&gt;, then
:&lt;math&gt;\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|&lt;/math&gt; or &lt;math&gt;1 \le c&lt;/math&gt;unless &lt;math&gt;x = y&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''3. Corollary (mean value inequality)''' ''Let &lt;math&gt;f : \mathbb{R}^n \to \mathbb{R}^m&lt;/math&gt; be differentiable. Then there exists some &lt;math&gt;z = (1 - t)x + ty&lt;/math&gt; for some &lt;math&gt;t \in [0, 1]&lt;/math&gt; such that''
:''&lt;math&gt;\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|&lt;/math&gt;''
where the equality holds if &lt;math&gt;n = m = 1&lt;/math&gt; (mean value theorem).&lt;br /&gt;
Proof:

'''Theorem''' ''Let &lt;math&gt;f: E \to \mathbb{R}&lt;/math&gt; where &lt;math&gt;E \subset \mathbb{R}^n&lt;/math&gt; and is open. If &lt;math&gt;D_1 f, D_2 f, ... D_n f&lt;/math&gt; are bounded in &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; and &lt;math&gt;x \in E&lt;/math&gt; be given. Using the assumption, we find a constant &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\sup_E | D_i f | &lt; M&lt;/math&gt; for &lt;math&gt;i = 1, 2, ... n&lt;/math&gt;.
Let &lt;math&gt;\delta = \epsilon (nM)^{-1}&lt;/math&gt;. Suppose &lt;math&gt;|h| &lt; \delta&lt;/math&gt; and &lt;math&gt;x + h \in E&lt;/math&gt;. Let
:&lt;math&gt;\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)&lt;/math&gt;.
Then by the mean value theorem, we have: for some &lt;math&gt;c \in (0, 1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| \phi_k(1) - \phi_k(0) |&lt;/math&gt;
|&lt;math&gt;= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| M&lt;/math&gt;.
|}
It thus follows: since &lt;math&gt;\phi_k(0) = \phi_{k-1}(1)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;| f(x+h) - f(x) |&lt;/math&gt;
|&lt;math&gt;= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; |h| nM &lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''Theorem (differentiation rules)'''' ''Given &lt;math&gt;f, g: \mathbb{R} \to \mathbb{R}&lt;/math&gt; differentiable,''
* ''(a) (Chain Rule) &lt;math&gt;D(g \circ f) = (D(g) \circ f)D(f)&lt;/math&gt;''.
* ''(b) (Product Rule) &lt;math&gt;D(fg) = D(f)g + fD(g)&lt;/math&gt;''.
* ''(b) (Quotient Rule) &lt;math&gt;D(f / g) = g^{-2} (D(f)g - fD(g))&lt;/math&gt;''.
Proof: (b) and (c) follows after we apply (a) to them with &lt;math&gt;log&lt;/math&gt;, &lt;math&gt;h(x) = x^{-1}&lt;/math&gt; and the implicit function theorem. &lt;math&gt;\square&lt;/math&gt;.

'''Theorem (Cauchy-Riemann equations)''' ''Let &lt;math&gt;\Omega \subset \mathbb{C}&lt;/math&gt; and &lt;math&gt;u:\Omega \to \mathbb{C}&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is differentiable if and only if &lt;math&gt;{\partial \over \partial x}u&lt;/math&gt; and &lt;math&gt;{\partial \over \partial y}u&lt;/math&gt; are continuous on &lt;math&gt;\Omega&lt;/math&gt; and &lt;math&gt;{\partial \over \partial z} u = 0&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u&lt;/math&gt; is differentiable. Let &lt;math&gt;z \in \Omega&lt;/math&gt; and &lt;math&gt;x = \mbox{Re}z&lt;/math&gt; and &lt;math&gt;y = \mbox{Im}z&lt;/math&gt;.
:{|
|-
|&lt;math&gt;u'(z)&lt;/math&gt;
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)&lt;/math&gt;
|}
Since &lt;math&gt;x = {z + \overline{z} \over 2}&lt;/math&gt; and &lt;math&gt;y = {z - \bar z \over 2i}&lt;/math&gt;, the Chain Rule gives:
:{|
|-
|&lt;math&gt;{\partial \over \partial \bar z} u&lt;/math&gt;
|&lt;math&gt;= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u&lt;/math&gt;
|-
|
|&lt;math&gt;=0&lt;/math&gt;.
|}
Conversely, let &lt;math&gt;z \in \Omega&lt;/math&gt;. It suffices to show that &lt;math&gt;u'(z) = {\partial \over \partial x} u(z)&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given and &lt;math&gt;x = \Re z&lt;/math&gt; and &lt;math&gt;y = \Im z&lt;/math&gt;. Since the continuity of the partial derivatives and that &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: &lt;math&gt;B(\delta, z)  \subset \Omega&lt;/math&gt; and for &lt;math&gt;s \in B(\delta, z)&lt;/math&gt; it holds:
:&lt;math&gt;\left| {\partial \over \partial x}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt; and &lt;math&gt;\left| {\partial \over \partial y}(u(s) - u(z)) \right| &lt; \epsilon / 2&lt;/math&gt;.
Let &lt;math&gt;h \in B(\delta, 0)&lt;/math&gt; be given and &lt;math&gt;h_1= \Re h&lt;/math&gt; and &lt;math&gt;h_2 = \Im h&lt;/math&gt;. Using the mean value theorem we have: for some &lt;math&gt;s_1, s_2 \in B(\delta, z)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;u(x+h_1, y+h_2) - u(x, y)&lt;/math&gt;
|&lt;math&gt;= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)&lt;/math&gt;
|-
|
|&lt;math&gt;= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)&lt;/math&gt;
|}
where &lt;math&gt;{\partial \over \partial y} u = i {\partial \over \partial x} u&lt;/math&gt; by assumption. Finally it now follows:
:{|
|-
|&lt;math&gt;\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|&lt;/math&gt;
|&lt;math&gt;= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;
|}

'''3 Corollary''' ''Let &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and suppose &lt;math&gt;\Omega&lt;/math&gt; is connected. Then the following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is constant.''
* ''(b) &lt;math&gt;\mbox{Re}u&lt;/math&gt; is constant.''
* ''(c) &lt;math&gt;|u|&lt;/math&gt; is constant.''
Proof: That (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b) is obvious. Suppose (b). Since we have some constant &lt;math&gt;M&lt;/math&gt; so that for all &lt;math&gt;z \in \Omega&lt;/math&gt;,
:&lt;math&gt;|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M&lt;/math&gt;,
clearly it holds that &lt;math&gt;|u| = |\log e^u| = M&lt;/math&gt;. Thus, (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). Then &lt;math&gt;M^2 = |u|^2 = u \overline{u}&lt;/math&gt;.
Differentiating both sides we get:
:&lt;math&gt;0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u&lt;/math&gt;.
Since &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt;, it follows that &lt;math&gt;{\partial \over \partial z} \overline{u} = 0&lt;/math&gt; and &lt;math&gt;\overline{u} {\partial \over \partial z}u = 0&lt;/math&gt;.
If &lt;math&gt;\overline{u} = 0&lt;/math&gt;, then &lt;math&gt;u = 0&lt;/math&gt;. If &lt;math&gt;{\partial \over \partial z}u = 0&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is constant since &lt;math&gt;\Omega&lt;/math&gt; is connected. Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (a). &lt;math&gt;\square&lt;/math&gt;

We say a function has the ''open mapping property'' if it maps open sets to open sets. The ''maximum principle'' states that equivalently
* if a function has a local maximum, then the function is constant.

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. The following are equivalent:''
* ''(a) &lt;math&gt;u&lt;/math&gt; is harmonic.''
* ''(b) &lt;math&gt;u&lt;/math&gt; has the mean value property.''

'''3 Theorem''' ''Let &lt;math&gt;u: \Omega \to \mathbb{C}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has the open mapping property, then the maximum principle holds.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u \in \mathcal{A}(\Omega)&lt;/math&gt; and &lt;math&gt;\Omega&lt;/math&gt; is open and connected. Let &lt;math&gt;\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}&lt;/math&gt;. If &lt;math&gt;u&lt;/math&gt; has a local maximum, then &lt;math&gt;\omega&lt;/math&gt; is nonempty. Also, &lt;math&gt;\Omega&lt;/math&gt; is closed in &lt;math&gt;\Omega&lt;/math&gt; since &lt;math&gt;\Omega = u^{-1}({\omega})&lt;/math&gt;. Let &lt;math&gt;a \in \omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is open, we can find a &lt;math&gt;r &gt; 0&lt;/math&gt; so that: &lt;math&gt;B = B(r, a) \subset \Omega&lt;/math&gt;. Since &lt;math&gt;u(\omega)&lt;/math&gt; is open by the open mapping property, we can find a &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; so that &lt;math&gt;B(\epsilon, u(a)) \subset u(\omega)&lt;/math&gt;. This is to say that &lt;math&gt;u(a) &lt; \epsilon + u(z)&lt;/math&gt; for some &lt;math&gt;z \subset B(r, a)&lt;/math&gt;. This is absurd since &lt;math&gt;a \in \omega&lt;/math&gt; and &lt;math&gt;u(z) \le u(a)&lt;/math&gt; for all &lt;math&gt;z \in \Omega&lt;/math&gt;. Thus, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; identically on &lt;math&gt;B(r, a)&lt;/math&gt; and it thus holds that &lt;math&gt;B(r, a) \subset \omega&lt;/math&gt; and &lt;math&gt;\omega&lt;/math&gt; is open in &lt;math&gt;\Omega&lt;/math&gt;. Since &lt;math&gt;\Omega&lt;/math&gt; is connected, &lt;math&gt;\Omega = \omega&lt;/math&gt;. Therefore, &lt;math&gt;u = \sup_\Omega |u|&lt;/math&gt; on &lt;math&gt;\Omega&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;


== Addendum ==
'''Exercise''' ''Let &lt;math&gt;f \in \mathcal{A}(\mathbb{C})0&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is a polynomial of degree &lt;math&gt; &gt; n&lt;/math&gt; if and only if there are constants &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;|f(z)| \le A + B|z|^n&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;.''

'''Exercise 2''' ''Let &lt;math&gt;f: A \to A&lt;/math&gt; be linear. Further suppose &lt;math&gt;A&lt;/math&gt; has dimension &lt;math&gt;n &lt; \infty&lt;/math&gt;. Then the following are equivalent:''
# ''&lt;math&gt;f^{-1}&lt;/math&gt; exists''
# ''&lt;math&gt;\det(f) \ne 0&lt;/math&gt; where &lt;math&gt;\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}&lt;/math&gt;''
# ''The set''
#:''&lt;math&gt;\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}&lt;/math&gt; has dimension &lt;math&gt;n&lt;/math&gt;.''

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      <text bytes="55483" sha1="r20ff6j1asz5zbenjcaa3w3249c82da" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' ''
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:
:&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent:
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55481" sha1="qrdiyj83owbdo7ucdd8z6zoc38z3o8u" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:
:&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent:
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55483" sha1="ha5o9tr9jn23civp39gp5hne3ve54q1" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:
:&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent:
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55486" sha1="kzozzwcgsbiumeg1yqcgsf1r5a7df8y" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:
:&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

'''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent'':
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55492" sha1="a6atckb5gasfc4vf26b17nxv04r2wiu" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:''
:''&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;''
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

'''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent'':
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55498" sha1="nxwbqk8ptlcbpwnr0vezbr833ux9cgo" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:''
:''&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;''
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

'''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent'':
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:''
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:''&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.''
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55500" sha1="pm2nmxb3ospxgy5eeg7f0dp2oezvxr2" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:''
:''&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;''
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

'''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent'':
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:''
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:''&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.''
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:''&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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      <text bytes="55502" sha1="fh1ppakh4zopbb34nqs7vz7l0mz3k9o" xml:space="preserve">The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

== Real numbers ==

Let &lt;math&gt;\mathbf{Q}&lt;/math&gt; be the set of rational numbers. A sequence &lt;math&gt;x_n&lt;/math&gt; in &lt;math&gt;\mathbf{Q}&lt;/math&gt; is said to be Cauchy if &lt;math&gt;|x_n - x_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;. Let &lt;math&gt;I&lt;/math&gt; be the set of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt;. &lt;math&gt;I&lt;/math&gt; is then an ideal of &lt;math&gt;\mathbf{Q}&lt;/math&gt;. It then follows that &lt;math&gt;I&lt;/math&gt; is a maximal ideal and &lt;math&gt;\mathbf{Q} / I&lt;/math&gt; is a field, which we denote by &lt;math&gt;\mathbf{R}&lt;/math&gt;.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a [[w:Absolute value (algebra)|valuation]] &lt;math&gt;| \cdot |&lt;/math&gt; ''could'' give rise to a different field. It turned out that every valuation for &lt;math&gt;\mathbf{Q}&lt;/math&gt; is either equivalent to the usual absolute or some p-adic absolute value, where &lt;math&gt;p&lt;/math&gt; is a prime. ([[w:Ostrowski's theorem]])  Define &lt;math&gt;\mathbf{Q}_p = \mathbf{Q} / I&lt;/math&gt; where &lt;math&gt;I&lt;/math&gt; is the maximal ideal of all sequences &lt;math&gt;x_n&lt;/math&gt; that converges to &lt;math&gt;0&lt;/math&gt; in &lt;math&gt;| \cdot |_p&lt;/math&gt;.

A p-adic absolute value is defined as follows. Given a prime &lt;math&gt;p&lt;/math&gt;, every rational number &lt;math&gt;x&lt;/math&gt; can be written as &lt;math&gt;x = p^n a / b&lt;/math&gt; where &lt;math&gt;a, b&lt;/math&gt; are integers not divisible by &lt;math&gt;p&lt;/math&gt;. We then let &lt;math&gt;|x|_p = p^{-n}&lt;/math&gt;. Most importantly, &lt;math&gt;|\cdot|_p&lt;/math&gt; is non-Archimedean; i.e.,
:&lt;math&gt;|x + y|_p \le \max \{ |x|_p, |y|_p \}&lt;/math&gt;
This is stronger than the triangular inequality since &lt;math&gt;\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p&lt;/math&gt;

Example: &lt;math&gt;|6|_2 = |18|_2 = {1 \over 2}&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;|x_n - x|_p \to 0&lt;/math&gt;, then &lt;math&gt;|x_n - x_m|_p&lt;/math&gt;''.&lt;br /&gt;
Proof:
:&lt;math&gt;|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;

Let &lt;math&gt;s_n = 1 + p + p^2 + .. + p^{n-1}&lt;/math&gt;. Since &lt;math&gt;(1 - p)s_n = 1 - p^n&lt;/math&gt;,
:&lt;math&gt;|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0&lt;/math&gt;
Thus, &lt;math&gt;\lim_{n \to \infty} s_n = {1 \over 1 - p}&lt;/math&gt;.

Let &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; be the closed unit ball of &lt;math&gt;\mathbf{Q}_p&lt;/math&gt;. That &lt;math&gt;\mathbf{Z}_p&lt;/math&gt; is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

'''2 Theorem''' 
:&lt;math&gt;\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;
''where the project maps &lt;math&gt;f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt; are such that &lt;math&gt;f_n \circ \pi_{n+1} = \pi_n&lt;/math&gt; with &lt;math&gt;\pi_n&lt;/math&gt; being the projection &lt;math&gt;\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;.''&lt;br /&gt;

The theorem implies that &lt;math&gt; \mathbf{Z}_p&lt;/math&gt; is a closed subspace of:
:&lt;math&gt;\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}&lt;/math&gt;,
which is a product of compact spaces and thus is compact.

== Sequences ==

We say a sequence of scalar-valued functions ''converges uniformly'' to another function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; if &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;

'''1 Theorem (iterated limit theorem)''' ''Let &lt;math&gt;f_n&lt;/math&gt; be a sequence of scalar-valued functions on &lt;math&gt;X&lt;/math&gt;. If &lt;math&gt;\lim_{y \to x} f_n(y)&lt;/math&gt; exists and if &lt;math&gt;f_n&lt;/math&gt; is uniformly convergent, then for each fixed &lt;math&gt;x \in X&lt;/math&gt;, we have:''
:''&lt;math&gt;\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)&lt;/math&gt;''
Proof:
Let &lt;math&gt;a_n = \lim_{n \to \infty} f_n(x)&lt;/math&gt;, and &lt;math&gt;f&lt;/math&gt; be a uniform limit of &lt;math&gt;f_n&lt;/math&gt;; i.e., &lt;math&gt;\sup |f_n - f| \to 0&lt;/math&gt;. Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. By uniform convergence, there exists &lt;math&gt;N &gt; 0&lt;/math&gt; such that
:&lt;math&gt;2\sup |f - f_N| &lt; \epsilon / 2&lt;/math&gt;
Then there is a neighborhood &lt;math&gt;U_x&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; such that:
:&lt;math&gt; |f_N(y) - f_N(x)| &lt; \epsilon / 2&lt;/math&gt; whenever &lt;math&gt;y \in U_x&lt;/math&gt;
Combine the two estimates:
:&lt;math&gt;|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| &lt; 2\sup |f - f_N| + \epsilon / 2 = \epsilon&lt;/math&gt;
Hence,
:&lt;math&gt;\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A uniform limit of a sequence of continuous functions is again continuous.''

=== Real and complex numbers ===

In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by &lt;math&gt;\mathbb{N}&lt;/math&gt; the set of natural numbers. The set &lt;math&gt;\mathbb{N}&lt;/math&gt; does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by &lt;math&gt;\mathbb{Z}&lt;/math&gt;. It forms an integral domain; thus, we may define the quotient field &lt;math&gt;\mathbb{Q}&lt;/math&gt; of &lt;math&gt;\mathbb{Z}&lt;/math&gt;, that is, the set of rational numbers, by
:&lt;math&gt;\mathbb{Q} = \left \{ {a \over b} :  \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}&lt;/math&gt;.
As usual, we say for two rationals &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; &lt;math&gt;x &lt; y&lt;/math&gt; if &lt;math&gt;x - y&lt;/math&gt; is positive.

We say &lt;math&gt;E \subset \mathbb{Q}&lt;/math&gt; is ''bounded above'' if there exists some &lt;math&gt;x&lt;/math&gt; in &lt;math&gt;\mathbb{Q}&lt;/math&gt; such that for any &lt;math&gt;y \in E&lt;/math&gt; &lt;math&gt;y \le x&lt;/math&gt;, and is ''bounded below'' if the reversed relation holds. Notice &lt;math&gt;E&lt;/math&gt; is bounded above and below if and only if &lt;math&gt;E&lt;/math&gt; is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with &lt;math&gt;\mathbb{R}&lt;/math&gt; instead of is a quite simple one:

'''2 Theorem''' ''Fundamental axiom of analysis fails in &lt;math&gt;\mathbb{Q}&lt;/math&gt;.''&lt;br /&gt;
Proof: &lt;math&gt;1, 1.4, 1.41, 1.414, ... \to 2^{1/2}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field &lt;math&gt;\mathbb{R}&lt;/math&gt; by completing &lt;math&gt;\mathbb{Q}&lt;/math&gt;.

We define the set of complex numbers &lt;math&gt;\mathbb{C} = \mathbb{R}[i] / &lt;i^2 + 1&gt;&lt;/math&gt; where &lt;math&gt;i&lt;/math&gt; is just a symbol. That &lt;math&gt;i^2 + 1&lt;/math&gt; is irreducible says the ideal generated by it is maximal, and the field theory tells that &lt;math&gt;C&lt;/math&gt; is a field. Every complex number &lt;math&gt;z&lt;/math&gt; has a form:&lt;br /&gt;
:&lt;math&gt;z = a + bi + &lt;i^2 + 1&gt;&lt;/math&gt;.
Though the square root of -1 does not exist, &lt;math&gt;i^2&lt;/math&gt; can be thought of as -1 since &lt;math&gt;i^2 + &lt;i^2 + 1&gt; = -1 + 1 + i^2 + &lt;i^2 + 1&gt; = -1&lt;/math&gt;. Accordingly, the term &lt;math&gt;&lt;i^2 + 1&gt;&lt;/math&gt;is usually omitted.

'''2 Exercise''' ''Prove that there exists irrational numbers &lt;math&gt;a, b&lt;/math&gt; such that &lt;math&gt;a^b&lt;/math&gt; is rational.''

=== Sequences ===

'''2 Theorem''' ''Let &lt;math&gt;s_n&lt;/math&gt; be a sequence of numbers, be they real or complex. Then the following are equivalent'':
* (a) &lt;math&gt;s_n&lt;/math&gt; converges.
* (b) There exists a cofinite subsequence in every open ball.

'''Theorem (Bolzano-Weierstrass)''' ''Every infinite bounded set has a non-isolated point.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;S&lt;/math&gt; is discrete. Then &lt;math&gt;S&lt;/math&gt; is closed; thus, &lt;math&gt;S&lt;/math&gt; is compact by Heine-Borel theorem. Since &lt;math&gt;S&lt;/math&gt; is discrete, there exists a collection of disjoint open balls &lt;math&gt;S_x&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; for each &lt;math&gt;x \in S&lt;/math&gt;. Since the collection is an open cover of &lt;math&gt;S&lt;/math&gt;, there exists a finite subcover &lt;math&gt;\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}&lt;/math&gt;.
But
:&lt;math&gt;S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} &lt;/math&gt;
This contradicts that &lt;math&gt;S&lt;/math&gt; is infinite. &lt;math&gt;\square&lt;/math&gt;.

'''2 Corollary''' ''Every bounded sequence has a convergent subsequence.''
Proof: 

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence &lt;math&gt;a_n&lt;/math&gt; of numbers, let &lt;math&gt;E_j = \{ a_j, a_{j+1}, \cdot \}&lt;/math&gt;. Then we have:
{|
|-
|&lt;math&gt;\limsup_{j \rightarrow \infty} a_n&lt;/math&gt;
|&lt;math&gt;= \limsup_{j \rightarrow \infty} E_j&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty}\bigcup_{k=j}^{\infty}E_k&lt;/math&gt;
|-
|
|&lt;math&gt;= \bigcap_{j=1}^{\infty} \sup \{ a_k, a_{k+1}, ... \}&lt;/math&gt;
|}

The similar case holds for liminf as well.

'''Theorem''' ''Let &lt;math&gt;s_j&lt;/math&gt; be a sequence. The following are equivalent:''
*(a) ''The sequence &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(b) ''The sequence &lt;math&gt;s_j&lt;/math&gt; is Cauchy; i.e., &lt;math&gt;|s_n - s_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''
*(c) ''Every convergent subsequence of &lt;math&gt;s_j&lt;/math&gt; converges to &lt;math&gt;s&lt;/math&gt;.''
*(d) ''&lt;math&gt;\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j&lt;/math&gt;.''
*(e) ''For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find some real number &lt;math&gt;N&lt;/math&gt; (i.e., N is a function of &lt;math&gt;\epsilon&lt;/math&gt;) so that''
*:''&lt;math&gt;|s_j - s| &lt; \epsilon&lt;/math&gt; for &lt;math&gt;j \ge N&lt;/math&gt;.''
Proof: From the triangular inequality it follows that:
:&lt;math&gt;|s_n - s_m| \le |s_n - n| + |s_m - m|&lt;/math&gt;.
Letting &lt;math&gt;n, m \to \infty&lt;/math&gt; gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, &lt;math&gt;s_k&lt;/math&gt;. Thus,
:{|
|-
|&lt;math&gt;|s_n - s|&lt;/math&gt;
|&lt;math&gt;= |s_n - s_m + s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;\le |s_n - s_m| + |s_m - s| &lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon \ 2 + \epsilon \ 2 = \epsilon&lt;/math&gt;
|}. Therefore, &lt;math&gt;s_j \to s&lt;/math&gt;. That (c) implies (d) is obvious by definition.

'''2 Theorem''' ''Let &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; converge to &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt;, respectively. Then we have:''
:(a) &lt;math&gt;\lim_{j \to \infty} (x_j + y_j) = x + y&lt;/math&gt;.
:(b) &lt;math&gt;\lim_{j \to \infty} (x_jy_j) = xy&lt;/math&gt;.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. (a) From the triangular inequality it follows:
:&lt;math&gt;|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| &lt; {\epsilon \over 2} + {\epsilon \over 2} = \epsilon&lt;/math&gt;
where the convergence of &lt;math&gt;x_j&lt;/math&gt; and &lt;math&gt;y_j&lt;/math&gt; tell that we can find some &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;|x_j - x| &lt; {\epsilon \over 2}&lt;/math&gt; and &lt;math&gt;|y_j - y| &lt; {\epsilon \over 2}&lt;/math&gt; for all &lt;math&gt;j &gt; N&lt;/math&gt;.
(b) Again from the triangular inequality, it follows:
:{|
|-
|&lt;math&gt;|x_j y_j - xy|&lt;/math&gt;
|&lt;math&gt;= |(x_j - x)(y_j - y + y) + x(y_j - y)|&lt;/math&gt;
|-
|
|&lt;math&gt;\le |x_j - x|(1 + |y|) + |x| |y_j - y|&lt;/math&gt;
|-
|
|&lt;math&gt;\to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;
|}
where we may suppose that &lt;math&gt;|y_j - y| &lt; 1&lt;/math&gt;.
&lt;math&gt;\square&lt;/math&gt;

Other similar cases follows from the theorem; for example, we can have by letting &lt;math&gt;y_j = \alpha&lt;/math&gt;
:&lt;math&gt;\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j&lt;/math&gt;.

'''2.7 Theorem''' ''Given &lt;math&gt;\sum a_n&lt;/math&gt; in a normed space:''&lt;br /&gt;
* ''(a) &lt;math&gt;\sum \left \Vert a_n \right \Vert&lt;/math&gt; converges.''
* ''(b) The sequence &lt;math&gt;\left \Vert a_n \right \Vert^{1 \over n}&lt;/math&gt; has the upper limit &lt;math&gt;a^*&lt; 1&lt;/math&gt;.'' (Archimedean property)
* ''(c) &lt;math&gt;\prod (a_n + 1)&lt;/math&gt; converges.''
Proof: If (b) is true, then we can find a &lt;math&gt;N &gt; 0&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all &lt;math&gt;n \ge N&lt;/math&gt;:
:&lt;math&gt;\left\| a_n \right\|^{1 \over n} &lt; b &lt; 1&lt;/math&gt;. Thus (a) is true since:,
:&lt;math&gt;\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}&lt;/math&gt; if &lt;math&gt;a = \sum_1^{N-1}&lt;/math&gt;.

=== Continuity ===

Let &lt;math&gt;f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}&lt;/math&gt;. We write &lt;math&gt;\{ f &gt; a \}&lt;/math&gt; to mean the set &lt;math&gt;\{ x : x \in \Omega, f(x) &gt; a \}&lt;/math&gt;. In the same vein we also use the notations like &lt;math&gt;\{f &lt; a\}, \{f = a\}&lt;/math&gt;, etc.

We say, &lt;math&gt;u&lt;/math&gt; is ''upper semicontinuous'' if the set &lt;math&gt;\{ f &lt; c \}&lt;/math&gt; is open for every &lt;math&gt;c&lt;/math&gt; and ''lower semicontinuous'' if &lt;math&gt;-f&lt;/math&gt; is upper semicontinuous.

'''2 Lemma''' ''The following are equivalent.''
* ''(i) &lt;math&gt;u&lt;/math&gt; is upper semicontinuous.''
* ''(ii) If &lt;math&gt;u(z) &lt; c&lt;/math&gt;, then there is a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;\delta} f(s) &lt; c&lt;/math&gt;.''
* ''(iii) &lt;math&gt;\limsup_{s \to z} u(s) \le u(z)&lt;/math&gt;''.
Proof: Suppose &lt;math&gt;u(z) &lt; c&lt;/math&gt;. Then we find a &lt;math&gt;c_2&lt;/math&gt; such that &lt;math&gt;u(z) &lt; c_2 &lt; c&lt;/math&gt;. If (i) is true, then we can find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that &lt;math&gt;\sup_{|s-z|&lt;\delta} u(s) \le c_2 &lt; c&lt;/math&gt;. Thus, the converse being clear, (i) &lt;math&gt;\iff&lt;/math&gt; (ii). Assuming (ii), for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, we can find a &lt;math&gt;\delta_\epsilon &gt; 0&lt;/math&gt; such that &lt;math&gt;\sup_{|s-z|&lt;{\delta_\epsilon}}u(s) &lt; u(z) + \epsilon&lt;/math&gt;. Taking inf over all &lt;math&gt;\epsilon&lt;/math&gt; gives (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii), whose converse is clear. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''If &lt;math&gt;u&lt;/math&gt; is upper semicontinuous and &lt;math&gt;&lt; \infty&lt;/math&gt; on a compact set &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;u&lt;/math&gt; is bounded from above and attains its maximum.''&lt;br /&gt;
Proof: Suppose &lt;math&gt;u(x) &lt; \sup_K u&lt;/math&gt; for all &lt;math&gt;x \in K&lt;/math&gt;. For each &lt;math&gt;x \in K&lt;/math&gt;, we can find &lt;math&gt;g(x)&lt;/math&gt; such that &lt;math&gt;u(x) &lt; g(x) &lt; \sup_K u&lt;/math&gt;. Since &lt;math&gt;u&lt;/math&gt; is upper semicontinuous, it follows that the sets &lt;math&gt;\{u &lt; g(x)\}&lt;/math&gt;, running over all &lt;math&gt;x \in K&lt;/math&gt;, form an open cover of &lt;math&gt;K&lt;/math&gt;. It admits a finite subcover since &lt;math&gt;K&lt;/math&gt; is compact. That is to say, there exist &lt;math&gt;x_1, x_2, ..., x_n \in K&lt;/math&gt; such that &lt;math&gt;K \subset \{u &lt; g(x_1)\} \cup \{u &lt; g(x_1)\} \cup ... \{u &lt; g(x_n)\}&lt;/math&gt; and so:
:&lt;math&gt;\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} &lt; \sup_K u&lt;/math&gt;,
which is a contradiction. Hence, there must be some &lt;math&gt;z \in K&lt;/math&gt; such that &lt;math&gt;u(z) = \sup_K u&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

If &lt;math&gt;f&lt;/math&gt; is continuous, then both &lt;math&gt;f^{-1} ( (\alpha, \infty) )&lt;/math&gt; and &lt;math&gt;f^{-1} ( (-\infty, \beta) )&lt;/math&gt; for all &lt;math&gt;\alpha, \beta \in \mathbb{R}&lt;/math&gt; are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

'''2 Lemma''' ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let &lt;math&gt;f&lt;/math&gt; be upper semicontinuous. Then there exists a decreasing sequence &lt;math&gt;f_j&lt;/math&gt; of Lipschitz continuous functions that converges to f.''&lt;br /&gt;
Proof: The first part is obvious. To show the second part, let &lt;math&gt;f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }&lt;/math&gt;. Then &lt;math&gt;f_j&lt;/math&gt; has the desired properties since the identity &lt;math&gt;|f_j(x) - f_j(y)| = j^{-1} |x - y|&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

We say a function is ''uniform continuous'' on &lt;math&gt;E&lt;/math&gt; if for each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for all &lt;math&gt;x, y \in E&lt;/math&gt; with &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt; we have &lt;math&gt;|f(x) - f(y)|&lt;/math&gt;.

Example: The function &lt;math&gt;f(x) = x&lt;/math&gt; is uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x - y| &lt; \delta = \epsilon&lt;/math&gt;
while the function &lt;math&gt;f(x) = x^2&lt;/math&gt; is not uniformly continuous on &lt;math&gt;\mathbb{R}&lt;/math&gt; since
:&lt;math&gt;|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|&lt;/math&gt;
and &lt;math&gt;|x + y|&lt;/math&gt; can be made arbitrary large.

'''2 Theorem''' ''Let &lt;math&gt;f: K \to \mathbb{R}&lt;/math&gt; for &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact. If &lt;math&gt;f&lt;/math&gt; is continuous on &lt;math&gt;K&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is uniformly continuous on &lt;math&gt;K&lt;/math&gt;.''&lt;br /&gt;
Proof:

The theorem has this interpretation; an uniform continuity is a ''global'' property in contrast to continuity, which is a local property.

'''2 Corollary''' ''A function is uniform continuous on a bounded set &lt;math&gt;E&lt;/math&gt; if and only if it has a continuos extension to &lt;math&gt;\overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof:

'''2 Theorem''' ''if the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; of continuos functions converges uniformly on a compact set &lt;math&gt;K&lt;/math&gt;, then the sequence &lt;math&gt;\{ f_j \}&lt;/math&gt; is equicontinuous.''&lt;br /&gt;
Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since &lt;math&gt;f_j \to f&lt;/math&gt; uniformly, we can find a &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;|f_j(x) - f(x) | &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;j &gt; N&lt;/math&gt; and any &lt;math&gt;x \in K&lt;/math&gt;.
Also, since &lt;math&gt;K&lt;/math&gt; is compact, &lt;math&gt;f&lt;/math&gt; and each &lt;math&gt;f_j&lt;/math&gt; are uniformly continuous on &lt;math&gt;K&lt;/math&gt; and so, we find a finite sequence &lt;math&gt;\{ \delta_0, \delta_1, ... \delta_N \}&lt;/math&gt; so that:
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_0&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;,
and for &lt;math&gt;i = 1, 2, ... N&lt;/math&gt;,
:&lt;math&gt;|f_i(x) - f_i(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta_i&lt;/math&gt; and &lt;math&gt;x, y \in E&lt;/math&gt;.
Let &lt;math&gt;\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}&lt;/math&gt;, and let &lt;math&gt;x, y \in K&lt;/math&gt; be given.
If &lt;math&gt;j \le N&lt;/math&gt;, then
:&lt;math&gt;|f_j(x) - f_j(y)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;.
If &lt;math&gt;j &gt; N&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;|f_j(x) - f_j(y)|&lt;/math&gt;
|&lt;math&gt;\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|&lt;/math&gt;
|-
|
|&lt;math&gt;&lt; \epsilon&lt;/math&gt;
|}
whenever &lt;math&gt;|x - y| &lt; \delta&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''A sequence &lt;math&gt;f_j&lt;/math&gt; of complex-valued functions converges uniformly on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;\sup_E | f_n - f_m | \to 0&lt;/math&gt;as &lt;math&gt;n, m \to \infty&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;\| \cdot \| = \sup_E | \cdot |&lt;/math&gt;. The direct part follows holds since the limit exists by assumption. To show the converse, let &lt;math&gt;f(x) = \lim_{j \to \infty} f_j(x)&lt;/math&gt; for each &lt;math&gt;x \in E&lt;/math&gt;, which exists since the completeness of &lt;math&gt;\mathbb{C}&lt;/math&gt;. Moreover, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Since from the hypothesis it follows that the sequence &lt;math&gt;\|f_j\|&lt;/math&gt; of real numbers is Cauchy, we can find some &lt;math&gt;N&lt;/math&gt; so that:
:&lt;math&gt;\|f_n - f_m\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;n, m &gt; N&lt;/math&gt;.
The theorem follows. Indeed, suppose &lt;math&gt;j &gt; N&lt;/math&gt;. For each &lt;math&gt;x&lt;/math&gt;, 
since the pointwise convergence, we can find some &lt;math&gt;M&lt;/math&gt; so that:
:&lt;math&gt;\|f_k(x) - f(x)\| &lt; \epsilon / 2&lt;/math&gt; for all &lt;math&gt;k &gt; M&lt;/math&gt;.
Thus, if &lt;math&gt;n = \max \{ N, M \} + 1&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| &lt; \epsilon.&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''A numerical sequence &lt;math&gt;a_j&lt;/math&gt; converges if and only if &lt;math&gt;|a_n - a_m| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;f_j&lt;/math&gt; in the theorem be constant functions. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem (Arzela)''' ''Let &lt;math&gt;K&lt;/math&gt; be compact and metric. If a family &lt;math&gt;\mathcal{F}&lt;/math&gt; of real-valued functions on K bestowed with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt; is pointwise bounded and equicontinuous on a compact set &lt;math&gt;K&lt;/math&gt;, then:
*(i) &lt;math&gt;\mathcal{F}&lt;/math&gt; is uniformly bounded.
*(ii) &lt;math&gt;\mathcal{F}&lt;/math&gt; is totally bounded.
Proof: Let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; be given. Then by equicontinuity we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that: for any &lt;math&gt;x, y \in K&lt;/math&gt; with &lt;math&gt;x \in B(\delta, y)&lt;/math&gt;
:&lt;math&gt;|f(x) - f(y)| &lt; \epsilon / 3&lt;/math&gt;
Since &lt;math&gt;K&lt;/math&gt; is compact, it admits a finite subset &lt;math&gt;K_2&lt;/math&gt; such that:
:&lt;math&gt;K \subset \bigcup_{x \in K_2} B(\delta, x)&lt;/math&gt;.
Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that &lt;math&gt;K_2&lt;/math&gt; is a singleton &lt;math&gt;\{ y \}&lt;/math&gt;. Since the pointwise boundedness we have: 
:&lt;math&gt;|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| &lt; \infty&lt;/math&gt; for any &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;,
showing that (i) holds.
To show (ii) let &lt;math&gt;A = \cup_{f \in \mathcal{F}} \{ f(y) \}&lt;/math&gt;. Then since &lt;math&gt;\overline{A}&lt;/math&gt; is compact, &lt;math&gt;A&lt;/math&gt; is totally bounded; hence, it admits a finite subset &lt;math&gt;A_2&lt;/math&gt; such that: for each &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;, we can find some &lt;math&gt;g(y) \in A_2&lt;/math&gt; so that:
:&lt;math&gt;|f(y) - g(y)| &lt; \epsilon / 3&lt;/math&gt;.
Now suppose &lt;math&gt;x \in K&lt;/math&gt; and &lt;math&gt;f \in \mathcal{F}&lt;/math&gt;. Finding some &lt;math&gt;g(y)&lt;/math&gt; in &lt;math&gt;A_2&lt;/math&gt; we have:
:&lt;math&gt;|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| &lt; \epsilon / 3&lt;/math&gt;.
Since there can be finitely many such &lt;math&gt;g&lt;/math&gt;, this shows (ii). &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary (Ascoli's theorem)''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued functions is equicontinuous and pointwise bounded on every compact subset of &lt;math&gt;\Omega&lt;/math&gt;, then &lt;math&gt;f_j&lt;/math&gt; admits a subsequence converging normally on &lt;math&gt;\Omega&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;K \subset \Omega&lt;/math&gt; be compact. By Arzela's theorem the sequence &lt;math&gt;f_j&lt;/math&gt; is totally bounded with &lt;math&gt;\| \cdot \| = \sup_K | \cdot |&lt;/math&gt;. It follows that it has an accumulation point and has a subsequence converging to it on &lt;math&gt;K&lt;/math&gt;. The application of Cantor's diagonal process on an exhaustion by compact subsets of &lt;math&gt;\Omega&lt;/math&gt; yields the desired subsequence. &lt;math&gt;\square&lt;/math&gt;

'''2 Corollary''' ''If a sequence &lt;math&gt;f_j&lt;/math&gt; of real-valued &lt;math&gt;\mathcal{C}^1&lt;/math&gt; functions on &lt;math&gt;\Omega&lt;/matH&gt; obeys: for some &lt;math&gt;c \in \Omega&lt;/math&gt;,
:&lt;math&gt;\sup_j |f_j(c) | &lt; \infty&lt;/math&gt; and &lt;math&gt;\sup_{x \in \Omega, j} |f_j'(x)| &lt; \infty&lt;/math&gt;,
then &lt;math&gt;f_j&lt;/math&gt; has a uniformly convergent subsequence.&lt;br /&gt;
Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let &lt;math&gt;M&lt;/math&gt; be the second sup in the condition. Since by the mean value theorem we have:
:&lt;math&gt;|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|&lt;/math&gt;,
and the hypothesis, the sup taken all over the sequence &lt;math&gt;f_j&lt;/math&gt; is finite. Using the mean value theorem again we also have: for any &lt;math&gt;j&lt;/math&gt; and &lt;math&gt;x, y \in \Omega&lt;/math&gt;,
:&lt;math&gt;|f_j(x) - f_j(y)| \le |x - y| M&lt;/math&gt;
showing the equicontinuity. &lt;math&gt;\square&lt;/math&gt;

=== First and second countability ===

'''2 Theorem (first countability)''' ''Let &lt;math&gt;E&lt;/math&gt; have a countable base at each point of &lt;math&gt;E&lt;/math&gt;. Then we have the following:
* ''(i) For &lt;math&gt;A \subset E&lt;/math&gt;, &lt;math&gt;x \in \overline{A}&lt;/math&gt; if and only if there is a sequence &lt;math&gt;x_j \in A&lt;/math&gt; such that &lt;math&gt;x_j \to x&lt;/math&gt;.
* ''(ii) A function is continuous on &lt;math&gt;E&lt;/math&gt; if and only if &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt; whenever &lt;math&gt;x_j \to x&lt;/math&gt;.
Proof: (i) Let &lt;math&gt;\{B_j\}&lt;/math&gt; be a base at &lt;math&gt;x&lt;/math&gt; such that &lt;math&gt;B_{j + 1} \subset B_j&lt;/math&gt;. If &lt;math&gt;x \in \overline{A}&lt;/math&gt;, then every &lt;math&gt;B_j&lt;/math&gt; intersects &lt;math&gt;A&lt;/math&gt;. Let &lt;math&gt;x_j \in B_j \cap E&lt;/math&gt;. It now follows: if &lt;math&gt;x \in G&lt;/math&gt;, then we find some &lt;math&gt;B_N \subset G&lt;/math&gt;. Then &lt;math&gt;x_k \in B_k \subset B_N&lt;/math&gt; for &lt;math&gt;k \ge N&lt;/math&gt;. Hence, &lt;math&gt;x_j \to x&lt;/math&gt;. Conversely, if &lt;math&gt;x \not\in \overline{A}&lt;/math&gt;, then &lt;math&gt;E \backslash \overline{E}&lt;/math&gt; is an open set containing &lt;math&gt;x&lt;/math&gt; and no &lt;math&gt;x_j \subset E&lt;/math&gt;. Hence, no sequence in &lt;math&gt;A&lt;/math&gt; converges to &lt;math&gt;x&lt;/math&gt;. That (ii) is valid follows since &lt;math&gt;f(x_j)&lt;/math&gt; is the composition of continuous functions &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt;, which is again continuous. In other words, (ii) is essentially the same as (i). &lt;math&gt;\square&lt;/math&gt;.

'''2. 2 Theorem''' ''&lt;math&gt;\Omega&lt;/math&gt; has a countable basis consisting of open sets with compact closure.''&lt;br /&gt;
Proof: Suppose the collection of interior of closed balls &lt;math&gt;G_{\alpha}&lt;/math&gt; in &lt;math&gt;\Omega&lt;/math&gt; for a rational coordinate &lt;math&gt;\alpha&lt;/math&gt;. It is countable since &lt;math&gt;\mathbb{Q}&lt;/math&gt; is. It is also a basis of &lt;math&gt;\Omega&lt;/math&gt;; since if not, there exists an interior point that is isolated, and this is absurd.

'''2.5 Theorem''' ''In &lt;math&gt;\mathbb{R}^k&lt;/math&gt; there exists a set consisting of uncountably many components.''&lt;br /&gt;

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

'''2 Theorem (uncountability of the reals)''' ''The set of all real numbers is never a sequence.''&lt;br /&gt;
Proof: See [http://www.dpmms.cam.ac.uk/~twk/Anal.pdf].

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

'''2 Theorem (continuous extension)''' ''Let &lt;math&gt;f, g: \overline{E} \to \mathbb{C}&lt;/math&gt; be continuous. If &lt;math&gt;f = g&lt;/math&gt; on &lt;math&gt;E&lt;/math&gt;, then &lt;math&gt;f = g on \overline{E}&lt;/math&gt;.''&lt;br /&gt;
Proof: Let &lt;math&gt;u = f - g&lt;/math&gt;. Then clearly &lt;math&gt;u&lt;/math&gt; is continuous on &lt;math&gt;\overline{E}&lt;/math&gt;. Since &lt;math&gt;{0}&lt;/math&gt; is closed in &lt;math&gt;\mathbb{C}&lt;/math&gt;, &lt;math&gt;u^{-1}(0)&lt;/math&gt; is also closed. Hence, &lt;math&gt;u = 0&lt;/math&gt; on &lt;math&gt;\overline{E}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

, and for each &lt;math&gt;j&lt;/math&gt;, &lt;math&gt;B_j = \overline{ \{ x_k : k \ge j \} }&lt;/math&gt;. Since for any subindex &lt;math&gt;j(1), j(2), ..., j(n)&lt;/math&gt;, we have:
:&lt;math&gt;x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}&lt;/math&gt;.
That is to say the sequence &lt;math&gt;B_j&lt;/math&gt; has the finite intersection property. Since &lt;math&gt;E&lt;/math&gt; is coun

Since &lt;math&gt;E&lt;/math&gt; is first countable, &lt;math&gt;E&lt;/math&gt; is also sequentially countable. Hence, (a) &lt;math&gt;\to&lt;/math&gt; (b).

Suppose &lt;math&gt;E&lt;/math&gt; is not bounded, then there exists some &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; such that: for any finite set &lt;math&gt;\{ x_1, x_2, ... x_n \} \subset E&lt;/math&gt;,
:&lt;math&gt;E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)&lt;/math&gt;.
Let recursively &lt;math&gt;x_1 \in E&lt;/math&gt; and &lt;math&gt;x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)&lt;/math&gt;. Since &lt;math&gt;d(x_n, x_m) &gt; \epsilon&lt;/math&gt; for any n, m, &lt;math&gt;x_j&lt;/math&gt; is not Cauchy. Hence, (a) implies that &lt;math&gt;E&lt;/math&gt; is totally bounded. Also, 

[[Image:Hausdorff_space.svg|thumb|right|separated by neighborhoods]]
'''3 Theorem''' ''the following are equivalent:''
* (a) &lt;math&gt;\| x \| = 0&lt;/math&gt; implies that &lt;math&gt;x = 0&lt;/math&gt;.
* (b) Two points can be separated by disjoint open sets.
* (c) Every limit is unique.

'''3 Theorem''' ''The separation by neighborhoods implies that a compact set &lt;math&gt;K&lt;/math&gt; is closed in E.''&lt;br /&gt;
Proof [http://planetmath.org/encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html]: If &lt;math&gt;K = E&lt;/math&gt;, then &lt;math&gt;K&lt;/math&gt; is closed. If not, there exists a &lt;math&gt;x \in E \backslash K&lt;/math&gt;. For each &lt;math&gt;y \in K&lt;/math&gt;, the hypothesis says there exists two disjoint open sets &lt;math&gt;A(y)&lt;/math&gt; containing &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;B(y)&lt;/math&gt; containing &lt;math&gt;y&lt;/math&gt;. The collection &lt;math&gt;\{ B(y) : y \in E \}&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;. Since the compactness, there exists a finite subcover &lt;math&gt;\{ B(y_1), B(y_2), ... B(y_n) \}&lt;/math&gt;. Let &lt;math&gt;A(x)&lt;/math&gt; = &lt;math&gt;A(y_1) \cap A(y_2) ... A(y_n)&lt;/math&gt;. If &lt;math&gt;z \in K&lt;/math&gt;, then &lt;math&gt;z \in B(y_k)&lt;/math&gt; for some &lt;math&gt;k&lt;/math&gt;. Hence, &lt;math&gt;z \not\in A(y_k) \subset A&lt;/math&gt;. Hence, &lt;math&gt;A(x) \subset E \backslash K&lt;/math&gt;. Since the finite intersection of open sets is again open, &lt;math&gt;A(x)&lt;/math&gt; is open. Since the union of open sets is open,
:&lt;math&gt;E \backslash K = \bigcup_{x \not\in K} A(x)&lt;/math&gt; is open. &lt;math&gt;\square&lt;/math&gt;

=== Seminorm ===

[[Image:Vector_norms.png|frame|right|Illustrations of unit circles in different norms.]] Let &lt;math&gt;G&lt;/math&gt; be a linear space. The ''seminorm'' of an element in &lt;math&gt;G&lt;/math&gt; is a nonnegative number, denoted by &lt;math&gt;\| \cdot \|&lt;/math&gt;, such that: for any &lt;math&gt;x, y \in G&lt;/math&gt;,
# &lt;math&gt;\|x\| \ge 0&lt;/math&gt;.
# &lt;math&gt;\| \alpha x \| = |\alpha| \|x\|&lt;/math&gt;.
# &lt;math&gt;\|x + y\| \le \|x\| + \|y\|&lt;/math&gt;. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be &quot;normed linear space&quot;.

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a ''sequence space'' &lt;math&gt;l^p&lt;/math&gt;, the set of sequences &lt;math&gt;x_j&lt;/math&gt; such that
:For &lt;math&gt;1 \le p &lt; \infty&lt;/math&gt;, &lt;math&gt;\sum_1^{\infty} |x_j|^p &lt; \infty&lt;/math&gt;
:For &lt;math&gt;p = \infty&lt;/math&gt;, &lt;math&gt;x_j&lt;/math&gt; is a bounded sequence.

In particular, a subspace of &lt;math&gt;l^p&lt;/math&gt; is said to have ''dimension'' ''n'' if in every sequence the terms after ''n''th are all zero, and if &lt;math&gt;n &lt; \infty&lt;/math&gt;, then the subspace is said to be an ''Euclidean space''.

An ''open ball'' centered at &lt;math&gt;a&lt;/math&gt; of radius &lt;math&gt;r&lt;/math&gt; is the set &lt;math&gt;\{ z : \| z - a \| &lt; r \}&lt;/math&gt;. An ''open set'' is then the union of open balls.

Continuity and convergence has close and reveling connection.

'''3 Theorem''' ''Let &lt;math&gt;L&lt;/math&gt; be a seminormed space and &lt;math&gt;f: \Omega \to L&lt;/math&gt;. The following are equivalent:''
* (a) &lt;math&gt;f&lt;/math&gt; is continuous.
* (b) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. If &lt;math&gt;f(x) \in G&lt;/math&gt;, then there exists a &lt;math&gt;\omega \subset \Omega&lt;/math&gt; such that &lt;math&gt;x \in \omega&lt;/math&gt; and &lt;math&gt;f(\omega) \subset G&lt;/math&gt;.
* (c) Let &lt;math&gt;x \in \Omega&lt;/math&gt;. For each &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that
:&lt;math&gt;|f(y) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;y \in \Omega&lt;/math&gt; and &lt;math&gt;|y - x|&lt; \delta&lt;/math&gt;
* (e) Let 
Proof: Suppose (c). Since continuity, there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that:
:&lt;math&gt;|f(x_j) - f(x)| &lt; \epsilon&lt;/math&gt; whenever &lt;math&gt;|x_j - x| &lt; \delta&lt;/math&gt;

The following special case is often useful; in particular, showing the sequence's failure to converge.

'''2 Theorem''' ''Let &lt;math&gt;x_j \in \Omega&lt;/math&gt; be a sequence that converges to &lt;math&gt;x \in \Omega&lt;/math&gt;. Then a function &lt;math&gt;f&lt;/math&gt; is continuous at &lt;math&gt;x&lt;/math&gt; if and only if the sequence &lt;math&gt;f(x_j)&lt;/math&gt; converges to &lt;math&gt;f(x)&lt;/math&gt;.''&lt;br /&gt;
Proof: The function &lt;math&gt;x&lt;/math&gt; of &lt;math&gt;j&lt;/math&gt; is continuous on &lt;math&gt;\mathbb{N}&lt;/math&gt;. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. &lt;math&gt;\square&lt;/math&gt;.

'''2 Theorem''' ''Let &lt;math&gt;u_j&lt;/math&gt; be continuous and suppose &lt;math&gt;u_j&lt;/math&gt; converges uniformly to a function &lt;math&gt;u&lt;/math&gt; (i.e., the sequence &lt;math&gt;\|u_j - u\| \to 0&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Then &lt;math&gt;u&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: In short, the theorem holds since
:&lt;math&gt;\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)&lt;/math&gt;.
But more rigorously, let &lt;math&gt;\epsilon &gt; 0&lt;/math&gt;. Since the convergence is uniform, we find a &lt;math&gt;N&lt;/math&gt; so that
:&lt;math&gt;\|u_N(x) - u(x)\| \le \|u_N - u\| &lt; \epsilon / 3&lt;/math&gt; for any &lt;math&gt;x&lt;/math&gt;.
Also, since &lt;math&gt;u_N&lt;/math&gt; is continuous, we find a &lt;math&gt;\delta &gt; 0&lt;/math&gt; so that
:&lt;math&gt;\|u_N(y) - u_N(x)\| &lt; \epsilon / 3&lt;/math&gt; whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt;
It then follows that &lt;math&gt;u&lt;/math&gt; is continuous since:
:&lt;math&gt;\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| &lt; \epsilon&lt;/math&gt;
whenever &lt;math&gt;|y - x| &lt; \delta&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''The set of all continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt; is complete if and only if &lt;math&gt;B&lt;/math&gt; is complete.''&lt;br /&gt;
Proof: Let &lt;math&gt;u_j&lt;/math&gt; be a Cauchy sequence of continuous linear operators from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;B&lt;/math&gt;. That is, if &lt;math&gt;x \in A&lt;/math&gt; and &lt;math&gt;\|x\| = 1&lt;/math&gt;, then
:&lt;math&gt;\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0&lt;/math&gt; as &lt;math&gt;n, m \to \infty&lt;/math&gt;
Thus, &lt;math&gt;u_j(x)&lt;/math&gt; is a Cauchy sequence in &lt;math&gt;B&lt;/math&gt;. Since B is complete, let
:&lt;math&gt;u(x) = \lim_{j \to \infty} u_j(x)&lt;/math&gt; for each &lt;math&gt;x \in A&lt;/math&gt;.
Then &lt;math&gt;u&lt;/math&gt; is linear since the limit operator is and also &lt;math&gt;u&lt;/math&gt; is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) &lt;math&gt;\square&lt;/math&gt;

== Metric spaces ==

We say a topological space is ''metric'' or ''metrizable'' if every open set in it is the union of open ''balls''; that is, the set of the form &lt;math&gt;\{ x; d(x, y) &lt; r \}&lt;/math&gt; with radius &lt;math&gt;r&lt;/math&gt; and center &lt;math&gt;y&lt;/math&gt; where &lt;math&gt;d&lt;/math&gt;, called ''metric'', is a real-valued function satisfying the axioms: for all &lt;math&gt;x, y, z&lt;/math&gt;
# &lt;math&gt;d(x, y) = d(y, x)&lt;/math&gt;
# &lt;math&gt;d(x, y) + d(y, z) \ge d(x, z)&lt;/math&gt;
# &lt;math&gt;d(x, y) = 0&lt;/math&gt; if and only if &lt;math&gt;x = y&lt;/math&gt;. ([[w:Identity of indiscernibles|Identity of indiscernibles]])

It follows immediately that &lt;math&gt;|d(x, y) - d(z, y)| \le d(x, y)&lt;/math&gt; for every &lt;math&gt;x, y, z&lt;/math&gt;. In particular, &lt;math&gt;d&lt;/math&gt; is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

'''2 Theorem''' ''Let &lt;math&gt;K, d)&lt;/math&gt; be a compact metric space. If &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt;, then there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that for any &lt;math&gt;E \subset K&lt;/math&gt; with &lt;math&gt;\sup_{x, y \in E} d(x, y) &lt; \delta&lt;/math&gt; &lt;math&gt;E \subset &lt;/math&gt; some member of &lt;math&gt;\Gamma&lt;/math&gt;''&lt;br /&gt;
Proof: Let &lt;math&gt;x, y&lt;/math&gt; be in &lt;math&gt;K&lt;/math&gt;, and suppose &lt;math&gt;x&lt;/math&gt; is in some member &lt;math&gt;S&lt;/math&gt; of &lt;math&gt;\Gamma&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; is in the complement of &lt;math&gt;S&lt;/math&gt;. If we define a function &lt;math&gt;\delta&lt;/math&gt; by for every &lt;math&gt;x&lt;/math&gt;
:&lt;math&gt;\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}&lt;/math&gt;,
then
:&lt;math&gt;\delta(x) \le d(x, y)&lt;/math&gt;
The theorem thus follows if we show the inf of &lt;math&gt;\delta&lt;/math&gt; over &lt;math&gt;K&lt;/math&gt; is positive. &lt;math&gt;\square&lt;/math&gt;

'''2 Lemma''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then every open cover of &lt;math&gt;K&lt;/math&gt; admits a countable subcover if and only if &lt;math&gt;K&lt;/math&gt; is separable.'' &lt;br /&gt;
Proof: To show the direct part, fix &lt;math&gt;n = 1, 2, ...&lt;/math&gt; and let &lt;math&gt;\Gamma&lt;/math&gt; be the collection of all open balls of radius &lt;math&gt;{1 \over n}&lt;/math&gt;. Then &lt;math&gt;\Gamma&lt;/math&gt; is an open cover of &lt;math&gt;K&lt;/math&gt; and admits a countable subcover &lt;math&gt;\gamma&lt;/math&gt;. Let &lt;math&gt;E_n&lt;/math&gt; be the centers of the members of &lt;math&gt;\gamma&lt;/math&gt;. Then &lt;math&gt;\cup_1^\infty E_n&lt;/math&gt; is a countable dense subset. Conversely, let &lt;math&gt;\Gamma&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; be a countable dense subset of &lt;math&gt;K&lt;/math&gt;. Since open balls of radii &lt;math&gt;1, {1 \over 2}, {1 \over 3}, ...&lt;/math&gt;, the centers lying in &lt;math&gt;E&lt;/math&gt;, form a countable base for &lt;math&gt;K&lt;/math&gt;, each member of &lt;math&gt;\Gamma&lt;/math&gt; is the union of some subsets of countably many open sets &lt;math&gt;B_1, B_2, ...&lt;/math&gt;. Since we may suppose that for each &lt;math&gt;n&lt;/math&gt; &lt;math&gt;B_n&lt;/math&gt; is contained in some &lt;math&gt;G_n \in \Gamma&lt;/math&gt; (if not remove it from the sequence) the sequence &lt;math&gt;G_1, ...&lt;/math&gt; is a countable subcover of &lt;math&gt;\Gamma&lt;/math&gt; of &lt;math&gt;K&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Remark: For more of this with relation to cardinality see [http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0570.0001].

'''2 Theorem''' ''Let &lt;math&gt;K&lt;/math&gt; be a metric space. Then the following are equivalent:''
* ''(i) &lt;math&gt;K&lt;/math&gt; is compact.''
* ''(ii) Every sequence of &lt;math&gt;K&lt;/math&gt; admits a convergent subsequence.'' (sequentially compact)
* ''(iii) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover.'' (countably compact)
Proof (from [http://www.mathreference.com/top-cs,count.html]): Throughout the proof we may suppose that &lt;math&gt;K&lt;/math&gt; is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) &lt;math&gt;\Rightarrow&lt;/math&gt; (ii). Supposing that (iii) is false, let &lt;math&gt;G_1, G_2, ...&lt;/math&gt; be an open cover of &lt;math&gt;K&lt;/math&gt; such that for each &lt;math&gt;n = 1, 2, ...&lt;/math&gt; we can find a point &lt;math&gt;x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c&lt;/math&gt;. It follows that &lt;math&gt;x_n&lt;/math&gt; does not have a convergent subsequence, proving (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii). Indeed, suppose it does. Then the sequence &lt;math&gt;x_n&lt;/math&gt; has a limit point &lt;math&gt;p&lt;/math&gt;. Thus, &lt;math&gt;p \in \cap_1^\infty {G_j}^c&lt;/math&gt;, a contradiction. To show (iii) &lt;math&gt;\Rightarrow&lt;/math&gt; (i), in view of the preceding lemma, it suffices to show that &lt;math&gt;K&lt;/math&gt; is separable. But this follows since if &lt;math&gt;K&lt;/math&gt; is not separable, we can find a countable discrete subset of &lt;math&gt;K&lt;/math&gt;, violating (iii). &lt;math&gt;\square&lt;/math&gt;

Remark: The proof for (ii) &lt;math&gt;\Rightarrow&lt;/math&gt; (iii) does not use the fact that the space is metric.

'''2 Theorem''' ''A subset &lt;math&gt;E&lt;/math&gt; of a metric space is precompact (i.e., its completion is compact) if and only if there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt; such that &lt;math&gt;E&lt;/math&gt; is contained in the union of finitely many open balls of radius &lt;math&gt;\delta&lt;/math&gt; with the centers lying in E.''&lt;br /&gt;

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

'''2 Theorem (Heine-Borel)''' ''If &lt;math&gt;E&lt;/math&gt; is a subset of &lt;math&gt;\mathbb{R}^n&lt;/math&gt;, then the following are equivalent.''
* ''(i) &lt;math&gt;E&lt;/math&gt; is closed and bounded.''
* ''(ii) &lt;math&gt;E&lt;/math&gt; is compact.''
* ''(iii) Every infinite subset of &lt;math&gt;E&lt;/math&gt; contains some of its limit points.''

'''2 Theorem''' ''Every metric space is perfectly and fully normal.''&lt;br /&gt;

'''2 Corollary''' ''Every metric space is normal and Hausdorff.''

'''2 Theorem''' ''If &lt;math&gt;x_j \to x&lt;/math&gt; in a metric space &lt;math&gt;X&lt;/math&gt; implies &lt;math&gt;f(x_n) \to f(x)&lt;/math&gt;, then &lt;math&gt;f&lt;/math&gt; is continuous.''&lt;br /&gt;
Proof: Let &lt;math&gt;E \subset X&lt;/math&gt; and &lt;math&gt;x \in &lt;/math&gt; the closure of &lt;math&gt;E&lt;/math&gt;. Then there exists a &lt;math&gt;x_j \in E \to x&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt; as &lt;math&gt;j \to \infty&lt;/math&gt;. Thus, by assumption &lt;math&gt;f(x_j) \to f(x)&lt;/math&gt;, and this means that &lt;math&gt;f(x)&lt;/math&gt; is in the closure of &lt;math&gt;f(E)&lt;/math&gt;. We conclude: &lt;math&gt;f(\overline{E}) \subset \overline{f(E)}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''2 Theorem''' ''Suppose that &lt;math&gt;f: (X_1, d_1) \to (X_2, d_2)&lt;/math&gt; is continuous and satisfies
:&lt;math&gt;d_2(f(x), f(y)) \ge d_1(x, y)&lt;/math&gt; for all &lt;math&gt;x, y&lt;/math&gt;
If &lt;math&gt;F \subset X_2&lt;/math&gt; and &lt;math&gt;(F, d_2)&lt;/math&gt; is a complete metric space, then &lt;math&gt;f(F)&lt;/math&gt; is closed.&lt;br /&gt;
Proof: Let &lt;math&gt;x_n&lt;/math&gt; be a sequence in &lt;math&gt;F&lt;/math&gt; such that &lt;math&gt;\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0&lt;/math&gt;. Then by hypothesis &lt;math&gt;\lim_{n, m \to \infty} d_1(x_n, x_m) = 0&lt;/math&gt;. Since &lt;math&gt;F&lt;/math&gt; is complete in &lt;math&gt;d_2&lt;/math&gt;, the limit &lt;math&gt;y = \lim_{n \to \infty} x_n&lt;/math&gt; exists. Finally, since &lt;math&gt;f&lt;/math&gt; is continuous, &lt;math&gt;\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0&lt;/math&gt;, completing the proof. &lt;math&gt;\square&lt;/math&gt;

== Measure ==

[[Image:Cantor.png|200px|right|Cantor set]]

A ''measure'', which we shall denote by &lt;math&gt;\mu&lt;/math&gt; throughout this section, is a function from a delta-ring &lt;math&gt;\mathcal{F}&lt;/math&gt; to the set of nonnegative real numbers and &lt;math&gt;\infty&lt;/math&gt; such that &lt;math&gt;\mu&lt;/math&gt; is countably additive; i.e., &lt;math&gt;\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)&lt;/math&gt; for &lt;math&gt;E_j&lt;/math&gt; distinct. We shall define an integral in terms of a measure.

'''4.1 Theorem''' ''A measure &lt;math&gt;\mu&lt;/math&gt; has the following properties: given measurable sets &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; such that &lt;math&gt;A \subset B&lt;/math&gt;,''
* (1) &lt;math&gt;\mu (B \backslash A) = \mu (B) - \mu (A)&lt;/math&gt;.
* (2) &lt;math&gt;\mu (\varnothing) = 0&lt;/math&gt;.
* (3) &lt;math&gt;\mu (A) \le \mu (B)&lt;/math&gt; where the equality holds for &lt;math&gt;A = B&lt;/math&gt;. (monotonicity)
Proof: (1) &lt;math&gt;\mu (B) - \mu (A) = \mu (B \backslash A \cup  A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)&lt;/math&gt;. (2) &lt;math&gt;\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0&lt;/math&gt; since measures are always nonnegative. Also, &lt;math&gt;\mu (B) - \mu (A) = 0&lt;/math&gt; if &lt;math&gt;A = B&lt;/math&gt; since (2).

One example would be a ''counting measure''; i.e., &lt;math&gt;\mu E&lt;/math&gt; = the number of elements in a finite set &lt;math&gt;E&lt;/math&gt;. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let &lt;math&gt;x_0 \in \mathcal{F}&lt;/math&gt; be fixed, and &lt;math&gt;\mu(E) = 1&lt;/math&gt; if &lt;math&gt;x_0&lt;/math&gt; is in &lt;math&gt;E&lt;/math&gt; and 0 otherwise. The measure &lt;math&gt;\mu&lt;/math&gt; is indeed countably additive since for the sequence &lt;math&gt;{E_j}&lt;/math&gt; arbitrary, &lt;math&gt;\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)&lt;/math&gt; for some &lt;math&gt;j&lt;/math&gt; if &lt;math&gt;x_0 \in \bigcup E_j&lt;/math&gt; and 0 otherwise.

We now study the notion of geometric convexity.

'''2 Lemma'''
:''&lt;math&gt;{x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)&lt;/math&gt; for any &lt;math&gt;x, y&lt;/math&gt;''.

'''2 Theorem''' ''Let &lt;math&gt;D&lt;/math&gt; be a convex subset of a vector space. If &lt;math&gt;0 &lt; a &lt; 1&lt;/math&gt; and &lt;math&gt;f(ax + (1-a)y) \le af(x) + (1-a)f(y)&lt;/math&gt; for &lt;math&gt;x, y \in D&lt;/math&gt;, then
:&lt;math&gt;f({x+y \over 2}) \le {f(x) + f(y) \over 2}&lt;/math&gt;  for &lt;math&gt;x, y \in D&lt;/math&gt;
Proof: From the lemma we have:
:&lt;math&gt;2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\mathcal{F}&lt;/math&gt; be a collection of subsets of a set &lt;math&gt;G&lt;/math&gt;. We say &lt;math&gt;\mathcal{F}&lt;/math&gt; is a ''delta-ring'' if &lt;math&gt;\mathcal{F}&lt;/math&gt; has the empty set, G and every countable union and countable intersection of members of &lt;math&gt;\mathcal{F}&lt;/math&gt;. a member of &lt;math&gt;\mathcal{F}&lt;/math&gt; is called a ''measurable set'' and G a ''measure space'', by analogy to a topological space and open sets in it.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then
:&lt;math&gt;\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}&lt;/math&gt;
Proof: By replacing &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; with &lt;math&gt;|f|&lt;/math&gt; and &lt;math&gt;|g|&lt;/math&gt;, respectively, we may assume that &lt;math&gt;f&lt;/math&gt; and &lt;math&gt;g&lt;/math&gt; are non-negative. Let &lt;math&gt;C = \int g^q d\mu&lt;/math&gt;. If &lt;math&gt;C = 0&lt;/math&gt;, then the inequality is obvious. Suppose not, and let &lt;math&gt;d\nu = {g^q d\mu \over C}&lt;/math&gt;. Then, since &lt;math&gt;x^p&lt;/math&gt; is increasing and convex for &lt;math&gt;x \ge 0&lt;/math&gt;, by Jensen's inequality,
:&lt;math&gt;\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C&lt;/math&gt;.
Since &lt;math&gt;1/q = 1 - 1/p&lt;/math&gt;, this is the desired inequality. &lt;math&gt;\square&lt;/math&gt;

'''4 Corollary (Minkowski's inequality)''' ''Let &lt;math&gt;p \ge 1&lt;/math&gt; and &lt;math&gt;\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}&lt;/math&gt;. If &lt;math&gt;f \ge 0&lt;/math&gt; and &lt;math&gt;\int \int f(x, y) dx dy &lt; \infty&lt;/math&gt;, then
:&lt;math&gt;\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy&lt;/math&gt;
Proof (from [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=2987]): If &lt;math&gt;p = 1&lt;/math&gt;, then the inequality is the same as Fubini's theorem. If &lt;math&gt;p &gt; 1&lt;/math&gt;,
:{|
|-
|&lt;math&gt;\int |\int f(x, y)dy|^p dx &lt;/math&gt;
|&lt;math&gt;= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Fubini)&lt;/small&gt;
|&lt;math&gt;= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz&lt;/math&gt;
|-
|style=&quot;text-align:right;&quot;|&lt;small&gt;(Hölder)&lt;/small&gt;
|&lt;math&gt;\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz&lt;/math&gt;
|}
By division we get the desired inequality, noting that &lt;math&gt;(p - 1)q = p&lt;/math&gt; and &lt;math&gt;1 - {1 \over q} = {1 \over p}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing &lt;math&gt;\int dy&lt;/math&gt; by a counting measure we get:
:&lt;math&gt;\|f+g\|_p \le \|f\|_p + \|g\|_p&lt;/math&gt;
under the same assumption and notation.

'''2 Lemma''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt; and &lt;math&gt;a, b &gt; 0&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;,
:&lt;math&gt;a^{1 \over p}b^{1 \over q} = \inf_{t &gt; 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)&lt;/math&gt;
Proof: Let &lt;math&gt;f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b&lt;/math&gt; for &lt;math&gt;t &gt; 0&lt;/math&gt;. Then the derivative
:&lt;math&gt;f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)&lt;/math&gt;
becomes zero only when &lt;math&gt;t = a/b&lt;/math&gt;. Thus, the minimum of &lt;math&gt;f&lt;/math&gt; is attained there and is equal to &lt;math&gt;a^{1 \over p}b^{1 \over q}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

When &lt;math&gt;t&lt;/math&gt; is taken to 1, the inequality is known as Young's inequality.

'''2 Theorem (Hölder's inequality)''' ''For &lt;math&gt;1 \le p \le \infty&lt;/math&gt;, if &lt;math&gt;1/p + 1/q = 1&lt;/math&gt;, then''
:&lt;math&gt;\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}&lt;/math&gt;
Proof: If &lt;math&gt;p = \infty&lt;/math&gt;, the inequality is clear. By the application of the preceding lemma we have: for any &lt;math&gt;t &gt; 0&lt;/math&gt;
:&lt;math&gt;\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|&lt;/math&gt;
Taking the infimum we see that the right-hand side becomes:
:&lt;math&gt;\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}&lt;/math&gt; &lt;math&gt;\square&lt;/math&gt;

The ''convex hull'' in &lt;math&gt;\Omega&lt;/math&gt; of a compact set ''K'', denoted by &lt;math&gt;\hat K&lt;/math&gt; is:
:&lt;math&gt;\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}&lt;/math&gt;.
When &lt;math&gt;f&lt;/math&gt; is linear (besides being analytic), we say the &lt;math&gt;\hat K&lt;/math&gt; is ''geometrically convex hull''. That &lt;math&gt;K&lt;/math&gt; is compact ensures that the definition is meaningful.

'''Theorem''' ''The closure of the convex hull of &lt;math&gt;E&lt;/math&gt; in &lt;math&gt;G&lt;/math&gt; is the intersection if all half-spaces containing &lt;math&gt;E&lt;/math&gt;.''&lt;br /&gt;
Proof: Let F = the collection of half-spaces containing &lt;math&gt;E&lt;/math&gt;. Then &lt;math&gt;\overline{co}(E) \subset \cap F&lt;/math&gt; since each-half space in &lt;math&gt;F&lt;/math&gt; is closed and convex. Yet, if &lt;math&gt;x \not\in \overline{co}(E)&lt;/math&gt;, then there exists a half-space &lt;math&gt;H&lt;/math&gt; containing &lt;math&gt;E&lt;/math&gt; which however does not contain x. Hence, &lt;math&gt;\overline{co}(E) = \cap F&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

Let &lt;math&gt;\Omega \subset \mathbb{R}^n&lt;/math&gt;. A function &lt;math&gt;f:\Omega \to \mathbb{R}^n&lt;/math&gt; is said to be ''convex'' if
:&lt;math&gt;\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |&lt;/math&gt;.
for some &lt;math&gt;K \subset \Omega&lt;/math&gt; compact and any &lt;math&gt;h&lt;/math&gt; harmonic on &lt;math&gt;K&lt;/math&gt; and continuous on &lt;math&gt;\overline{K}&lt;/math&gt;.

'''Theorem''' ''The following are equivalent'':
*(a) &lt;math&gt;f&lt;/math&gt; is convex on some &lt;math&gt;K \subset \mathbb{R}&lt;/math&gt; compact.
*(b) if &lt;math&gt;[a, b] \subset K&lt;/math&gt;, then
*:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt; for &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;.
*(c) The difference quotient
*:&lt;math&gt;{f(x + h) - f(x) \over h}&lt;/math&gt; increases as &lt;math&gt;h&lt;/math&gt; does.
*(d) &lt;math&gt;f&lt;/math&gt; is measurable and we have:
*:&lt;math&gt;f \left( {a + b \over 2} \right) \le {a + b \over 2}&lt;/math&gt; for any &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
*(e) The set &lt;math&gt;\{ (x, y) : x \in [a, b], f(x) \le y \}&lt;/math&gt; is convex for &lt;math&gt;[a, b] \subset K&lt;/math&gt;.
Proof: Suppose (a). For each &lt;math&gt;x \in [a, b]&lt;/math&gt;, there exists some &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt; such that &lt;math&gt;x = \lambda a + (1 - \lambda) b&lt;/math&gt;. Let &lt;math&gt;A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;, and then since &lt;math&gt;\mbox{b}[a, b] = \{a, b\}&lt;/math&gt; and &lt;math&gt;(f - A)(a) = 0 = (f - A)(b)&lt;/math&gt;,
:{|
|-
|&lt;math&gt;|f(x)| = |f(x) - A(x) + A(x)|&lt;/math&gt;
|&lt;math&gt;\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)&lt;/math&gt;
|-
|
|&lt;math&gt;=\lambda f(a) + (1-\lambda) f(b)&lt;/math&gt;
|}
Thus, (a) &lt;math&gt;\Rightarrow&lt;/math&gt; (b). Now suppose (b). Since &lt;math&gt;\lambda = {x - a \over b - a}&lt;/math&gt;, for &lt;math&gt;\lambda \in (0, 1)&lt;/math&gt;, (b) says:
:{|
|-
|&lt;math&gt;(b - x)f(x) + (x - a)f(x)&lt;/math&gt;
|&lt;math&gt;\le (b -x)f(a) + (x - a)f(b)&lt;/math&gt;
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|}
Since &lt;math&gt;a - x &lt; b - x&lt;/math&gt;, we conclude (b) &lt;math&gt;\Rightarrow&lt;/math&gt; (c). Suppose (c). The continuity follows since we have:
:&lt;math&gt;\lim_{h &gt; 0, h \to 0}f(x + h) = f(x) + \lim_{h &gt; 0, h \to 0}h{f(x + h) - f(x) \over h}&lt;/math&gt;.
Also, let &lt;math&gt;x = 2^{-1}(a + b)&lt;/math&gt; such that &lt;math&gt;a &lt; b&lt;/math&gt;, for &lt;math&gt;a, b \in K&lt;/math&gt;. Then we have:
:{|
|-
|&lt;math&gt;{f(a) - f(x) \over a - x}&lt;/math&gt;
|&lt;math&gt;\le {f(b) - f(x) \over b - x}&lt;/math&gt;
|-
|&lt;math&gt;f(x) - f(a)&lt;/math&gt;
|&lt;math&gt;\le f(b) - f(x)&lt;/math&gt;
|-
|&lt;math&gt;f \left( {a + b \over 2} \right)&lt;/math&gt;
|&lt;math&gt;\le {f(a) + f(b) \over 2}&lt;/math&gt;
|}
Thus, (c) &lt;math&gt;\Rightarrow&lt;/math&gt; (d). Now suppose (d), and let &lt;math&gt;E = \{ (x, y) : x \in [a, b], y \ge f(x) \}&lt;/math&gt;. First we want to show
:&lt;math&gt;f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)&lt;/math&gt;.
If &lt;math&gt;n = 0&lt;/math&gt;, then the inequality holds trivially.
if the inequality holds for some &lt;math&gt;n - 1&lt;/math&gt;, then
:{|
|-
|&lt;math&gt;f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)&lt;/math&gt;
|&lt;math&gt;=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)&lt;/math&gt;
|-
|
|&lt;math&gt;\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) &lt;/math&gt;
|-
|
|&lt;math&gt;= {1 \over 2^n} \sum_1^{2^n}f(x_j)&lt;/math&gt;
|}

Let &lt;math&gt;x_1, x_2 \in [a, b]&lt;/math&gt; and &lt;math&gt;\lambda \in [0, 1]&lt;/math&gt;. There exists a sequence of rationals number such that:
:&lt;math&gt;\lim_{j \to \infty} {p_j \over 2q_j} = \lambda&lt;/math&gt;.
It then follows that:
:{|
|-
|&lt;math&gt;f(\lambda x_1 + (1 - \lambda) x_2)&lt;/math&gt;
|&lt;math&gt;\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;= \lambda f(x_1) + (1 - \lambda) f(x_2)&lt;/math&gt;
|-
|
|&lt;math&gt;\le \lambda y_1 + (1 - \lambda) y_2&lt;/math&gt;.
|}
Thus, (d) &lt;math&gt;\Rightarrow&lt;/math&gt; (e). Finally, suppose (e); that is, &lt;math&gt;E&lt;/math&gt; is convex. Also suppose &lt;math&gt;K&lt;/math&gt; is an interval for a moment. Then
:&lt;math&gt;f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

'''4. Corollary (inequality between geometric and arithmetic means)'''
:''&lt;math&gt;\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j&lt;/math&gt; if &lt;math&gt;a_j \ge 0&lt;/math&gt;, &lt;math&gt;\lambda_j \ge 0&lt;/math&gt; and &lt;math&gt;\sum_1^n \lambda_j = 1&lt;/math&gt;.''
Proof: If some &lt;math&gt;a_j = 0&lt;/math&gt;, then the inequality holds trivially; so, suppose &lt;math&gt;a_j &gt; 0&lt;/math&gt;. The function &lt;math&gt;e^x&lt;/math&gt; is convex since its second derivative, again &lt;math&gt;e^x&lt;/math&gt;, is &lt;math&gt;&gt; 0&lt;/math&gt;. It thus follows:
:&lt;math&gt;\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt;

The convex hull of a finite set &lt;math&gt;E&lt;/math&gt; is said to be a ''convex polyhedron''. Clearly, the set of the extremely points is the subset of &lt;math&gt;E&lt;/math&gt;.

'''4 Theorem (general Hölder's inequality)''' ''If &lt;math&gt;a_{jk} &gt; 0&lt;/math&gt; and &lt;math&gt;p_j &gt; 1&lt;/math&gt; for &lt;math&gt;j = 1, ... n_j&lt;/math&gt; and &lt;math&gt;k = 1, ... n_k&lt;/math&gt; and &lt;math&gt;\sum_1^{n_j} p_j = 1&lt;/math&gt;, then:''
:&lt;math&gt;\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}&lt;/math&gt; (Hörmander 11)
Proof: Let &lt;math&gt;A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}&lt;/math&gt;.

'''4. Theorem''' ''A convex polyhedron is the intersection of a finite number of closed half-spaces.''&lt;br /&gt;
Proof: Use induction. &lt;math&gt;\square&lt;/math&gt;

'''4. Theorem''' ''The convex hull of a compact set is compact.''&lt;br /&gt;
Proof: Let &lt;math&gt;f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j&lt;/math&gt;. Then &lt;math&gt;f&lt;/math&gt; is continuous since it is the finite sum of continuous functions &lt;math&gt;\lambda_j x_j&lt;/math&gt;. Since the intersection of compact sets is compact and 
:&lt;math&gt;\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)&lt;/math&gt;,
&lt;math&gt;\mbox{co}(K)&lt;/math&gt; is compact. &lt;math&gt;\square&lt;/math&gt;

Example: Let &lt;math&gt;p(z) = (z - s_1)(z - s_2) ... (z - s_n)&lt;/math&gt;. Then the derivative of &lt;math&gt;p&lt;/math&gt; has zeros in &lt;math&gt;\mbox{co}(\{s_1, s_2, ... s_n\})&lt;/math&gt;.

== Addendum ==

# Show the following are equivalent with assuming that &lt;math&gt;K&lt;/math&gt; is second countable.
#* (1) &lt;math&gt;K&lt;/math&gt; is compact.
#* (2) Every countable open cover of &lt;math&gt;K&lt;/math&gt; admits a finite subcover (countably compact)
#* (3) &lt;math&gt;K&lt;/math&gt; is sequentially compact.

Nets are, so to speak, generalized sequences.

# Show the following are equivalent with assuming Axiom of Choice:
#* (1) &lt;math&gt;K&lt;/math&gt; is compact; i.e, every open cover admits a finite subcover.
#* (2) In &lt;math&gt;K&lt;/math&gt; every net has a convergent subnet.
#* (3) In &lt;math&gt;K&lt;/math&gt; every ultrafilter has an accumulation point (Bourbaki compact)
#* (4) There is a subbase &lt;math&gt;\tau&lt;/math&gt; for &lt;math&gt;K&lt;/math&gt; such that every open cover that is a subcollection of &lt;math&gt;\tau&lt;/math&gt; admits a finite subcover. (subbase compact)
#* (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
#* (6) Every infinite subset of &lt;math&gt;K&lt;/math&gt; has a complete accumulation point (Alexandroff-Urysohn compact)

{{Subject|An Introduction to Analysis}}</text>
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  </page>
  <page>
    <title>Wikibooks:Reading room/Technical Assistance</title>
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      <text bytes="4621" sha1="q0uuuqb3i58zea92xb7hrtmq2qhwb19" xml:space="preserve">__NEWSECTIONLINK__ {{Discussion Rooms}} {{Shortcut|WB:TECH}} {{TOC left}}
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== Templates: Refbegin and Refend. ==

These DO NOT seem to work as the documentation indicates throwing lints when used with the form indicated. 

These templates need to be behave consistently, or thrown out and re-written entirely.  

It would be NICE to have ONE set of templates and documentation to accompany them, currently this does not apparently exist, with the relevant templates getting confused between definition lists &lt;nowiki&gt;:&lt;/nowiki&gt; and standard lists {{*}}. 

CONSISTENT behaviour would also be nice.

[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:39, 19 March 2026 (UTC)

:@[[User:ShakespeareFan00|ShakespeareFan00]] I temporarily reduced the protection down for these templates (semi-protection), so you can do the necessary changes. :) [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:38, 21 March 2026 (UTC)

: Thats the thing I'm not sure what the 'stable' repair is, without breaking other stuff :( [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:44, 22 March 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:09, 3 April 2026 (UTC)
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== plainlist and unbulleted list ==

Hi,&lt;br&gt;
The plainlist and unbulleted list templates don't work in a sandbox. [[User:PeterEasthope/sandbox|PeterEasthope/sandbox]] for example. What is the simplest soluttion?&lt;br&gt;
Thanks, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 17:04, 18 April 2026 (UTC)</text>
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      <comment>/* plainlist and unbulleted list */Fixed typo.</comment>
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      <model>wikitext</model>
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      <text bytes="4620" sha1="e4fynd6il8a2euc5uvqashfjca505sa" xml:space="preserve">__NEWSECTIONLINK__ {{Discussion Rooms}} {{Shortcut|WB:TECH}} {{TOC left}}
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== Templates: Refbegin and Refend. ==

These DO NOT seem to work as the documentation indicates throwing lints when used with the form indicated. 

These templates need to be behave consistently, or thrown out and re-written entirely.  

It would be NICE to have ONE set of templates and documentation to accompany them, currently this does not apparently exist, with the relevant templates getting confused between definition lists &lt;nowiki&gt;:&lt;/nowiki&gt; and standard lists {{*}}. 

CONSISTENT behaviour would also be nice.

[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:39, 19 March 2026 (UTC)

:@[[User:ShakespeareFan00|ShakespeareFan00]] I temporarily reduced the protection down for these templates (semi-protection), so you can do the necessary changes. :) [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:38, 21 March 2026 (UTC)

: Thats the thing I'm not sure what the 'stable' repair is, without breaking other stuff :( [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:44, 22 March 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:09, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Technical_Village_Pumps_distribution_list&amp;oldid=29905755 --&gt;

== plainlist and unbulleted list ==

Hi,&lt;br&gt;
The plainlist and unbulleted list templates don't work in a sandbox. [[User:PeterEasthope/sandbox|PeterEasthope/sandbox]] for example. What is the simplest solution?&lt;br&gt;
Thanks, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 17:04, 18 April 2026 (UTC)</text>
      <sha1>e4fynd6il8a2euc5uvqashfjca505sa</sha1>
    </revision>
    <revision>
      <id>4631266</id>
      <parentid>4631264</parentid>
      <timestamp>2026-04-18T18:47:42Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>3441010</id>
      </contributor>
      <comment>/* plainlist and unbulleted list */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>4631266</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4965" sha1="jbal4bj7uyzaepz0lh545txo0d2yhto" xml:space="preserve">__NEWSECTIONLINK__ {{Discussion Rooms}} {{Shortcut|WB:TECH}} {{TOC left}}
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Welcome to the '''Technical Assistance reading room'''. Get assistance on questions related to [[w:MediaWiki|MediaWiki]] markup, CSS, JavaScript, and such as they relate to Wikibooks. '''This is not a general-purpose technical support room'''.

To submit a ''bug notice or feature request'' for the MediaWiki software, visit [[phabricator:|Phabricator]].

To get more information about the ''MediaWiki software'', or to download your own copy, visit [[mw:|MediaWiki]]

There are also two IRC channels for technical help: {{Channel|mediawiki}} for issues about the software, and {{channel|mediawiki-core}} for [[m:WMF|WMF]] server or configuration issues.
{{clear}}
[[Category:Reading room]]

== Templates: Refbegin and Refend. ==

These DO NOT seem to work as the documentation indicates throwing lints when used with the form indicated. 

These templates need to be behave consistently, or thrown out and re-written entirely.  

It would be NICE to have ONE set of templates and documentation to accompany them, currently this does not apparently exist, with the relevant templates getting confused between definition lists &lt;nowiki&gt;:&lt;/nowiki&gt; and standard lists {{*}}. 

CONSISTENT behaviour would also be nice.

[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:39, 19 March 2026 (UTC)

:@[[User:ShakespeareFan00|ShakespeareFan00]] I temporarily reduced the protection down for these templates (semi-protection), so you can do the necessary changes. :) [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:38, 21 March 2026 (UTC)

: Thats the thing I'm not sure what the 'stable' repair is, without breaking other stuff :( [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:44, 22 March 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:09, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Technical_Village_Pumps_distribution_list&amp;oldid=29905755 --&gt;

== plainlist and unbulleted list ==

Hi,&lt;br&gt;
The plainlist and unbulleted list templates don't work in a sandbox. [[User:PeterEasthope/sandbox|PeterEasthope/sandbox]] for example. What is the simplest solution?&lt;br&gt;
Thanks, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 17:04, 18 April 2026 (UTC)
: The unbulleted list relies on a module, and the plainlist relies on a TemplateStyles CSS page. What were you trying to test on your sandbox? [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:47, 18 April 2026 (UTC)</text>
      <sha1>jbal4bj7uyzaepz0lh545txo0d2yhto</sha1>
    </revision>
    <revision>
      <id>4631271</id>
      <parentid>4631266</parentid>
      <timestamp>2026-04-18T21:29:26Z</timestamp>
      <contributor>
        <username>PeterEasthope</username>
        <id>660399</id>
      </contributor>
      <comment>/* plainlist and unbulleted list */ Reply</comment>
      <origin>4631271</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5325" sha1="kvddnol370558s44ft66m2w892k46q5" xml:space="preserve">__NEWSECTIONLINK__ {{Discussion Rooms}} {{Shortcut|WB:TECH}} {{TOC left}}
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Welcome to the '''Technical Assistance reading room'''. Get assistance on questions related to [[w:MediaWiki|MediaWiki]] markup, CSS, JavaScript, and such as they relate to Wikibooks. '''This is not a general-purpose technical support room'''.

To submit a ''bug notice or feature request'' for the MediaWiki software, visit [[phabricator:|Phabricator]].

To get more information about the ''MediaWiki software'', or to download your own copy, visit [[mw:|MediaWiki]]

There are also two IRC channels for technical help: {{Channel|mediawiki}} for issues about the software, and {{channel|mediawiki-core}} for [[m:WMF|WMF]] server or configuration issues.
{{clear}}
[[Category:Reading room]]

== Templates: Refbegin and Refend. ==

These DO NOT seem to work as the documentation indicates throwing lints when used with the form indicated. 

These templates need to be behave consistently, or thrown out and re-written entirely.  

It would be NICE to have ONE set of templates and documentation to accompany them, currently this does not apparently exist, with the relevant templates getting confused between definition lists &lt;nowiki&gt;:&lt;/nowiki&gt; and standard lists {{*}}. 

CONSISTENT behaviour would also be nice.

[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:39, 19 March 2026 (UTC)

:@[[User:ShakespeareFan00|ShakespeareFan00]] I temporarily reduced the protection down for these templates (semi-protection), so you can do the necessary changes. :) [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:38, 21 March 2026 (UTC)

: Thats the thing I'm not sure what the 'stable' repair is, without breaking other stuff :( [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:44, 22 March 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:09, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Technical_Village_Pumps_distribution_list&amp;oldid=29905755 --&gt;

== plainlist and unbulleted list ==

Hi,&lt;br&gt;
The plainlist and unbulleted list templates don't work in a sandbox. [[User:PeterEasthope/sandbox|PeterEasthope/sandbox]] for example. What is the simplest solution?&lt;br&gt;
Thanks, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 17:04, 18 April 2026 (UTC)
: The unbulleted list relies on a module, and the plainlist relies on a TemplateStyles CSS page. What were you trying to test on your sandbox? [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:47, 18 April 2026 (UTC)
::Aiming to make a list without bullets. Whether it works by a module or a template or some other device is probably not a significant concern. Certainly a list is needed rather than empty space. Thx, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 21:29, 18 April 2026 (UTC)</text>
      <sha1>kvddnol370558s44ft66m2w892k46q5</sha1>
    </revision>
    <revision>
      <id>4631273</id>
      <parentid>4631271</parentid>
      <timestamp>2026-04-18T22:11:48Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>3441010</id>
      </contributor>
      <comment>/* plainlist and unbulleted list */ reply to PeterEasthope ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>4631273</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5652" sha1="7xrdthqtxllxwjs3w8xhclbp31qm5pl" xml:space="preserve">__NEWSECTIONLINK__ {{Discussion Rooms}} {{Shortcut|WB:TECH}} {{TOC left}}
{{User:MiszaBot/config
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|counter = 1
|minthreadstoarchive = 1
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Welcome to the '''Technical Assistance reading room'''. Get assistance on questions related to [[w:MediaWiki|MediaWiki]] markup, CSS, JavaScript, and such as they relate to Wikibooks. '''This is not a general-purpose technical support room'''.

To submit a ''bug notice or feature request'' for the MediaWiki software, visit [[phabricator:|Phabricator]].

To get more information about the ''MediaWiki software'', or to download your own copy, visit [[mw:|MediaWiki]]

There are also two IRC channels for technical help: {{Channel|mediawiki}} for issues about the software, and {{channel|mediawiki-core}} for [[m:WMF|WMF]] server or configuration issues.
{{clear}}
[[Category:Reading room]]

== Templates: Refbegin and Refend. ==

These DO NOT seem to work as the documentation indicates throwing lints when used with the form indicated. 

These templates need to be behave consistently, or thrown out and re-written entirely.  

It would be NICE to have ONE set of templates and documentation to accompany them, currently this does not apparently exist, with the relevant templates getting confused between definition lists &lt;nowiki&gt;:&lt;/nowiki&gt; and standard lists {{*}}. 

CONSISTENT behaviour would also be nice.

[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:39, 19 March 2026 (UTC)

:@[[User:ShakespeareFan00|ShakespeareFan00]] I temporarily reduced the protection down for these templates (semi-protection), so you can do the necessary changes. :) [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:38, 21 March 2026 (UTC)

: Thats the thing I'm not sure what the 'stable' repair is, without breaking other stuff :( [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:44, 22 March 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:09, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Technical_Village_Pumps_distribution_list&amp;oldid=29905755 --&gt;

== plainlist and unbulleted list ==

Hi,&lt;br&gt;
The plainlist and unbulleted list templates don't work in a sandbox. [[User:PeterEasthope/sandbox|PeterEasthope/sandbox]] for example. What is the simplest solution?&lt;br&gt;
Thanks, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 17:04, 18 April 2026 (UTC)
: The unbulleted list relies on a module, and the plainlist relies on a TemplateStyles CSS page. What were you trying to test on your sandbox? [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:47, 18 April 2026 (UTC)
::Aiming to make a list without bullets. Whether it works by a module or a template or some other device is probably not a significant concern. Certainly a list is needed rather than empty space. Thx, ... [[User:PeterEasthope|PeterEasthope]] ([[User talk:PeterEasthope|discuss]] • [[Special:Contributions/PeterEasthope|contribs]]) 21:29, 18 April 2026 (UTC)
::: The plain list works; the thing is, I believe the span class tags might be conflicting with that plain list, possibly... [[User:Codename Noreste|&lt;span style=&quot;color:#0024FF&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:11, 18 April 2026 (UTC)</text>
      <sha1>7xrdthqtxllxwjs3w8xhclbp31qm5pl</sha1>
    </revision>
  </page>
  <page>
    <title>Wikijunior:Shapes/Basic Shapes/Square</title>
    <ns>110</ns>
    <id>180919</id>
    <revision>
      <id>4631268</id>
      <parentid>4408433</parentid>
      <timestamp>2026-04-18T20:47:26Z</timestamp>
      <contributor>
        <username>Ziv</username>
        <id>3267536</id>
      </contributor>
      <comment>→ File replacement: jpg/png/gif to svg vector image ([[:c:GR]])</comment>
      <origin>4631268</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="506" sha1="s60a4ewlt82fjj91fopv3robe336ffq" xml:space="preserve">[[File:Solid black.svg|border|200px|center]]


[[File:2005-06-25_Tiles_together.jpg|200px|center]]

&lt;div style=&quot;font-weight:bold; text-align:center; font-size:1.2em;&quot;&gt;
A square has 4 equal sides and 4 right angles.  It also has 2 sets of parallel sides.

Did you know??? A square can also be called a [[{{BOOKNAME}}/Basic Shapes/Rectangle|rectangle]].  See the definition of a rectangle to find out why.  It also can be called a quadrilateral, a parallelogram, a rhombus and a polygon. 
&lt;/div&gt;

{{BookCat}}</text>
      <sha1>s60a4ewlt82fjj91fopv3robe336ffq</sha1>
    </revision>
  </page>
  <page>
    <title>OpenSCAD User Manual/General</title>
    <ns>0</ns>
    <id>219725</id>
    <revision>
      <id>4631245</id>
      <parentid>4609055</parentid>
      <timestamp>2026-04-18T16:10:33Z</timestamp>
      <contributor>
        <username>~2026-24036-08</username>
        <id>3577453</id>
      </contributor>
      <origin>4631245</origin>
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      <format>text/x-wiki</format>
      <text bytes="30076" sha1="doeqgdp8k2if6cn7xlz8xatxxgsrurj" xml:space="preserve">{{incomplete}}

OpenSCAD is a [[w:2D_computer_graphics|2D]]/[[w:3D computer graphics software|3D]] and [[w:Solid modeling|solid modeling]] program that is based on a [[w:Functional programming|Functional programming]] [[w:Procedural modeling|language]] used to create [[w:Polygonal modeling|models]] that are  previewed on the screen, and rendered into 3D [[w:Polygon mesh|mesh]] that can be exported in a variety of 2D/3D file formats.

== Introduction to Syntax ==
A script in the OpenSCAD language is used to create 2D or 3D models. This script is a free format list of statements, e.g.:
  variable = value;
  object();
  operator() object();
  operator() {
      variable = value;
      object();
  }
  operator() operator() {
      object();
      object();
  }
  operator() {
      operator() object();
      operator() {
          object();
          object();
      }
  }

;Assignments
:Assignment statements associate a value with a name.
$fn=200;

// Ring parameters

inner_dia = 19.84;        // Size 10 inner diameter mm

band_width = 7.0;         // slightly wider near stone

band_thickness = 1.7;     // adjusted for ~5g weight

pearl_dia = 7.93;         // measured pearl diameter

bezel_thickness = 1.5;    // bezel wall thickness

bezel_height = band_thickness + 1.5;  // bezel height above band

// Ring band with comfort fit (slightly beveled inner edge)

module band() {

    difference() {

        // Outer ring band

        rotate_extrude(angle=360)

        translate([inner_dia/2 + band_thickness/2, 0])

        circle(d = band_thickness);

        // Inner hole - comfort fit radius 0.4mm bevel

        rotate_extrude(angle=360)

        translate([inner_dia/2, 0])

        offset(r=-0.4)

        circle(d=0.1);

    }

}

// Bezel mount for pearl

module bezel() {

    // Bezel outer cylinder

    translate([0,0,band_thickness])

    cylinder(d = pearl_dia + 2*bezel_thickness, h = bezel_height, center=false);

    // Pearl seat (inner hollow)

    translate([0,0,band_thickness])

    cylinder(d=pearl_dia + 0.1, h=bezel_height + 0.2, center=false);

}

// Pearl (simplified as sphere)

module pearl() {

    translate([0,0,band_thickness + bezel_height/2])

    sphere(r = pearl_dia/2);

}

// Compose ring

union() {

    band();

    bezel();

    pearl();

}
;Objects
:Objects are the building blocks for models, created by 2D and 3D primitives and user-defined modules. Objects end in a semicolon ';'.
:Examples are: cube(), sphere(), polygon(), circle(), etc.

;Operators
:Operators, or transformations, modify the location, color and other properties of objects. Operators use braces '{}' when their scope covers more than one action. More than one operator may be used for the same object or group of objects. Multiple operators are processed Right to Left, that is, the operator closest to the object is processed first.
:Examples:   
    cube(5);
    x = 4+y;
    rotate(40) square(5,10);
    translate([10,5]) {
        circle(5);
        square(4);
    }
    rotate(60) color(&quot;red&quot;) {
        circle(5);
        square(4);
    }
    color(&quot;blue&quot;) {
        translate([5,3,0]) sphere(5);
        rotate([45,0,45]) {
            cylinder(10);
            cube([5,6,7]);
        }
    }

== Comments ==
Comments are a way of leaving notes within the script, or code, (either to yourself or to future programmers) describing how the code works, or what it does. Comments are not evaluated by the compiler, and should not be used to describe self-evident code. 
 
OpenSCAD uses C++-style comments:
 // This is a comment
   
 myvar = 10; // The rest of the line is a comment
   
 /*
    Multi-line comments
    can span multiple lines.
 */

== Values and Data Types ==
A value in OpenSCAD is either a Number (like 42), a Boolean (like true), a String (like &quot;foo&quot;), a Range (like [0: 1: 10]), a Vector (like [1,2,3]), or the Undefined value (undef). Values can be stored in variables, passed as function arguments, and returned as function results.

[OpenSCAD is a dynamically typed language with a fixed set of data types. There are no type names, and no user defined types.]

=== Numbers ===

Numbers are the most important type of value in OpenSCAD, and they are written in the familiar decimal notation used in other languages, e.g., -1, 42, 0.5, 2.99792458e+8. 

{{requires|Development snapshot}}
Hexadecimal constants are allowed in the C style, &lt;code&gt;0x&lt;/code&gt; followed by hexadecimal digits.  [OpenSCAD does not support octal notation for numbers.]

In addition to decimal numerals, the following name for a special number is defined:
* &lt;code&gt;PI&lt;/code&gt;

OpenSCAD has only a single kind of number, which is a 64 bit IEEE floating point number. OpenSCAD does not distinguish integers and floating point numbers as two different types, nor does it support complex numbers. Because OpenSCAD uses the IEEE floating point standard, there are a few deviations from the behavior of numbers in mathematics:
* We use binary floating point. A fractional number is not represented exactly unless the denominator is a power of 2. For example, 0.2 (2/10) does not have an exact internal representation, but 0.25 (1/4) and 0.375 (3/8) are represented exactly.
* The largest representable number is about 1e308. If a numeric result is too large, then the result can be infinity (printed as inf by echo).
* The smallest representable number is about -1e308. If a numeric result is too small, then the result can be -infinity (printed as -inf by echo).
* Values are precise to about 16 decimal digits.
* If a numeric result is invalid, then the result can be Not A Number (printed as nan by echo).
* If a non-zero numeric result is too close to zero to be representable, then the result is -0 if the result is negative, otherwise it is 0. Zero (0) and negative zero (-0) are treated as two distinct numbers by some of the math operations, and are printed differently by 'echo', although they compare as equal.

The constants &lt;code&gt;inf&lt;/code&gt; and &lt;code&gt;nan&lt;/code&gt; are not supported as numeric constants by OpenSCAD, even though you can compute numbers that are printed this way by 'echo'. You can define variables with these values by using:
 inf = 1e200 * 1e200;
 nan = 0 / 0;
 echo(inf,nan);

The value &lt;code&gt;nan&lt;/code&gt; is the only OpenSCAD value that is not equal to any other value, including itself. Although you can test if a variable 'x' has the undefined value using 'x == undef', you can't use 'x == 0/0' to test if x is Not A Number. Instead, you must use 'x != x' to test if x is nan.

=== Boolean values ===
Booleans are variables with two states, typically denoted in OpenSCAD as true and false.  
Boolean variables are typically generated by conditional tests and are employed by conditional statement 'if()'. conditional operator '? :', 
and generated by logical operators &lt;code&gt;!&lt;/code&gt; (not), &lt;code&gt;&amp;&amp;&lt;/code&gt; (and), and &lt;code&gt;||&lt;/code&gt; (or).  Statements such as &lt;code&gt;if()&lt;/code&gt; actually accept non-boolean variables, but most values are converted to &lt;code&gt;true&lt;/code&gt; in a boolean context. The values that count as &lt;code&gt;false&lt;/code&gt; are:
* &lt;code&gt;false&lt;/code&gt;
* &lt;code&gt;0&lt;/code&gt; and &lt;code&gt;-0&lt;/code&gt;
* &lt;code&gt;&quot;&quot;&lt;/code&gt;
* &lt;code&gt;[]&lt;/code&gt;
* &lt;code&gt;undef&lt;/code&gt;

Note that &lt;code&gt;&quot;false&quot;&lt;/code&gt; (the string), &lt;code&gt;[0]&lt;/code&gt; (a numeric vector), 
&lt;code&gt;[ [] ]&lt;/code&gt; (a vector containing an empty vector), &lt;code&gt;[false]&lt;/code&gt; 
(a vector containing the Boolean value false) and &lt;code&gt;0/0&lt;/code&gt; (not a number) all count as true.

=== Strings ===

A string is a sequence of zero or more unicode characters. String values are used to specify file names when importing a file, and to display text for debugging purposes when using echo(). Strings can also be used with the [[OpenSCAD_User_Manual/Text |'''text()''' primitive]], added in version '''2015.03'''.

A string literal is written as a sequence of characters enclosed in quotation marks &lt;code&gt;&quot;&lt;/code&gt;, like this: &lt;code&gt;&quot;&quot;&lt;/code&gt; (an empty string), or &lt;code&gt;&quot;this is a string&quot;&lt;/code&gt;.

To include a &lt;code&gt;&quot;&lt;/code&gt; character in a string literal, use &lt;code&gt;\&quot;&lt;/code&gt;. To include a &lt;code&gt;\&lt;/code&gt; character in a string literal, use &lt;code&gt;\\&lt;/code&gt;. The following escape sequences beginning with &lt;code&gt;\&lt;/code&gt; can be used within string literals:
* \&quot; → &quot;
* \\ → \
* \t → tab
* \n → newline
* \r → carriage return
* \x21 → ! - valid only in the range from \x01 to \x7f, \x00 produces a space
* \u03a9 → Ω - 4 digit unicode code point, see &lt;b&gt;text()&lt;/b&gt; for further information on unicode characters
* \U01f600 → 😀 - 6 digit unicode code point
This behavior is new since OpenSCAD-2011.04. You can upgrade old files using the following sed command: &lt;code&gt;sed 's/\\/\\\\/g' non-escaped.scad &gt; escaped.scad&lt;/code&gt;

'''Example:'''
   
  echo(&quot;The quick brown fox \tjumps \&quot;over\&quot; the lazy dog.\rThe quick brown fox.\nThe \\lazy\\ dog.&quot;);
   
  '''result'''&lt;br/&gt;
    ECHO: &quot;The quick brown fox     jumps &quot;over&quot; the lazy dog.
    The quick brown fox.
    The \lazy\ dog.&quot;
   
  '''old result'''
    ECHO: &quot;The quick brown fox \tjumps \&quot;over\&quot; the lazy dog.
    The quick brown fox.\nThe \\lazy\\ dog.&quot;

=== Ranges ===

Ranges are used by [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#For_Loop| for() loops]] and [[OpenSCAD_User_Manual/User-Defined_Functions_and_Modules#Children|children()]]. They have 2 varieties:
: [&lt;''start''&gt;:&lt;''end''&gt;]
: [&lt;''start''&gt;:&lt;''increment''&gt;:&lt;''end''&gt;]

A missing &lt;''increment''&gt; defaults to 1.

Although enclosed in square brackets [] , they are not vectors. They use colons : for separators rather than commas.
 r1 = [0:10];
 r2 = [0.5:2.5:20];
 echo(r1); // ECHO: [0: 1: 10]
 echo(r2); // ECHO: [0.5: 2.5: 20]

'''In version 2021.01 and earlier''', a range in the form  [&lt;''start''&gt;:&lt;''end''&gt;] with &lt;''start''&gt; greater than &lt;''end''&gt; generates a warning and is equivalent to [&lt;''end''&gt;: 1: &lt;''start''&gt;].

A range in the form  [&lt;''start''&gt;:1:&lt;''end''&gt;] with &lt;''start''&gt; greater than &lt;''end''&gt; does not generate a warning and is equivalent to []. {{requires|Development snapshot}} This also applies when the increment is omitted.

The &lt;''increment''&gt; in a range may be negative (for versions after 2014).  A start value less than the end value, with a negative increment, does not generate a warning and is equivalent to [].

You should take care with step values that cannot be represented exactly as binary floating point numbers. Integers are okay, as are fractional values whose denominator is a power of two. For example, 0.25 (1/4) and 0.375 (3/8) are safe, but 0.2 (2/10) may cause problems. The problem with these step values is that your range may have more or fewer elements than you expect, due to inexact arithmetic.

=== The Undefined Value ===
The undefined value is a special value written as &lt;code&gt;undef&lt;/code&gt;. It is the initial value of a variable that hasn't been assigned a value, and it is often returned as a result by functions or operations that are passed illegal arguments. Finally, &lt;code&gt;undef&lt;/code&gt; can be used as a null value, equivalent to &lt;code&gt;null&lt;/code&gt; or &lt;code&gt;NULL&lt;/code&gt; in other programming languages.

All arithmetic expressions containing &lt;code&gt;undef&lt;/code&gt; values evaluate as &lt;code&gt;undef&lt;/code&gt;. In logical expressions, &lt;code&gt;undef&lt;/code&gt; is equivalent to &lt;code&gt;false&lt;/code&gt;. Relational operator expressions with &lt;code&gt;undef&lt;/code&gt; evaluate as &lt;code&gt;false&lt;/code&gt; except for &lt;code&gt;undef==undef&lt;/code&gt;, which is &lt;code&gt;true&lt;/code&gt;.

Note that numeric operations may also return 'nan' (not-a-number) to indicate an illegal argument. For example, &lt;code&gt;0/false&lt;/code&gt; is &lt;code&gt;undef&lt;/code&gt;, but &lt;code&gt;0/0&lt;/code&gt; is 'nan'. Relational operators like &lt; and &gt; return &lt;code&gt;false&lt;/code&gt; if passed illegal arguments. Although &lt;code&gt;undef&lt;/code&gt; is a language value, 'nan' is not.

==Variables==

OpenSCAD variables are created by a statement with a name or [[w:Identifier (computer programming)|identifier]], assignment via an expression and a semicolon. The role of arrays, found in many imperative languages, is handled in OpenSCAD via vectors. Valid identifiers are composed of simple characters and underscores [a-zA-Z0-9_]. High_ASCII and Unicode characters are not allowed.

'''Before Development snapshot''', variable names can begin with digits.  In '''Development snapshot''', names starting with &lt;code&gt;0x&lt;/code&gt; and followed by hexadecimal digits represent a hexadecimal constant, and other variable names beginning with digits will trigger a warning.

 var = 25;
 xx = 1.25 * cos(50);
 y = 2*xx+var;
 logic = true;
 MyString = &quot;This is a string&quot;;
 a_vector = [1,2,3];
 rr = a_vector[2];      // member of vector
 range1 = [-1.5:0.5:3]; // for() loop range
 xx = [0:5];            // alternate for() loop range

OpenSCAD is a [[w:Functional programming| Functional]] programming language, so such [[w:Variable_(computer_science%29)|variables]] are bound to expressions and keep a single value during their entire lifetime due to the requirements of [[w:Referential transparency (computer science)|referential transparency]]. In [[w:Imperative programming|imperative languages]], such as C, the same behavior is seen as constants, which are typically contrasted with normal variables. 

In other words, OpenSCAD variables are more like constants, but with an important difference. If variables are assigned a value multiple times, only the last assigned value is used in all places in the code. See further discussion at [[#Variables cannot be changed]].  This behavior is due to the need to supply variable input on the [[OpenSCAD User Manual/Using OpenSCAD in a command line environment|command line]], via the use of ''-D variable=value'' option. OpenSCAD currently places that assignment at the end of the source code, and thus must allow a variable's value to be changed for this purpose.

Values cannot be modified during run time; all variables are effectively constants that do not change once created. Each variable retains its last assigned value, in line with [[w:Functional programming|Functional]] programming languages. Unlike [[w:Imperative programming|Imperative]] languages, such as C, OpenSCAD is not an iterative language, and therefore the concept of ''x''&amp;nbsp;=&amp;nbsp;''x''&amp;nbsp;+&amp;nbsp;1 is not valid. The only way an expression like this works in OpenSCAD is if the ''x'' on the right comes from a parent scope, such as an argument list, and the assignment operation creates a new ''x'' in the current scope. Understanding this concept leads to understanding the beauty of OpenSCAD.

'''Before version 2015.03''', it was not possible to do assignments at any place except the file top-level and module top-level. Inside an ''if/else''&amp;nbsp; or ''for''&amp;nbsp; loop, assign() was needed.

'''Since version 2015.03''', variables can now be assigned in any scope. Note that assignments are valid only within the scope in which they are defined - it is still not possible to leak values to an outer scope. See [[#Scope of variables|Scope of variables]] for more details.

 a=0;
 if (a==0) 
   {
  a=1; //  before 2015.03 this line would generate a Compile Error
       //  since 2015.03  no longer an error, but the value a=1 is confined to within the braces {}
   }
=== Undefined variable ===

Referring to a variable that has not had a value assigned triggers a warning, and yields the special value '''undef'''.

It is possible to detect an undefined variable using `is_undef(var)`.  (Note that the simpler `var == undef` will yield a warning.)

  '''Example'''
   
  echo(&quot;Variable a is &quot;, a);                // Variable a is undef, triggers a warning
  if (is_undef(a)) {                        // does not trigger a warning
    echo(&quot;Variable a is tested undefined&quot;); // Variable a is tested undefined
  }

=== Scope of variables ===

When operators such as translate() and color() need to encompass more than one action ( actions end in ;), braces {} are needed to group the actions, creating a new, inner scope.
When there is only one semicolon, braces are usually optional.

Each pair of braces grouping &quot;children&quot; of an operator creates a new scope inside the scope where they were used. '''Since 2015.03''', new variables can be created within this new scope. New values can be given to variables that were created in an outer scope. 
These variables and their values are also available to further inner scopes created within this scope, but are '''not available''' to anything outside this scope. Variables still have only the last value assigned within a scope.

                        // scope 1
  a = 6;                // create a
  echo(a,b);            //                6, undef
  translate([5,0,0]){   // scope 1.1
    a= 10;
    b= 16;              // create b
    echo(a,b);          //              100, 16   a=10; was overridden by later a=100;
    color(&quot;blue&quot;) {     // scope 1.1.1
      echo(a,b);        //              100, 20
      cube();
      b=20;
    }                   // back to 1.1
    echo(a,b);          //              100, 16
    a=100;              // override a in 1.1
  }                     // back to 1   
  echo(a,b);            //                6, undef
  color(&quot;red&quot;){         // scope 1.2
    cube();
    echo(a,b);          //                6, undef
  }                     // back to 1
  echo(a,b);            //                6, undef
   
  //In this example, scopes 1 and 1.1 are outer scopes to 1.1.1 but 1.2 is not.

: '''Anonymous scopes''' are not considered scopes:
  {
    angle = 45;
  }
  rotate(angle) square(10);

For() loops are not an exception to the rule about variables having only one value within a scope. A copy of loop contents is created for each pass. Each pass is given its own scope, allowing any variables to have unique values for that pass. No, you still can't do a = a + 1.

=== Variables cannot be changed ===

The simplest description of OpenSCAD variables is that an assignment creates a new variable in the current scope, and that it's not legal to set a variable that has already been set in the current scope.  In a lot of ways, it's best to think of them as named constants, calculated on entry to the scope, but there's a catch.  If you set a variable twice in the same scope, the second assignment triggers a warning (which may abort the program, depending on preferences settings).  It does *not* then replace the value of the variable - rather, it replaces the original assignment, at its position in the list of assignments.  The original assignment is never executed.

 a = 1;   // never executed
 echo(a); // 2
 a = 2;   // executed at the position of the original assignment
 echo(a); // 2

That's still not the complete story.  There are two special cases that do not trigger warnings:
* if the first assignment is in the top level of an `include` file, and the second assignment is in the including file.
* If the first assignment is in the top level of the program source, and the second assignment comes from a `-D` option or from the Customizer.

While this appears to be counter-intuitive, it allows you to do some interesting things: for instance, if you set up your shared library files to have default values defined as variables at their root level, when you include that file in your own code you can 're-define' or override those constants by simply assigning a new value to them - and other variables based on that variable are based on the value from the main program.

 // main.scad
 include &lt;lib.scad&gt;
 a = 2;
 echo(b);

 // lib.scad
 a = 1;
 b = a + 1;

will produce 3.

=== Special variables ===
Special variables provide an alternate means of passing arguments to modules and functions.
All variables starting with a '$' are special variables, similar to special variables in lisp.
As such they are more dynamic than regular variables.
(for more details see [[OpenSCAD_User_Manual/Other_Language_Features#Special_variables|Other Language Features]])

== Vectors ==

A vector or list is a sequence of zero or more OpenSCAD values. Vectors are collections of numeric or boolean values, variables, vectors, strings or any combination thereof. They can also be expressions that evaluate to one of these. Vectors handle the role of arrays found in many imperative languages.
The information here also applies to lists and tables that use vectors for their data.
   
A vector has square brackets, [] enclosing zero or more items (elements or members), separated by commas. A vector can contain vectors, which can contain vectors, etc.

'''Examples'''

    [1,2,3]
    [a,5,b]
    []
    [5.643]
    [&quot;a&quot;,&quot;b&quot;,&quot;string&quot;]
    [[1,r],[x,y,z,4,5]]
    [3, 5, [6,7], [[8,9],[10,[11,12],13], c, &quot;string&quot;]
    [4/3, 6*1.5, cos(60)]

use in OpenSCAD:
   cube( [width,depth,height] );           // optional spaces shown for clarity
   translate( [x,y,z] )
   polygon( [ [x&lt;sub&gt;0&lt;/sub&gt;,y&lt;sub&gt;0&lt;/sub&gt;],  [x&lt;sub&gt;1&lt;/sub&gt;,y&lt;sub&gt;1&lt;/sub&gt;],  [x&lt;sub&gt;2&lt;/sub&gt;,y&lt;sub&gt;2&lt;/sub&gt;] ] );

=== Creation ===

Vectors are created by writing the list of elements, separated by commas, and enclosed in square brackets. Variables are replaced by their values.
   cube([10,15,20]);
   a1 = [1,2,3];
   a2 = [4,5];
   a3 = [6,7,8,9];
   b  = [a1,a2,a3];    // [ [1,2,3], [4,5], [6,7,8,9] ]  note increased nesting depth
Vectors can be initialized using a for loop enclosed in square brackets. 

The following example initializes the vector ''result'' with a length ''n'' of 10 values to the value of ''a''.&lt;syntaxhighlight lang=&quot;openscad&quot;&gt;n = 10;
a = 0;

result = [ for (i=[0:n-1]) a ];
echo(result); //ECHO: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]&lt;/syntaxhighlight&gt;The following example shows a vector ''result'' with a ''n'' length of 10 initialized with values that are alternatively ''a'' or ''b'' respectively if the index position ''i'' is an even or an odd number.&lt;syntaxhighlight&gt;
n = 10;
a = 0;
b = 1;
result = [ for (i=[0:n-1]) (i % 2 == 0) ? a : b ];
echo(result); //ECHO: [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
&lt;/syntaxhighlight&gt;

=== Indexing elements within vectors ===
 
Elements within vectors are numbered from 0 to n - 1 where n is the length returned by [[#len|len()]].
Address elements within vectors with the following notation:
 e[5]           // element no 5 (sixth) at   1st nesting level
 e[5][2]        // element 2 of element 5    2nd nesting level
 e[5][2][0]     // element 0 of 2 of 5       3rd nesting level
 e[5][2][0][1]  // element 1 of 0 of 2 of 5  4th nesting level

&lt;div id= &quot;example1&quot;&gt; example elements with lengths from len()&lt;/div&gt;
 e = [ [1], [], [3,4,5], &quot;string&quot;, &quot;x&quot;, [[10,11],[12,13,14],[[15,16],[17]]] ];  // length 6
 
 address       length  element
 e[0]          1       [1]
 e[1]          0       []
 e[5]          3       [ [10,11], [12,13,14], [[15,16],[17]] ]
 e[5][1]       3       [ 12, 13, 14 ]
 e[5][2]       2       [ [15,16], [17] ]
 e[5][2][0]    2       [ 15, 16 ]
 e[5][2][0][1] undef   16
     
 e[3]          6       &quot;string&quot;
 e[3 ][2]      1       &quot;r&quot;
   
 s = [2,0,5]; a = 2;
 s[a]          undef   5
 e[s[a]]       3       [ [10,11], [12,13,14], [[15,16],[17]] ]

==== String indexing ====

The elements (characters) of a string can be accessed:
 &quot;string&quot;[2]    //resolves to &quot;r&quot;
 

==== Dot notation indexing ====

The first three elements of a vector can be accessed with an alternate dot notation:
 e.x    //equivalent to e[0]
 e.y    //equivalent to e[1]
 e.z    //equivalent to e[2]

=== Vector operators ===

==== concat ====
{{requires|2015.03}}
  
&lt;code&gt;concat()&lt;/code&gt; combines the elements of 2 or more vectors into a single vector. No change in nesting level is made.
  vector1 = [1,2,3]; vector2 = [4]; vector3 = [5,6];
  new_vector = concat(vector1, vector2, vector3); // [1,2,3,4,5,6]
   
  string_vector = concat(&quot;abc&quot;,&quot;def&quot;);                 // [&quot;abc&quot;, &quot;def&quot;]
  one_string = str(string_vector[0],string_vector[1]); // &quot;abcdef&quot;

==== len ====
&lt;code&gt;len()&lt;/code&gt; is a function that returns the length of vectors or strings. 
Indices of elements are from [0] to [length-1].
: '''vector'''
:: Returns the number of elements at this level.
:: Single values, which are '''not''' vectors, raise an error.
: '''string'''
:: Returns the number of characters in a string.
  a = [1,2,3]; echo(len(a));   //  3

[[#example1|See example elements with lengths]]

=== Matrix ===

A matrix is a vector of vectors.

 Example that defines a 2D rotation matrix
 mr = [
      [cos(angle), -sin(angle)],
      [sin(angle),  cos(angle)]
     ];

== Objects ==

{{requires|Development snapshot}}

Objects store collections of data, like vectors, but the individual members are accessed by string names rather than by numeric indexes.  They are analogous to JavaScript objects or Python dictionaries.

=== Creating an object ===

The &lt;code&gt;object()&lt;/code&gt; function can be used to create objects.  &lt;code&gt;object()&lt;/code&gt; processes its arguments left to right, adding them to the object under construction.  It accepts three different styles of argument, which can be mixed as desired:

The first form is this:

    &lt;name&gt;=&lt;value&gt;

This form is convenient for cases where the name is constant and suitable for use as an identifier.

The second form is this:

    [
        [ &lt;name-string&gt;, &lt;value&gt; ],
        ...
    ]

This form allows calculated names and names that are not suitable as identifiers. It also allows lists of members that are derived from other processing - from a list comprehension, perhaps.

The third form is this:

    &lt;object&gt;

If an argument is itself an object, its members are added to the object under construction.

If a member is mentioned more than once, the last mention wins.  Thus you can copy an object, replacing one member, using &lt;code&gt;object(obj, name=newvalue)&lt;/code&gt;.

You can remove a member from the object under construction using the list form, supplying only a name and no value - e.g., &lt;code&gt;object(obj, [ [ &quot;unwanted&quot; ] ])&lt;/code&gt;

=== Retrieving a value from an object ===

    obj.name

Retrieves the named value from the object, for a name that is constant and syntactically suitable for use as an identifier.

    obj[&quot;name&quot;]

Retrieves the named value from the object, for a name that is an arbitrary string expression.

Note that members with identifier-like names can be accessed using either mechanism.  The choice depends on the particular use case.

If the specified member is not present in the object, yields &lt;code&gt;undef&lt;/code&gt;.

=== Iterating over object members ===

   for (name = obj) { ... }

iterates over the members of the object, in an unspecified order, setting &lt;code&gt;name&lt;/code&gt; to the name of each member.  It is then typically desirable to access the value using &lt;code&gt;obj[name]&lt;/code&gt;.

This construct works for [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#For_Loop|flow-control &lt;code&gt;for&lt;/code&gt;]], [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#Intersection_For_Loop|intersection_for()]], and [[OpenSCAD_User_Manual/List_Comprehensions#for|list-comprehension &lt;code&gt;for&lt;/code&gt;]].

=== Examples ===

    // Create an object with a few members.
    o1 = object(a = 1, b = &quot;hello&quot;, c = true);

    // Retrieve a member using a constant name.
    echo(o1.b);      // &quot;hello&quot;

    // Retrieve members using a name that varies.
    for (k = [ &quot;a&quot;, &quot;b&quot; ])
        echo(o1[k]);    // 1; &quot;hello&quot;

    // Copy the object, changing &quot;c&quot;, adding &quot;d&quot;, and removing &quot;a&quot;.
    o2 = object(o1, c = false, d = &quot;dog&quot;, [[&quot;a&quot;]]);
    echo(o2);        // { b = &quot;hello&quot;; c = false; d = &quot;dog&quot;; }

    // Create an object, calculating the names to use.
    o3 = object([
        for (i = [1:3])
            [ str(&quot;x&quot;, i), i*i ]
    ]);
    echo(o3);        // { x1 = 1; x2 = 4; x3 = 9; }

    // Merge the three objects together.
    o4 = object(o1, o2, o3);
    // Separately print each member of the result
    for (name = o4)
        echo(name = name, value = o4[name]);
    // name = &quot;a&quot;, value = 1
    // name = &quot;b&quot;, value = &quot;hello&quot;
    // name = &quot;c&quot;, value = false
    // name = &quot;d&quot;, value = &quot;dog&quot;
    // name = &quot;x1&quot;, value = 1
    // name = &quot;x2&quot;, value = 4
    // name = &quot;x3&quot;, value = 9

== Getting input ==

There is no mechanism for variable input from keyboard or reading from arbitrary files.  There is no prompting mechanism, no input window, or input fields or any way to manually enter data while the script is running.

Variables can be set via:
* assignments in the script
* [[OpenSCAD_User_Manual/Customizer|the Customizer]]
* &lt;code&gt;-D&lt;/code&gt; at the [[OpenSCAD_User_Manual/Using_OpenSCAD_in_a_command_line_environment|command line interface]]
* accessing data in a few file formats (stl, dxf, png, etc).

With the exception of DXF files, data from files is not accessible to the script, although to a limited extent the script may be able to manipulate the data as a whole.  For example, STL files can be rendered in OpenSCAD, translated, clipped, etc.  But the internal data that constitutes the STL file is inaccessible.

=== Getting a point from a drawing ===

Getting a point is useful for reading an origin point in a 2D view in a technical drawing. The function dxf_cross reads the intersection of two lines on a layer you specify and returns the intersection point. This means that the point must be given with two lines in the DXF file, and not a point entity.

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
OriginPoint = dxf_cross(file=&quot;drawing.dxf&quot;, layer=&quot;SCAD.Origin&quot;, 
                        origin=[0, 0], scale=1);
&lt;/syntaxhighlight&gt;

=== Getting a dimension value ===

You can read dimensions from a technical drawing. This can be useful to read a rotation angle, an extrusion height, or spacing between parts. In the drawing, create a dimension that does not show the dimension value, but an identifier. To read the value, you specify this identifier from your program:

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
TotalWidth = dxf_dim(file=&quot;drawing.dxf&quot;, name=&quot;TotalWidth&quot;,
                        layer=&quot;SCAD.Origin&quot;, origin=[0, 0], scale=1);
&lt;/syntaxhighlight&gt;

For a nice example of both functions, see Example009 and the image on the [http://www.openscad.org/ homepage of OpenSCAD].

{{BookCat}}</text>
      <sha1>doeqgdp8k2if6cn7xlz8xatxxgsrurj</sha1>
    </revision>
    <revision>
      <id>4631250</id>
      <parentid>4631245</parentid>
      <timestamp>2026-04-18T16:23:03Z</timestamp>
      <contributor>
        <username>Tp42</username>
        <id>1412063</id>
      </contributor>
      <comment>Rejected the last text change (by [[Special:Contributions/~2026-24036-08|~2026-24036-08]]) and restored revision 4609055 by MathXplore</comment>
      <origin>4631250</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28643" sha1="n2floxih36nfpu0ac0cqvzk4v3j589k" xml:space="preserve">{{incomplete}}

OpenSCAD is a [[w:2D_computer_graphics|2D]]/[[w:3D computer graphics software|3D]] and [[w:Solid modeling|solid modeling]] program that is based on a [[w:Functional programming|Functional programming]] [[w:Procedural modeling|language]] used to create [[w:Polygonal modeling|models]] that are  previewed on the screen, and rendered into 3D [[w:Polygon mesh|mesh]] that can be exported in a variety of 2D/3D file formats.

== Introduction to Syntax ==
A script in the OpenSCAD language is used to create 2D or 3D models. This script is a free format list of statements, e.g.:
  variable = value;
  object();
  operator() object();
  operator() {
      variable = value;
      object();
  }
  operator() operator() {
      object();
      object();
  }
  operator() {
      operator() object();
      operator() {
          object();
          object();
      }
  }

;Assignments
:Assignment statements associate a value with a name.

;Objects
:Objects are the building blocks for models, created by 2D and 3D primitives and user-defined modules. Objects end in a semicolon ';'.
:Examples are: cube(), sphere(), polygon(), circle(), etc.

;Operators
:Operators, or transformations, modify the location, color and other properties of objects. Operators use braces '{}' when their scope covers more than one action. More than one operator may be used for the same object or group of objects. Multiple operators are processed Right to Left, that is, the operator closest to the object is processed first.
:Examples:   
    cube(5);
    x = 4+y;
    rotate(40) square(5,10);
    translate([10,5]) {
        circle(5);
        square(4);
    }
    rotate(60) color(&quot;red&quot;) {
        circle(5);
        square(4);
    }
    color(&quot;blue&quot;) {
        translate([5,3,0]) sphere(5);
        rotate([45,0,45]) {
            cylinder(10);
            cube([5,6,7]);
        }
    }

== Comments ==
Comments are a way of leaving notes within the script, or code, (either to yourself or to future programmers) describing how the code works, or what it does. Comments are not evaluated by the compiler, and should not be used to describe self-evident code. 
 
OpenSCAD uses C++-style comments:
 // This is a comment
   
 myvar = 10; // The rest of the line is a comment
   
 /*
    Multi-line comments
    can span multiple lines.
 */

== Values and Data Types ==
A value in OpenSCAD is either a Number (like 42), a Boolean (like true), a String (like &quot;foo&quot;), a Range (like [0: 1: 10]), a Vector (like [1,2,3]), or the Undefined value (undef). Values can be stored in variables, passed as function arguments, and returned as function results.

[OpenSCAD is a dynamically typed language with a fixed set of data types. There are no type names, and no user defined types.]

=== Numbers ===

Numbers are the most important type of value in OpenSCAD, and they are written in the familiar decimal notation used in other languages, e.g., -1, 42, 0.5, 2.99792458e+8. 

{{requires|Development snapshot}}
Hexadecimal constants are allowed in the C style, &lt;code&gt;0x&lt;/code&gt; followed by hexadecimal digits.  [OpenSCAD does not support octal notation for numbers.]

In addition to decimal numerals, the following name for a special number is defined:
* &lt;code&gt;PI&lt;/code&gt;

OpenSCAD has only a single kind of number, which is a 64 bit IEEE floating point number. OpenSCAD does not distinguish integers and floating point numbers as two different types, nor does it support complex numbers. Because OpenSCAD uses the IEEE floating point standard, there are a few deviations from the behavior of numbers in mathematics:
* We use binary floating point. A fractional number is not represented exactly unless the denominator is a power of 2. For example, 0.2 (2/10) does not have an exact internal representation, but 0.25 (1/4) and 0.375 (3/8) are represented exactly.
* The largest representable number is about 1e308. If a numeric result is too large, then the result can be infinity (printed as inf by echo).
* The smallest representable number is about -1e308. If a numeric result is too small, then the result can be -infinity (printed as -inf by echo).
* Values are precise to about 16 decimal digits.
* If a numeric result is invalid, then the result can be Not A Number (printed as nan by echo).
* If a non-zero numeric result is too close to zero to be representable, then the result is -0 if the result is negative, otherwise it is 0. Zero (0) and negative zero (-0) are treated as two distinct numbers by some of the math operations, and are printed differently by 'echo', although they compare as equal.

The constants &lt;code&gt;inf&lt;/code&gt; and &lt;code&gt;nan&lt;/code&gt; are not supported as numeric constants by OpenSCAD, even though you can compute numbers that are printed this way by 'echo'. You can define variables with these values by using:
 inf = 1e200 * 1e200;
 nan = 0 / 0;
 echo(inf,nan);

The value &lt;code&gt;nan&lt;/code&gt; is the only OpenSCAD value that is not equal to any other value, including itself. Although you can test if a variable 'x' has the undefined value using 'x == undef', you can't use 'x == 0/0' to test if x is Not A Number. Instead, you must use 'x != x' to test if x is nan.

=== Boolean values ===
Booleans are variables with two states, typically denoted in OpenSCAD as true and false.  
Boolean variables are typically generated by conditional tests and are employed by conditional statement 'if()'. conditional operator '? :', 
and generated by logical operators &lt;code&gt;!&lt;/code&gt; (not), &lt;code&gt;&amp;&amp;&lt;/code&gt; (and), and &lt;code&gt;||&lt;/code&gt; (or).  Statements such as &lt;code&gt;if()&lt;/code&gt; actually accept non-boolean variables, but most values are converted to &lt;code&gt;true&lt;/code&gt; in a boolean context. The values that count as &lt;code&gt;false&lt;/code&gt; are:
* &lt;code&gt;false&lt;/code&gt;
* &lt;code&gt;0&lt;/code&gt; and &lt;code&gt;-0&lt;/code&gt;
* &lt;code&gt;&quot;&quot;&lt;/code&gt;
* &lt;code&gt;[]&lt;/code&gt;
* &lt;code&gt;undef&lt;/code&gt;

Note that &lt;code&gt;&quot;false&quot;&lt;/code&gt; (the string), &lt;code&gt;[0]&lt;/code&gt; (a numeric vector), 
&lt;code&gt;[ [] ]&lt;/code&gt; (a vector containing an empty vector), &lt;code&gt;[false]&lt;/code&gt; 
(a vector containing the Boolean value false) and &lt;code&gt;0/0&lt;/code&gt; (not a number) all count as true.

=== Strings ===

A string is a sequence of zero or more unicode characters. String values are used to specify file names when importing a file, and to display text for debugging purposes when using echo(). Strings can also be used with the [[OpenSCAD_User_Manual/Text |'''text()''' primitive]], added in version '''2015.03'''.

A string literal is written as a sequence of characters enclosed in quotation marks &lt;code&gt;&quot;&lt;/code&gt;, like this: &lt;code&gt;&quot;&quot;&lt;/code&gt; (an empty string), or &lt;code&gt;&quot;this is a string&quot;&lt;/code&gt;.

To include a &lt;code&gt;&quot;&lt;/code&gt; character in a string literal, use &lt;code&gt;\&quot;&lt;/code&gt;. To include a &lt;code&gt;\&lt;/code&gt; character in a string literal, use &lt;code&gt;\\&lt;/code&gt;. The following escape sequences beginning with &lt;code&gt;\&lt;/code&gt; can be used within string literals:
* \&quot; → &quot;
* \\ → \
* \t → tab
* \n → newline
* \r → carriage return
* \x21 → ! - valid only in the range from \x01 to \x7f, \x00 produces a space
* \u03a9 → Ω - 4 digit unicode code point, see &lt;b&gt;text()&lt;/b&gt; for further information on unicode characters
* \U01f600 → 😀 - 6 digit unicode code point
This behavior is new since OpenSCAD-2011.04. You can upgrade old files using the following sed command: &lt;code&gt;sed 's/\\/\\\\/g' non-escaped.scad &gt; escaped.scad&lt;/code&gt;

'''Example:'''
   
  echo(&quot;The quick brown fox \tjumps \&quot;over\&quot; the lazy dog.\rThe quick brown fox.\nThe \\lazy\\ dog.&quot;);
   
  '''result'''&lt;br/&gt;
    ECHO: &quot;The quick brown fox     jumps &quot;over&quot; the lazy dog.
    The quick brown fox.
    The \lazy\ dog.&quot;
   
  '''old result'''
    ECHO: &quot;The quick brown fox \tjumps \&quot;over\&quot; the lazy dog.
    The quick brown fox.\nThe \\lazy\\ dog.&quot;

=== Ranges ===

Ranges are used by [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#For_Loop| for() loops]] and [[OpenSCAD_User_Manual/User-Defined_Functions_and_Modules#Children|children()]]. They have 2 varieties:
: [&lt;''start''&gt;:&lt;''end''&gt;]
: [&lt;''start''&gt;:&lt;''increment''&gt;:&lt;''end''&gt;]

A missing &lt;''increment''&gt; defaults to 1.

Although enclosed in square brackets [] , they are not vectors. They use colons : for separators rather than commas.
 r1 = [0:10];
 r2 = [0.5:2.5:20];
 echo(r1); // ECHO: [0: 1: 10]
 echo(r2); // ECHO: [0.5: 2.5: 20]

'''In version 2021.01 and earlier''', a range in the form  [&lt;''start''&gt;:&lt;''end''&gt;] with &lt;''start''&gt; greater than &lt;''end''&gt; generates a warning and is equivalent to [&lt;''end''&gt;: 1: &lt;''start''&gt;].

A range in the form  [&lt;''start''&gt;:1:&lt;''end''&gt;] with &lt;''start''&gt; greater than &lt;''end''&gt; does not generate a warning and is equivalent to []. {{requires|Development snapshot}} This also applies when the increment is omitted.

The &lt;''increment''&gt; in a range may be negative (for versions after 2014).  A start value less than the end value, with a negative increment, does not generate a warning and is equivalent to [].

You should take care with step values that cannot be represented exactly as binary floating point numbers. Integers are okay, as are fractional values whose denominator is a power of two. For example, 0.25 (1/4) and 0.375 (3/8) are safe, but 0.2 (2/10) may cause problems. The problem with these step values is that your range may have more or fewer elements than you expect, due to inexact arithmetic.

=== The Undefined Value ===
The undefined value is a special value written as &lt;code&gt;undef&lt;/code&gt;. It is the initial value of a variable that hasn't been assigned a value, and it is often returned as a result by functions or operations that are passed illegal arguments. Finally, &lt;code&gt;undef&lt;/code&gt; can be used as a null value, equivalent to &lt;code&gt;null&lt;/code&gt; or &lt;code&gt;NULL&lt;/code&gt; in other programming languages.

All arithmetic expressions containing &lt;code&gt;undef&lt;/code&gt; values evaluate as &lt;code&gt;undef&lt;/code&gt;. In logical expressions, &lt;code&gt;undef&lt;/code&gt; is equivalent to &lt;code&gt;false&lt;/code&gt;. Relational operator expressions with &lt;code&gt;undef&lt;/code&gt; evaluate as &lt;code&gt;false&lt;/code&gt; except for &lt;code&gt;undef==undef&lt;/code&gt;, which is &lt;code&gt;true&lt;/code&gt;.

Note that numeric operations may also return 'nan' (not-a-number) to indicate an illegal argument. For example, &lt;code&gt;0/false&lt;/code&gt; is &lt;code&gt;undef&lt;/code&gt;, but &lt;code&gt;0/0&lt;/code&gt; is 'nan'. Relational operators like &lt; and &gt; return &lt;code&gt;false&lt;/code&gt; if passed illegal arguments. Although &lt;code&gt;undef&lt;/code&gt; is a language value, 'nan' is not.

==Variables==

OpenSCAD variables are created by a statement with a name or [[w:Identifier (computer programming)|identifier]], assignment via an expression and a semicolon. The role of arrays, found in many imperative languages, is handled in OpenSCAD via vectors. Valid identifiers are composed of simple characters and underscores [a-zA-Z0-9_]. High_ASCII and Unicode characters are not allowed.

'''Before Development snapshot''', variable names can begin with digits.  In '''Development snapshot''', names starting with &lt;code&gt;0x&lt;/code&gt; and followed by hexadecimal digits represent a hexadecimal constant, and other variable names beginning with digits will trigger a warning.

 var = 25;
 xx = 1.25 * cos(50);
 y = 2*xx+var;
 logic = true;
 MyString = &quot;This is a string&quot;;
 a_vector = [1,2,3];
 rr = a_vector[2];      // member of vector
 range1 = [-1.5:0.5:3]; // for() loop range
 xx = [0:5];            // alternate for() loop range

OpenSCAD is a [[w:Functional programming| Functional]] programming language, so such [[w:Variable_(computer_science%29)|variables]] are bound to expressions and keep a single value during their entire lifetime due to the requirements of [[w:Referential transparency (computer science)|referential transparency]]. In [[w:Imperative programming|imperative languages]], such as C, the same behavior is seen as constants, which are typically contrasted with normal variables. 

In other words, OpenSCAD variables are more like constants, but with an important difference. If variables are assigned a value multiple times, only the last assigned value is used in all places in the code. See further discussion at [[#Variables cannot be changed]].  This behavior is due to the need to supply variable input on the [[OpenSCAD User Manual/Using OpenSCAD in a command line environment|command line]], via the use of ''-D variable=value'' option. OpenSCAD currently places that assignment at the end of the source code, and thus must allow a variable's value to be changed for this purpose.

Values cannot be modified during run time; all variables are effectively constants that do not change once created. Each variable retains its last assigned value, in line with [[w:Functional programming|Functional]] programming languages. Unlike [[w:Imperative programming|Imperative]] languages, such as C, OpenSCAD is not an iterative language, and therefore the concept of ''x''&amp;nbsp;=&amp;nbsp;''x''&amp;nbsp;+&amp;nbsp;1 is not valid. The only way an expression like this works in OpenSCAD is if the ''x'' on the right comes from a parent scope, such as an argument list, and the assignment operation creates a new ''x'' in the current scope. Understanding this concept leads to understanding the beauty of OpenSCAD.

'''Before version 2015.03''', it was not possible to do assignments at any place except the file top-level and module top-level. Inside an ''if/else''&amp;nbsp; or ''for''&amp;nbsp; loop, assign() was needed.

'''Since version 2015.03''', variables can now be assigned in any scope. Note that assignments are valid only within the scope in which they are defined - it is still not possible to leak values to an outer scope. See [[#Scope of variables|Scope of variables]] for more details.

 a=0;
 if (a==0) 
   {
  a=1; //  before 2015.03 this line would generate a Compile Error
       //  since 2015.03  no longer an error, but the value a=1 is confined to within the braces {}
   }
=== Undefined variable ===

Referring to a variable that has not had a value assigned triggers a warning, and yields the special value '''undef'''.

It is possible to detect an undefined variable using `is_undef(var)`.  (Note that the simpler `var == undef` will yield a warning.)

  '''Example'''
   
  echo(&quot;Variable a is &quot;, a);                // Variable a is undef, triggers a warning
  if (is_undef(a)) {                        // does not trigger a warning
    echo(&quot;Variable a is tested undefined&quot;); // Variable a is tested undefined
  }

=== Scope of variables ===

When operators such as translate() and color() need to encompass more than one action ( actions end in ;), braces {} are needed to group the actions, creating a new, inner scope.
When there is only one semicolon, braces are usually optional.

Each pair of braces grouping &quot;children&quot; of an operator creates a new scope inside the scope where they were used. '''Since 2015.03''', new variables can be created within this new scope. New values can be given to variables that were created in an outer scope. 
These variables and their values are also available to further inner scopes created within this scope, but are '''not available''' to anything outside this scope. Variables still have only the last value assigned within a scope.

                        // scope 1
  a = 6;                // create a
  echo(a,b);            //                6, undef
  translate([5,0,0]){   // scope 1.1
    a= 10;
    b= 16;              // create b
    echo(a,b);          //              100, 16   a=10; was overridden by later a=100;
    color(&quot;blue&quot;) {     // scope 1.1.1
      echo(a,b);        //              100, 20
      cube();
      b=20;
    }                   // back to 1.1
    echo(a,b);          //              100, 16
    a=100;              // override a in 1.1
  }                     // back to 1   
  echo(a,b);            //                6, undef
  color(&quot;red&quot;){         // scope 1.2
    cube();
    echo(a,b);          //                6, undef
  }                     // back to 1
  echo(a,b);            //                6, undef
   
  //In this example, scopes 1 and 1.1 are outer scopes to 1.1.1 but 1.2 is not.

: '''Anonymous scopes''' are not considered scopes:
  {
    angle = 45;
  }
  rotate(angle) square(10);

For() loops are not an exception to the rule about variables having only one value within a scope. A copy of loop contents is created for each pass. Each pass is given its own scope, allowing any variables to have unique values for that pass. No, you still can't do a = a + 1.

=== Variables cannot be changed ===

The simplest description of OpenSCAD variables is that an assignment creates a new variable in the current scope, and that it's not legal to set a variable that has already been set in the current scope.  In a lot of ways, it's best to think of them as named constants, calculated on entry to the scope, but there's a catch.  If you set a variable twice in the same scope, the second assignment triggers a warning (which may abort the program, depending on preferences settings).  It does *not* then replace the value of the variable - rather, it replaces the original assignment, at its position in the list of assignments.  The original assignment is never executed.

 a = 1;   // never executed
 echo(a); // 2
 a = 2;   // executed at the position of the original assignment
 echo(a); // 2

That's still not the complete story.  There are two special cases that do not trigger warnings:
* if the first assignment is in the top level of an `include` file, and the second assignment is in the including file.
* If the first assignment is in the top level of the program source, and the second assignment comes from a `-D` option or from the Customizer.

While this appears to be counter-intuitive, it allows you to do some interesting things: for instance, if you set up your shared library files to have default values defined as variables at their root level, when you include that file in your own code you can 're-define' or override those constants by simply assigning a new value to them - and other variables based on that variable are based on the value from the main program.

 // main.scad
 include &lt;lib.scad&gt;
 a = 2;
 echo(b);

 // lib.scad
 a = 1;
 b = a + 1;

will produce 3.

=== Special variables ===
Special variables provide an alternate means of passing arguments to modules and functions.
All variables starting with a '$' are special variables, similar to special variables in lisp.
As such they are more dynamic than regular variables.
(for more details see [[OpenSCAD_User_Manual/Other_Language_Features#Special_variables|Other Language Features]])

== Vectors ==

A vector or list is a sequence of zero or more OpenSCAD values. Vectors are collections of numeric or boolean values, variables, vectors, strings or any combination thereof. They can also be expressions that evaluate to one of these. Vectors handle the role of arrays found in many imperative languages.
The information here also applies to lists and tables that use vectors for their data.
   
A vector has square brackets, [] enclosing zero or more items (elements or members), separated by commas. A vector can contain vectors, which can contain vectors, etc.

'''Examples'''

    [1,2,3]
    [a,5,b]
    []
    [5.643]
    [&quot;a&quot;,&quot;b&quot;,&quot;string&quot;]
    [[1,r],[x,y,z,4,5]]
    [3, 5, [6,7], [[8,9],[10,[11,12],13], c, &quot;string&quot;]
    [4/3, 6*1.5, cos(60)]

use in OpenSCAD:
   cube( [width,depth,height] );           // optional spaces shown for clarity
   translate( [x,y,z] )
   polygon( [ [x&lt;sub&gt;0&lt;/sub&gt;,y&lt;sub&gt;0&lt;/sub&gt;],  [x&lt;sub&gt;1&lt;/sub&gt;,y&lt;sub&gt;1&lt;/sub&gt;],  [x&lt;sub&gt;2&lt;/sub&gt;,y&lt;sub&gt;2&lt;/sub&gt;] ] );

=== Creation ===

Vectors are created by writing the list of elements, separated by commas, and enclosed in square brackets. Variables are replaced by their values.
   cube([10,15,20]);
   a1 = [1,2,3];
   a2 = [4,5];
   a3 = [6,7,8,9];
   b  = [a1,a2,a3];    // [ [1,2,3], [4,5], [6,7,8,9] ]  note increased nesting depth
Vectors can be initialized using a for loop enclosed in square brackets. 

The following example initializes the vector ''result'' with a length ''n'' of 10 values to the value of ''a''.&lt;syntaxhighlight lang=&quot;openscad&quot;&gt;n = 10;
a = 0;

result = [ for (i=[0:n-1]) a ];
echo(result); //ECHO: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]&lt;/syntaxhighlight&gt;The following example shows a vector ''result'' with a ''n'' length of 10 initialized with values that are alternatively ''a'' or ''b'' respectively if the index position ''i'' is an even or an odd number.&lt;syntaxhighlight&gt;
n = 10;
a = 0;
b = 1;
result = [ for (i=[0:n-1]) (i % 2 == 0) ? a : b ];
echo(result); //ECHO: [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
&lt;/syntaxhighlight&gt;

=== Indexing elements within vectors ===
 
Elements within vectors are numbered from 0 to n - 1 where n is the length returned by [[#len|len()]].
Address elements within vectors with the following notation:
 e[5]           // element no 5 (sixth) at   1st nesting level
 e[5][2]        // element 2 of element 5    2nd nesting level
 e[5][2][0]     // element 0 of 2 of 5       3rd nesting level
 e[5][2][0][1]  // element 1 of 0 of 2 of 5  4th nesting level

&lt;div id= &quot;example1&quot;&gt; example elements with lengths from len()&lt;/div&gt;
 e = [ [1], [], [3,4,5], &quot;string&quot;, &quot;x&quot;, [[10,11],[12,13,14],[[15,16],[17]]] ];  // length 6
 
 address       length  element
 e[0]          1       [1]
 e[1]          0       []
 e[5]          3       [ [10,11], [12,13,14], [[15,16],[17]] ]
 e[5][1]       3       [ 12, 13, 14 ]
 e[5][2]       2       [ [15,16], [17] ]
 e[5][2][0]    2       [ 15, 16 ]
 e[5][2][0][1] undef   16
     
 e[3]          6       &quot;string&quot;
 e[3 ][2]      1       &quot;r&quot;
   
 s = [2,0,5]; a = 2;
 s[a]          undef   5
 e[s[a]]       3       [ [10,11], [12,13,14], [[15,16],[17]] ]

==== String indexing ====

The elements (characters) of a string can be accessed:
 &quot;string&quot;[2]    //resolves to &quot;r&quot;
 

==== Dot notation indexing ====

The first three elements of a vector can be accessed with an alternate dot notation:
 e.x    //equivalent to e[0]
 e.y    //equivalent to e[1]
 e.z    //equivalent to e[2]

=== Vector operators ===

==== concat ====
{{requires|2015.03}}
  
&lt;code&gt;concat()&lt;/code&gt; combines the elements of 2 or more vectors into a single vector. No change in nesting level is made.
  vector1 = [1,2,3]; vector2 = [4]; vector3 = [5,6];
  new_vector = concat(vector1, vector2, vector3); // [1,2,3,4,5,6]
   
  string_vector = concat(&quot;abc&quot;,&quot;def&quot;);                 // [&quot;abc&quot;, &quot;def&quot;]
  one_string = str(string_vector[0],string_vector[1]); // &quot;abcdef&quot;

==== len ====
&lt;code&gt;len()&lt;/code&gt; is a function that returns the length of vectors or strings. 
Indices of elements are from [0] to [length-1].
: '''vector'''
:: Returns the number of elements at this level.
:: Single values, which are '''not''' vectors, raise an error.
: '''string'''
:: Returns the number of characters in a string.
  a = [1,2,3]; echo(len(a));   //  3

[[#example1|See example elements with lengths]]

=== Matrix ===

A matrix is a vector of vectors.

 Example that defines a 2D rotation matrix
 mr = [
      [cos(angle), -sin(angle)],
      [sin(angle),  cos(angle)]
     ];

== Objects ==

{{requires|Development snapshot}}

Objects store collections of data, like vectors, but the individual members are accessed by string names rather than by numeric indexes.  They are analogous to JavaScript objects or Python dictionaries.

=== Creating an object ===

The &lt;code&gt;object()&lt;/code&gt; function can be used to create objects.  &lt;code&gt;object()&lt;/code&gt; processes its arguments left to right, adding them to the object under construction.  It accepts three different styles of argument, which can be mixed as desired:

The first form is this:

    &lt;name&gt;=&lt;value&gt;

This form is convenient for cases where the name is constant and suitable for use as an identifier.

The second form is this:

    [
        [ &lt;name-string&gt;, &lt;value&gt; ],
        ...
    ]

This form allows calculated names and names that are not suitable as identifiers. It also allows lists of members that are derived from other processing - from a list comprehension, perhaps.

The third form is this:

    &lt;object&gt;

If an argument is itself an object, its members are added to the object under construction.

If a member is mentioned more than once, the last mention wins.  Thus you can copy an object, replacing one member, using &lt;code&gt;object(obj, name=newvalue)&lt;/code&gt;.

You can remove a member from the object under construction using the list form, supplying only a name and no value - e.g., &lt;code&gt;object(obj, [ [ &quot;unwanted&quot; ] ])&lt;/code&gt;

=== Retrieving a value from an object ===

    obj.name

Retrieves the named value from the object, for a name that is constant and syntactically suitable for use as an identifier.

    obj[&quot;name&quot;]

Retrieves the named value from the object, for a name that is an arbitrary string expression.

Note that members with identifier-like names can be accessed using either mechanism.  The choice depends on the particular use case.

If the specified member is not present in the object, yields &lt;code&gt;undef&lt;/code&gt;.

=== Iterating over object members ===

   for (name = obj) { ... }

iterates over the members of the object, in an unspecified order, setting &lt;code&gt;name&lt;/code&gt; to the name of each member.  It is then typically desirable to access the value using &lt;code&gt;obj[name]&lt;/code&gt;.

This construct works for [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#For_Loop|flow-control &lt;code&gt;for&lt;/code&gt;]], [[OpenSCAD_User_Manual/Conditional_and_Iterator_Functions#Intersection_For_Loop|intersection_for()]], and [[OpenSCAD_User_Manual/List_Comprehensions#for|list-comprehension &lt;code&gt;for&lt;/code&gt;]].

=== Examples ===

    // Create an object with a few members.
    o1 = object(a = 1, b = &quot;hello&quot;, c = true);

    // Retrieve a member using a constant name.
    echo(o1.b);      // &quot;hello&quot;

    // Retrieve members using a name that varies.
    for (k = [ &quot;a&quot;, &quot;b&quot; ])
        echo(o1[k]);    // 1; &quot;hello&quot;

    // Copy the object, changing &quot;c&quot;, adding &quot;d&quot;, and removing &quot;a&quot;.
    o2 = object(o1, c = false, d = &quot;dog&quot;, [[&quot;a&quot;]]);
    echo(o2);        // { b = &quot;hello&quot;; c = false; d = &quot;dog&quot;; }

    // Create an object, calculating the names to use.
    o3 = object([
        for (i = [1:3])
            [ str(&quot;x&quot;, i), i*i ]
    ]);
    echo(o3);        // { x1 = 1; x2 = 4; x3 = 9; }

    // Merge the three objects together.
    o4 = object(o1, o2, o3);
    // Separately print each member of the result
    for (name = o4)
        echo(name = name, value = o4[name]);
    // name = &quot;a&quot;, value = 1
    // name = &quot;b&quot;, value = &quot;hello&quot;
    // name = &quot;c&quot;, value = false
    // name = &quot;d&quot;, value = &quot;dog&quot;
    // name = &quot;x1&quot;, value = 1
    // name = &quot;x2&quot;, value = 4
    // name = &quot;x3&quot;, value = 9

== Getting input ==

There is no mechanism for variable input from keyboard or reading from arbitrary files.  There is no prompting mechanism, no input window, or input fields or any way to manually enter data while the script is running.

Variables can be set via:
* assignments in the script
* [[OpenSCAD_User_Manual/Customizer|the Customizer]]
* &lt;code&gt;-D&lt;/code&gt; at the [[OpenSCAD_User_Manual/Using_OpenSCAD_in_a_command_line_environment|command line interface]]
* accessing data in a few file formats (stl, dxf, png, etc).

With the exception of DXF files, data from files is not accessible to the script, although to a limited extent the script may be able to manipulate the data as a whole.  For example, STL files can be rendered in OpenSCAD, translated, clipped, etc.  But the internal data that constitutes the STL file is inaccessible.

=== Getting a point from a drawing ===

Getting a point is useful for reading an origin point in a 2D view in a technical drawing. The function dxf_cross reads the intersection of two lines on a layer you specify and returns the intersection point. This means that the point must be given with two lines in the DXF file, and not a point entity.

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
OriginPoint = dxf_cross(file=&quot;drawing.dxf&quot;, layer=&quot;SCAD.Origin&quot;, 
                        origin=[0, 0], scale=1);
&lt;/syntaxhighlight&gt;

=== Getting a dimension value ===

You can read dimensions from a technical drawing. This can be useful to read a rotation angle, an extrusion height, or spacing between parts. In the drawing, create a dimension that does not show the dimension value, but an identifier. To read the value, you specify this identifier from your program:

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
TotalWidth = dxf_dim(file=&quot;drawing.dxf&quot;, name=&quot;TotalWidth&quot;,
                        layer=&quot;SCAD.Origin&quot;, origin=[0, 0], scale=1);
&lt;/syntaxhighlight&gt;

For a nice example of both functions, see Example009 and the image on the [http://www.openscad.org/ homepage of OpenSCAD].

{{BookCat}}</text>
      <sha1>n2floxih36nfpu0ac0cqvzk4v3j589k</sha1>
    </revision>
  </page>
  <page>
    <title>Lentis</title>
    <ns>0</ns>
    <id>238420</id>
    <revision>
      <id>4631243</id>
      <parentid>4629647</parentid>
      <timestamp>2026-04-18T15:55:40Z</timestamp>
      <contributor>
        <username>Daask</username>
        <id>2788005</id>
      </contributor>
      <comment>Link to Wikipedia articles on various meanings of &quot;Lens&quot;.</comment>
      <origin>4631243</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20285" sha1="hu87x7k3jdd96awl7k0x58kvae5frs1" xml:space="preserve">__NOTOC__
[[Image:UVa Rotunda.jpg|thumb|196px|right|The Rotunda at The University of Virginia]]

''Lentis: The Social Interface of Technology'' is a guidebook to the realm where technological phenomena and social phenomena intersect. If we think of technology and society as circular domains that overlap, the common domain they share is a [[w:Lens (geometry)|lens]] in shape. Hence the short title of the book, ''Lentis,'' which is Latin for &quot;of [or about] the lens.&quot; If the title (with its association with [[w:Lens|lenses]]) also suggests means of viewing, of examining, of magnifying, and of discovering, so much the better. The lens-shaped realm is called the &quot;social interface of technology.&quot;

The chief authors of ''Lentis'' are students at the University of Virginia's School of Engineering and Applied Science. The authors are engineers representing diverse fields of engineering.

As a wikibook, ''Lentis'' will accept contributions from authors and editors all over the world, but the student authors will take particular responsibility to produce a complete, well documented, well written and useful book. This is a student project. Until December 20, 2024, would-be contributors who are not students in the class are asked to consider editing sparingly, but are invited to comment freely on discussion pages, where their suggestions and advice will be welcomed and appreciated. No one's right to edit is in question.

''Lentis'' is intended to serve a general purpose and a specific purpose. The general purpose is to present to interested readers worldwide illuminating cases with practical lessons for those who navigate the dangerous channels of the social interface of technology. The book begins with the premise that success in technological and social endeavors often depends upon the skillful negotiation of sociotechnical factors, where technological techniques alone, or social techniques alone, are insufficient. A second premise is that case studies offer generalizable lessons that can guide people who work where technology and society overlap. They are, in effect, &quot;true fables&quot; that offer &quot;morals&quot; of practical value in diverse endeavors.

More specifically, ''Lentis'' is a book written by and for engineers. Here the premise is that engineers by definition are problem solvers whose instruments may include social as well as technological tools, whose work ultimately serves non-engineers, and who must therefore inevitably venture into the social interface of technology, where these non-engineers dwell. Too often, engineers have had to leave this territory to managers, policymakers, clients and others who lack the technical expertise for success in this zone. If engineers can develop the social expertise they need at the social interface of technology, they can lead there.

If &quot;those who cannot remember the past are condemned to repeat it,&quot; then those who recover the past can best lead us out of it. In the history of technology countless technological innovations succeeded until they met the social interface, where social phenomena interact with technological phenomena in surprising ways. This book will be a success if it helps engineers anticipate these effects.

Most of the chapters in ''Lentis'' are examinations of cases. The authors will attempt to derive practical lessons from these cases; the most valuable lessons will be generalizable. If a lesson is generalizable, it is applicable in cases and situations that may be far removed in time, space or engineering field. A case from American transportation engineering in the 1990s, for example, may have lessons useful to biomedical engineers in 2024. The authors have endeavored to find such lessons in the cases they investigated. Because social theories are also useful navigational aids in the social interface of technology, some chapters examine such theories. The authors have sought not only to explain these theories, but to show how they can be of practical value.

== Table of Contents ==

=== '''Preliminaries''' ===
* [[/Chapters: Active and Candidates/]]

=== '''Lentis: The Social Interface of Technology''' ===

==== Food and Energy ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[Lentis/Public Health, Sugary Drinks, and the US Beverage Lobby|Public Health, Sugary Drinks, and the US Beverage Lobby]]

* [[/Biofuels Vs. Food in Developing Countries/]]
* [[/Politics of Biofuels/]]
* [[/Opposition to GMOs in Europe/]]
* [[/Patenting of GM Seeds/]]
* [[/Corn, Beef and Feedlots/]]
* [[/Dakota Access Pipeline/]]
* [[/High-Fructose Corn Syrup/]]
* [[/The Organic Foods Movement/]]
* [[/Local Food as a Case of Disintermediation/]]
* [[/Local Food as a Social Movement/]]
* [[/Marketing of Natural Foods/]]
* [[/Corn Ethanol in the United States/]]
* [[/Popular Perceptions of Nuclear Power/]]
* [[/Nuclear Meltdown: Is Nuclear Energy Socially Viable Following the 2011 Japanese Earthquake?/]]
* [[/Fracking/]]
* [[/Wind Energy/]] 
* [[/Rare Earth Metals/]]
* [[/Carbon Offsets/]]
* [[/Clean Coal/]]
* [[/Food Waste in the United States/]]
* [[/Genetically Modified Food Controversy in the United States/]]
*[[/Solar Energy Policy in Germany/]]
*[[/Peak Oil/]]
*[[/U.S. Arctic Oil Mining/]]
*[[/How Energy Companies Rebrand Themselves/]]
* [[/Gluten-Free: Nutritional Principle or Social Value/]]
*[[/Vegan and Vegetarian Diets: Nutritional and Social Values/]]
*[[/Energy from Trash/]]
*[[/Life Off the Grid/]]
* [[/Soylent/]]
* [[/Expansion of Solar Farms in the Rural United States/]]
* [[/Urban Farming/]]
* [[/Oil Palm Plantations/]]
* [[/Golden Rice/]]
* [[/Miracle Rice/]]
*[[/The Cavendish Banana, Monoculture, and Blight/]]
*[[/Atlantic Coast Pipeline/]]
* [[/Cooking with Wood Fuel/]]
* [[Lentis/Solar Panel Recycling in the United States|Solar Panel Recycling in the United States]]
* [[Lentis/Line 3 Pipeline Controversy|Line 3 Pipeline Controversy]]
* [[/Data Centers and Energy/]]
* [[/Conflicts of Interest in US Nutrition Research/]]
&lt;/div&gt;

==== Environmental Values and Climate Change ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[Lentis/8 House|8 House]]
* [[Lentis/Plastic Bags|Plastic Bags]]
* [[Lentis/Water Bottles|Water Bottles]]
* [[Lentis/Competition for Water in California|Competition for Water in California]]
* [[Lentis/Green Roofing|Green Roofing]]
* [[Lentis/Hypoxic Zones|Hypoxic Zones]]
* [[Lentis/Unnatural Selection: Explaining Strange Pet Breeds|Unnatural Selection: Explaining Strange Pet Breeds]]
* [[Lentis/Masdar City|Masdar City]]
* [[Lentis/World Trade as an Invasive Species Vector|World Trade as an Invasive Species Vector]]
* [[Lentis/Noise pollution|Noise pollution]]
* [[Lentis/Ecovillages|Ecovillages]]
* [[Lentis/Climate Change Denial|Climate Change Denial]]
* [[Lentis/Lawn Care in America: Intensive Agriculture, No Harvest|Lawn Care in America: Intensive Agriculture, No Harvest]]
* [[Lentis/Marine Waste|Marine Waste]]
* [[Lentis/Gold, Mercury, and Madre de Dios, Peru|Gold, Mercury, and Madre de Dios, Peru]]
* [[Lentis/The Amazon Basin Fires of 2019|The Amazon Basin Fires of 2019]]
* [[Lentis/Lithium-Ion Batteries in Electric Vehicles|Lithium-Ion Batteries in Electric Vehicles]]
* [[Lentis/BedZED|BedZED]]
* [[Lentis/Small Island Countries and Sea Level Rise|Small Island Countries and Sea Level Rise]]
* [[Lentis/Light pollution|Light pollution]]
* [[Lentis/The 2020 Western Wildfire Season in the U.S.|The 2020 Western Wildfire Season in the U.S.]]
* [[Lentis/les Zadistes|les Zadistes]]
*[[/Flight Shaming/]]
*[[/Ecological Implications of Commercial Marine Fishing/]]
*[[/Zero-Plastic Retailing/]]
*[[/The PFAS Controversy/]]
&lt;/div&gt;

==== Health and Medicine ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[AI &amp; Medical Imaging]]
* [[Antimicrobial Agents in Consumer Products]]
* [[Mental Health as a Pharmacological Growth Market]]
* [[/Nicotine Addictions/]]
* [[/Thinking Small: Appropriate Technology for Developing Countries/]]
* [[/Water Supply, Sanitation, and Public Health in Haiti/]]
* [[/Fluoridation/]]
* [[/Medicine and Disgust/]]
* [[/Popular Hygiene: Perceptions and Practices/]]
* [[/Bedside Manner in the High-Tech Hospital/]]
* [[/Technology and Quality of Life for the Terminally Ill/]]
* [[/Ellie, the Microsoft Kinect, and Psychotherapy/]]
* [[/Placebos/]]
* [[/Baby Formula/]]
* [[/Sick Building Syndrome/]]
* [[/Football and Concussions/]]
* [[/The Dietary and Bodybuilding Supplement Industry in the United States  /|The Dietary and Bodybuilding Supplement Industry in the United States]]
* [[/Obesity and Diets in Economic Classes in the United States/]]
* [[/Steroids and Baseball/]]
* [[/Nanotechnology and Health/]]
* [[/Dissociative Identity Disorder (Multiple Personality Disorder)/]]
* [[/Malaria and Mosquito Nets/]]
* [[/International Air Travel as a Disease Vector/]]
* [[/Social Resistance to Vaccination: Thiomersal and Autism/]]
* [[/Religious Opposition to Vaccination/]]
* [[/Physician-Assisted Suicide/]]
* [[/Power Balance, Magnetic Bracelets and Other Strange Cures/]]
* [[/The D.A.R.E. Program/]]
* [[/Social Obstacles to Public Health in Developing Countries/]]
* [[/Athletes, Superstition, and Performance/]]
* [[/Gattaca Revisited/]]
* [[/Artificial Wombs/]]
* [[/The Weight Loss Industry in the United States/]]
* [[/Direct-to-Consumer Personal Genomics/]]
* [[/Vaping/]]
* [[/Neuroprosthetics/]]
* [[/Detoxing as a Social Phenomenon/]]
* [[/Antibiotics in India/]]
* [[/Public Health: Fear Appeals vs Self-Efficacy and Social Norms Campaigns/]]
* [[/Public Health Responds to Physical Inactivity/]]
* [[/Mobility and Access for the Disabled/]]
* [[/Power Lines and Public Health/]]
* [[/The HPV Vaccine/]]
* [[/Medication Overload/]]
* [[/The 2020 Pandemic Response in Italy/]]
* [[/Healthcare in U.S. Prisons/]]
* [[/Antimaskers in the U.S. during the 2020 Pandemic/]]
*[[/Pain Scales/]]
*[[/Augmented Reality in Medicine/]]
*[[/COVID-19 Vaccine Distribution/]]
*[[/Chloramination of Drinking Water/]]
*[[/Manual Water Collection in Developing Countries/]]
*[[/The U.S. Pandemic Response: Influenza, 1918-1919/]]
*[[Lentis/Robotic Pets for Psychosocial Therapeutics|Robotic Pets for Psychosocial Therapeutics]]
*[[Lentis/Caffeine Addiction|Caffeine Addiction]]
*[[Lentis/The Cultural Politics of Obesity Drugs in the US|The Cultural Politics of Obesity Drugs in the US]]
*[[Lentis/Pharma’s influence in FDA|Pharma’s influence in FDA]]
&lt;/div&gt;


==== Mobility and Land Use ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[/Cascadia Earthquake Preparation/]]
* [[/Bicyclists in Cities/]]
* [[/Drivers’ and Bicyclists’ Perceptions of Each Other/]]
* [[/American Automobility and the Car Counter-Culture/]]
* [[/Congestion Pricing/]]
* [[/Urban Sprawl/]]
* [[/Planned Communities/]]
* [[/Slugging/]]
* [[/Bicycling in the Netherlands/]]
* [[/Tata Nano and Mobility in India/]]
* [[/How Cars Became Dining Rooms: Drive-Thrus, Cupholders and American Culture/]]
* [[/Autonomous Vehicles/]]
* [[/The Disappearing American Streetcar/]]
* [[/Pedestrians and Walkability in Cities and Suburbs/]]
* [[/Real-time Ridesharing/]]
* [[/Hitchhiking in the Digital Age /]]
* [[/Arcology/]]
* [[/Lowriding/]]
* [[/California High Speed Rail/]]
* [[/Self-Driving Cars/]]
* [[/The Future of U.S. Civil Aviation in 1945/]]
* [[/Road Rage/]]
* [[/The Ogallala Aquifer/]]
* [[/Rail in America/]]
* [[/The Belt and Road Initiative/]]
* [[/Car Dependency in the U.S./]]
* [[/Guerrilla Urbanism/]]
*[[/The Transformation of Times Square/]]
*[[/David Engwicht and Street Reclaiming/]]
*[[/Vision Zero/]]
*[[/Shared Space and Woonerven/]]
*[[/Eyjafjallajökull 2010/]]
*[[Lentis/Driving Speed Enforcement|Driving Speed Enforcement]]
*[[/E-bikes and Personal Mobility/]]
*[[Lentis/Zoning Laws in the United States|Zoning Laws in the United States]]
*[[Lentis/Carpooling|Carpooling]]
*[[The Politics of Electric Vehicle Subsidies]]
*[[Lentis/Freeway Removal Movements in US Cities|Freeway Removal Movements in US Cities]]
*[[/New Urbanism/]]
&lt;/div&gt;

==== Computers and the Internet ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[/Antipiracy/]]
* [[/Amazon and the Ecommerce Evolution/]]
* [[/Compulsive Connectivity/]]
* [[/Screen-Free Child Rearing/]]
* [[/Crowdsourcing Higher Education/]]
* [[/Cryptocurrency/]] 
* [[/&quot;Data is the new oil&quot;/]]
* [[/Deepfakes/]]
* [[/Fake Users/]]
* [[/Hacker Culture/]]
* [[/Human Flesh Search Engine/]]
* [[/Social Engineering/]]
* [[/Internet Memes/]]
* [[/Internet Subcultures/]]
* [[/The Open-Source Movement/]]
* [[/Electronic Voting/]]
* [[/Online Consumer Reviews/]]
* [[/Online Dating Scams/]]
* [[/Online Shopping/]]
* [[/Online Reputation Management/]]
* [[/Online Recruitment by Extremist Groups/]]
* [[/Peer-to-Peer Media Sharing/]]
* [[/Program and High Frequency Trading/]]
* [[/Social Networks/]]
* [[/Social Media and the Arab Spring/]]
* [[/Social Norms in Virtual Worlds/]]
* [[/Software Journalism: When Programs Write the News/]]
* [[/Street View/]]
* [[/Second Life/]]
* [[/User-Generated Content in the Internet Age/]]
* [[/password1234: Internet Security and Password Culture/]]
* [[/Reddit: Anonymity and Social Norms/]]
* [[/Wikipedia/]]
* [[/Cyber-Attacks on Cyber-Physical Systems/]]
* [[/Cyberterrorism and Cyberwarfare/]]
* [[/Mass Collaboration/]]
* [[/Net Neutrality/]]
* [[/Mass Control of a Single Gamer/]]
* [[/Harmonious Society: Internet Censorship in China/]]
* [[/Web Induced Risk Taking/]]
* [[/Where It Goes: Electronic Waste and Salvage/]]
* [[/Working Conditions at Apple Hardware Factories in China/]]
* [[/Facebook Cheating/]]
* [[/Intellectual Property in the Internet Age/]]
* [[/The Social Psychology of YouTube/]]
* [[/Learning from a Distance/]]
* [[/Communication Technology and Interpersonal Relationships/]]
* [[/Identity Theft/]]
* [[/Higher Education Online/]]
* [[/The Culture of Instagram/]]
* [[/New Media and the United States Presidential Election of 2008/]]
* [[/Targeted Advertising/]]
* [[/Viral Marketing/]]
* [[/Web Tracking/]]
* [[/Internet Anonymity/]]
* [[/The Culture of Snapchat/]]
* [[/Snopes, PolitiFact, and Other Fact-Checking Websites/]]
* [[/Twitter and other social networks in the Iranian protests of 2009/]]
* [[/The Internet Strategy of White Supremacists/]]
* [[/Google Translate|Google Translate]]
* [[/The Deep Web/]]
* [[/Social Media Mining/]]
* [[/Internet Witch Hunts/]]
* [[/News Echo Chambers/]]
* [[/Featuritis/]]
* [[/Content Moderation/]]
* [[/Virtual Reality/]]
*
* [[/Social Media Shaming Campaigns/]]
* [[Lentis/The Geopolitics of TikTok|The Geopolitics of TikTok]]
* [[/AI Music, Creativity, and Intellectual Property/]]
&lt;/div&gt;

==== Portable Electronics ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[Lentis/Smartphones and Cognitive Offloading|Smartphones and Cognitive Offloading]]
* [[Cell Phones and Cancer in Britain]]
* [[/Driving while Texting/]]
* [[/GPS and Driving/]]
* [[/Sociology of Texting/]]
* [[/The Text Effect/]]
* [[/Norms of Handheld Device Use/]]
* [[/Happy Slapping/]]
* [[/The Walkman Effect/]]
* [[/Electronically Enabled Test Cheating/]]
* [[/Cell Phones versus Face-to-Face Interaction/]]
* [[/Children and Cell Phones/]]
* [[/Amazon, E-readers and the Future of the Publishing Industry/]]
* [[/Social Aspects of Cell Phone Cameras/]]
* [[/Airline Passengers and Portable Electronics/]]
* [[/Pokémon Go/]]
* [[/Phone Cinematography/]]
* [[/Cell Phones in Developing Countries/]]
* [[/Wearable Activity Trackers/]]
* [[/Handheld Electronics in South Korean Society/]]
* [[/The Looking Glass/]]
&lt;/div&gt;

==== Entertainment and Media ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[/Massively Multiplayer Online Role-Playing Games/]]
* [[/Gambling/]]
* [[/Game Addictions/]]
* [[/The Psychology and Technology of Game Immersion/]]
* [[/The Proliferation of Music Production Capability/]]
* [[/Grand Theft Auto: Violent Video Games and Controversy/]]
* [[/Doom: Violent Video Games and Controversy/]]
* [[/Implementation of Technology in Sports: Historical Successes and Failures, and Modern Discussion/]]
* [[/Portrayal of Women in Video Games/]]
* [[/Children,Video Games and Obesity/]]
* [[/Electronic Sports (eSports)/]]
* [[/Electronic Music Popular/]]
* [[/From Cronkite to Stewart: TV News during and after Network Hegemony/]]
* [[/The Impact of Fans on Technological Innovation in the NFL/]]
* [[/Gold Farming/]]
* [[/Jackass: Media Driven Risk Propagation/]]
* [[/Anti-TV Social Movements/]]
* [[/Media Format Wars/]]
* [[/Microtransactions in Videogames/]]
* [[/Moe Anthropomorphism/]] 
* [[/Hello Kitty: Identity Crisis, Kawaii Culture, and More/]] 
* [[/Technology and Conventional Norms of Personal Beauty/]]
* [[/Dance Dance Revolution/]]
* [[/Super Smash Bros./]]
* [[/Twitch/]]
* [[/Instant Replay in International Soccer/]]
* [[/Among Us: Social Behavior in a Virtual World/]]
* [[Lentis/TikTok|TikTok]]
&lt;/div&gt;

==== Security, Official Violence, Freedom, Privacy ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[/JEDI Cloud/]]
* [[/Digital Rights Management/]]
* [[/Military Industrial Complex/]]
* [[/Tasers and Stun Guns/]]
* [[/Probation Technology/]]
* [[/International Drug Trafficking and Law Enforcement/]]
* [[/Air Travel Security/]]
* [[/The United States - Mexico Border/]]
* [[/Recording Police Activity/]]
* [[/Video Surveillance/]]
* [[/Shopkeepers and Shoplifters: Technology and the Changing Balance of Power/]]
* [[/Cell Phones in Prison/]]
* [[/Cell Phone Jamming in the United States/]]
* [[/Freedom of Information: WikiLeaks/]]
* [[/Playing Games at Work: Employees versus Employers, Surveillance and Stealth/]]
* [[/Cyberslacking/]]
* [[/Mashups and Remixes: Between Creativity and Theft/]]
* [[/Video Surveillance in Great Britain/]]
* [[/Technology and Incarceration in the United States/]]
* [[/Additive Manufacturing/]]
* [[/Law Enforcement Access to Encrypted Data/]]
* [[/Amateurs with Drones/]]
* [[/Body Cameras/]]
* [[/Human Terrain System: Military Meets Cultural Mindfulness/]]
* [[/Law Enforcement and Social Media/]]
* [[/Cyber-attack Attribution/]]
* [[/China’s Social Credit System/]]
* [[/Technology in the 2019 Hong Kong Protests/]]
* [[/8chan/]]
* [[/Lockheed Martin F-35/]]
*[[/Capital Punishment in the United States/]]
* [[/The Yellow Vests Movement/]]
* [[Lentis/The 2018 U.S. Prison Strike|The 2018 U.S. Prison Strike]]
* [[Lentis/Office Productivity in the Changing Workplace|Office Productivity in the Changing Workplace]]
* [[Lentis/The Geopolitics of Asymmetric War: The Case of Ukraine|The Geopolitics of Asymmetric War: The Case of Ukraine]]
&lt;/div&gt;

==== Systemic Racism in the U.S. ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[/Gentrification/]]
* [[/Predictive Policing/]]
*[[The Prison-Industrial Complex]]
*[[/The War on Drugs/]]
*[[Lentis/Algorithmic Bias|Algorithmic Bias]]
*[[Lentis/The School to Prison Pipeline|The School to Prison Pipeline]]
*[[Lentis/TheHBCURenaissance|The HBCU Renaissance]]
&lt;/div&gt;

==== Technology and Gender ====
* [[Lentis/Algorithmic_bias_by_gender|Algorithmic Bias by Gender]]

==== History of Technology ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[The Decline of Public Transport in the U.S., 1945-1975]]

* [[/The Pill, the Vatican, and American Catholics/]]
* [[/Education and the Space Race in the United States/]]
* [[/Technology, Organized Crime, and Law Enforcement in the early 20th-Century United States/]]
* [[/Disease Prevention in the First World War/]]
* [[/Atomic Age Optimism: 1930s - 1960s/]]
* [[/Abortion in America as a Sociotechnical Controversy/]]
* [[/Phreaking/]]
* [[/Rachel Carson, Silent Spring, and the Development of Environmental Values, 1950-1970/]]
*[[/The Legacy of the Donora Smog of 1946/]]
*[[/COINTELPRO: The FBI, Civil Rights, and Domestic Surveillance/]]
&lt;/div&gt;

==== Sociotechnical Theories and Movements ====
&lt;div style=&quot;column-count:3&quot;&gt;
* [[Lentis/Protection Motivation Theory|Protection Motivation Theory]]
* [[/The Singularity/]]
* [[/Conversion to the Metric Standard in the United States/]]
* [[/Disintermediation/]]
* [[/Jevons Paradox/]]
* [[/Path Dependence/]]
* [[/Neoluddism and Technophilia/]]
* [[/Emergent Behavior/]]
* [[/Free Range Kids/|Free Range Kids: Children's Independent Mobility]]
* [[/User Trust/]]
* [[/The Panopticon/]]
* [[/Planned Obsolescence/]]
* [[/Fake News/]]
*[[/Iron Triangles in the U.S. Federal Government/]]
*[[/Risk Compensation/]]
&lt;/div&gt;

{{BookCat}}
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  <page>
    <title>Lentis/Conversion to the Metric Standard in the United States</title>
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      <comment>/* References */ Close references tag. See [[mediawikiwiki:Help:Cite]].</comment>
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      <text bytes="18348" sha1="q75t0xfrjfr5gqtgvizzhyf8aqsa81l" xml:space="preserve">[http://en.wikipedia.org/wiki/Metric_system The Metric System] is a decimalized unit system used near globally.  The United States is the only non-metric industrialized nation. The focus of this chapter is on how the actions of interested parties have effected the adoption. [http://en.wikipedia.org/wiki/History_of_the_metric_system A primer on the global history of the metric system can be found here].

==Political History==
[[File:JAKasson.jpg|John A. Kasson|thumb|150px|right]]
The history of the metric system in the United States can be traced to the Metric Act of 1866, signed by Andrew Johnson, which made legal the use of metric units in commerce. Congressman John Kasson, chairman of the House Committee on Coinage, Weights, and Measures, exemplified the positivity towards the adoption, hoping that &quot;a further act of Congress can fix a date for its exclusive adoption as a legal system&quot;. &lt;ref&gt;{{cite web |url=http://lamar.colostate.edu/~hillger/laws/metric-act-bill.html |title=Text of Metric Act of 1866}} &lt;/ref&gt; In 1875, the United States attended the Metric Convention even though the nation had not converted.  The U.S. did however, sign the treaty, that created the International Bureau of Weights and Measures to regulate the metric system.&lt;ref&gt;{{cite web |url=http://gsi.nist.gov/global/docs/Omnibus.pdf |title=USMA}}&lt;/ref&gt; On April 5, 1893, T.C. Mendenhall, the Superintendent of Weights and Measures, decided that the metric system would become the standard unit system in the U.S. and since then the American unit system has been defined in terms of the metric system.&lt;ref&gt;{{cite web |url=http://gsi.nist.gov/global/docs/Omnibus.pdf |title=USMA}}&lt;/ref&gt; In 1902, a bill supporting full conversion of the Federal government from U.S. Customary to the metric system was introduced to Congress.  The bill lost by a single vote.&lt;ref name = &quot;How soon?&quot;&gt;{{cite web |url=http://gsi.nist.gov/global/docs/Omnibus.pdf |title=Omnibus Trade and Competitiveness Act of 1988}}&lt;/ref&gt; Early metrication efforts were high-minded, but little came of them.

In 1975 the [http://en.wikipedia.org/wiki/94th_United_States_Congress 94th U.S. Congress] passed [http://en.wikipedia.org/wiki/Metric_Conversion_Act The Metric Conversion Act].  This bill created the [http://en.wikipedia.org/wiki/United_States_Metric_Board United States Metric Board] and initiated planning and preparation for [http://en.wikipedia.org/wiki/Metric_system metric system] conversion.  In 1988, President Reagan signed the [http://en.wikipedia.org/wiki/Omnibus_Trade_and_Competitiveness_Act_of_1988 Omnibus Foreign Trade and Competitiveness Act].  This bill required all federal agencies convert, &quot;to the extent economically possible&quot;, to the metric system by the end of the 1992 fiscal year, but disbanded the USMB. &lt;ref name = &quot;Omnibus&quot;&gt;{{cite web |url=http://gsi.nist.gov/global/docs/Omnibus.pdf |title=Omnibus Trade and Competitiveness Act of 1988}} &lt;/ref&gt; In 1991 President [http://en.wikipedia.org/wiki/George_H.W._Bush George H.W. Bush] signed [http://en.wikipedia.org/wiki/Executive_order Executive Order] 12770, reiterating this requirement.  While some agencies did convert, the executive order stated that if conversion could &quot;cause significant inefficiencies or loss of markets to United States firms,&quot; the requirement was waived. Since this time there have been no further requirements for federal agencies or other groups to convert to the metric system. 

Recently, there have even been some political movements away from the metric system. In 2004, the California Department of Transportation initiated movements back to the customary system. In a letter to the department, acting chief engineer Mike Leonardo establishes a &quot;Metric to English Transition Team&quot; to help. He cites a benefits of aligning with commercial partners, in addition to 'long-term financial advantages to the department.' &lt;ref&gt;{{cite web |url=http://www.dot.ca.gov/hq/oppd/design/metric-to-english-transition.pdf |title=Metric to U.S. Customary Units (English) Transition}} &lt;/ref&gt;

==Proponents==
Proponents of U.S. metrication commonly cite the fact that the United States is the only industrialized country in the world that has not yet converted to the metric system.  By converting, it would unify measurement systems around the world and improve efficiency of international trade and commerce.  Currently, some industries in the United States do use the metric system.  These include the medical industry, sciences, and the military.

===Governmental Organizations===
The [http://lamar.colostate.edu/~hillger/ United States Metric Association] was created to encourage the voluntary use and conversion to the metric system in the United States. &lt;ref&gt;&lt;/ref&gt;  Created in 1916 as the American Metric Association, the USMA has been educating people about the metric system and advocating for its use.  It has worked closely with the United States Metric Board and the U.S. government proposing laws and studying metric system usage. &lt;ref name = &quot;USMA History&quot;&gt;{{cite web |url=http://lamar.colostate.edu/~hillger/history.htm |title=A Brief History of the U.S. Metric Association (1916 to the 2000s)}} &lt;/ref&gt; The USMA attempted to personalize losses caused by unit confusions by publishing &quot;Metric Mixups.&quot; &lt;ref name = &quot;mixups&quot;&gt;{{cite web |url=http://lamar.colostate.edu/~hillger/unit-mixups.html|title=Metric Mixups, USMA}}&lt;/ref&gt; The implicit point of compiling the events is that if there were just one system, such accidents would not occur.

The [http://en.wikipedia.org/wiki/United_States_Metric_Board United States Metric Board] was created by the Metric Conversion Act of 1975.  Its role was to facilitate the voluntary metric conversion by government agencies and other organizations.  By 1981 the USMB had produced TV public service announcements (PSAs) and 27 radio PSAs.  These PSAs served to encourage conversion to metric units.

One of the roles of the USMB was to publicize programs and allow interest groups to submit programs and comments on programs.  In 1981 the United States Metric Council submitted to the USMB two metric conversion plans for the shipping and billing of industrial chemicals and for instrument charts, scales, and mounting dimensions.  The USMB met to discuss each of these plans and voted to approve both.  The board passed two resolutions in accordance with these plans and stated in both that, &quot;This endorsement is not a mandate to proceed, since the USMB does not have such powers... The implementation of the plan is now a matter for decision by individual companies, organizations, and persons...&quot; &lt;ref name = &quot;USMB History&quot;&gt;{{cite web |url=http://lamar.colostate.edu/~hillger/laws/usmb.html |title=History of The United States Metric Board}} &lt;/ref&gt;  By passing these resolutions, the board acknowledged beneficial plans, but could do nothing to actually implement the plans.

The American National Metric Council was established by the American National Standards Institute in 1973.&lt;ref&gt;{{cite web |url=http://lamar.colostate.edu/~hillger/laws/anmc.html |title=American National Metric Council}} &lt;/ref&gt;  It was the view of the council that U.S. metrication was going to be a complex process involving hundreds of different organizations.  The council's role was to coordinate conversion efforts in order to ensure that the metric system was applied consistently.
The ANMC proposed two metric conversion papers to the USMB in 1981 and both were approved.  They also published two newsletters: the Metric Reporter, and Metric Messenger for Consumers.  The Metric Reporter included editorials, stories, and updates about the progress of metrication both in the government and private industries.  The Metric Messenger for Consumers was written for the general public and included various articles about the metric system and its usage.

==Opponents==

====Organized Groups====
The non-mainstream arguments are varied, but usually attempt to vilify the metric system or glorify the English units.  There have been attempts to associate the metric system with Dictatorial regimes, and thus shed English units in a more positive light:  &quot;Most Americans can remember, from the late 1970s, when U.S. metrication (metric conversion) was proceeding like a five-year plan commanded by the Kremlin.&quot;&lt;ref name = &quot;Americans for Customary Weights and Measures&quot;&gt;{{cite web |url=http://www.bwmaonline.com/ACWM.htm | title = Americans for Customary Weights and Measures}}&lt;/ref&gt; The English units have been associated with being patriotic, which attempts to show opposition to the &quot;American&quot; units as being un-patriotic.&lt;ref name = &quot;Freedom2Measure&quot;&gt;{{cite web |url=http://www.freedom2measure.org/ | title = Freedom2Measure}}&lt;/ref&gt;  

====Popular Culture====
Pop culture media often presents metric conversion as being absurd and presents the topic as a joke, or to be humorous.  Grandpa Simpson presents the argument via song that the metric system is something to be opposed, and likens it to the Illuminati or Masons.&lt;ref name = &quot;Gandpa Simpson&quot;&gt;{{cite web |url=http://www.youtube.com/watch?v=_ZI_aEalijE | title = Grandpa Simpson}}&lt;/ref&gt;   In the T.V. show [http://en.wikipedia.org/wiki/The_Big_Bang_Theory The Big Bang Theory] the metric system is shown as something that only nerds and scientists care about.  Saturday Night Live has presented the base 10 quality as applied to an alphabet, or &quot;Decibet&quot;, a play on words originating from a metric prefix.&lt;ref name = &quot;Decibet Transcript&quot;&gt;{{cite web |url=http://snltranscripts.jt.org/75/75rdecabet.phtml | title = Decibet Transcript}}&lt;/ref&gt;

==Economics==
The two main economic factors associated with US metrication are the cost of conversion and the cost of mistakes.
===Space Exploration===

Given that space flight is an increasingly international enterprise, it has received particular attention regarding its varied use of the metric and customary systems.

=====Mars Climate Orbiter=====

[[File:Mars_Climate_Orbiter_-_mishap_diagram.png|Intended vs. actual trajectories|thumb|250px|right]]

An expensive effect of metrication was the loss of NASA's [http://en.wikipedia.org/wiki/Mars_Climate_Orbiter Mars Climate Orbiter].  In 1999 NASA launched a probe to Mars, assembled with the assistance of multiple contractors.  Upon reaching Mars, it did not enter orbit, but entered the Martian atmosphere where it was destroyed.  A single contractor failed to abide by NASA's metric requirement.  This contractor used imperial pound-force rather than metric Newtons.  The resulting miscalculation of thrust led to the probe's destruction,&lt;ref name = &quot;NASA Mars&quot;&gt;{{cite web |url=http://mars.jpl.nasa.gov/msp98/news/mco991110.html | title= Mars Climate Orbiter Failure Board Releases Report, Numerous NASA Actions Underway in Response}} &lt;/ref&gt; a $327 million mistake. &lt;ref&gt;http://mars.jpl.nasa.gov/msp98/orbiter/fact.html title|title = Mars Climate Orbiter Fact Sheet |publisher = NASA / JPL&lt;/ref&gt;

=====Constellation=====

The Constellation Program,&lt;ref name = &quot;Constellation&quot;&gt;{{cite web |url=http://en.wikipedia.org/wiki/Constellation_program | title = Constellation}}&lt;/ref&gt; a 2010 NASA project for human spaceflight, was granted permission to not be metric-compliant, for which NASA has received criticism&lt;ref name=&quot;NewScientist&gt;{{cite web |url=http://www.newscientist.com/article/dn17350-nasa-criticised-for-sticking-to-imperial-units.html | title=NASA criticized for sticking to imperial units}}&lt;/ref&gt;.  The primary reason for this was the $368 million&lt;ref name = &quot;NASA OIG&quot;&gt;{{cite web |url=http://nasawatch.com/archives/2010/03/nasa-oig-368-mi.html | title = NASA OIG}}&lt;/ref&gt; price tag required for contractors to convert all of the prior work and documentation to metric units.  After many years of regular business in U.S. customary units it is more difficult every day to perform the conversion to the metric system.

=====Lunar Base=====

The Vision for Space Exploration expects astronauts to return to the Moon by 2020 to set up a manned lunar outpost. NASA and 13 other space agencies discussed methods to coordinate their lunar exploration programs. The decision was unanimous to use metric units exclusively so that parts and supplies would be interchangeable between habitats. &lt;ref name = &quot;Metric Moon&quot;&gt;{{cite web |url=http://science.nasa.gov/science-news/science-at-nasa/2007/08jan_metricmoon/ | title = Metric Moon}}&lt;/ref&gt;

===Medicine===
The medical community recognizes the need to switch to the metric system, but remains very dependent on English units. Paramedics calculate dosages completely in metric units, but doctors can prescribe drug dosages in any units they want. &lt;ref name = &quot;Paramedic Pharmacology&quot;&gt;{{cite web |url=http://www.dhss.delaware.gov/dph/ems/files/pharmacology08.pdf | title = Paramedic Pharmacology}}&lt;/ref&gt; There have been cases where patients have underdosed and overdosed because of unit mix-ups. Overdosed patients have been at risk for seizure, stroke and even death. In underdosed patients the original illness will not be prevented and may further progress.&lt;ref name = &quot;Medical Unit Mix-ups&quot;&gt;{{cite web |url=http://www.pharmacyerrorlawfirm.com/practice_areas/cvs-claims.cfm | title = Medical Unit Mix-ups}}&lt;/ref&gt;

===Conversion Opportunities===
Gasoline pumps offered a unique chance for metric conversion. Businesses were wary of having to retrofit their pumps to allow for gas prices greater than 99.9¢ per gallon. The USMB found that selling gas by the liter, would save $94 million dollars -- nearly $300 million 2012 dollars. Gas would not have exceeded $1/liter until the mid 2000's.

==Psychology==

===Path Dependence===
Citizens of the United States, especially the older generations, have spent lots of time working with English units. Complete conversion to metric units is viewed by these people to be a hassle. This is a case of path dependency. The United States is losing its dominance in world trade. Before, other countries were forced to convert to English units for trading, but now they are forcing the United States to convert to metric. This incentivizes businesses to convert to metric units to keep their trading partners. &lt;ref name = &quot;Why Hasn't the U.S. Gone Metric?&quot;&gt;{{cite web |url=http://www.slate.com/articles/news_and_politics/explainer/1999/10/why_hasnt_the_us_gone_metric.html | title = Why Hasn't the U.S. Gone Metric?}}&lt;/ref&gt;

===Patriotism===
The United States is the only industrialized nation that has not converted to the metric system. Some researchers believe that patriotism is to blame. In a report by Kidela Capital Group, Jane McFadyen writes:

::“The United States breeds an almost unparalleled patriotic fever among its citizens, the likes of which is rarely found outside North Korea, and it is perhaps this strong sense of national pride and hint of superiority that prevents Americans from appearing to be the last to jump on the bandwagon.” &lt;ref name = &quot;The U.S. and the metric system&quot;&gt;{{cite web |url=http://www.kidela.com/columns/daily-debate/the-u-s-and-the-metric-system/ | title = The U.S. and the metric system}}&lt;/ref&gt;

Patriotism stems from our need to honor our parents by being loyal to their nation. This can almost directly be related to English units; people think that they are honoring their parents by using English units. &lt;ref name = &quot;Patriotism&quot;&gt;{{cite web |url=http://www.psychologytoday.com/blog/who-we-are/200907/patriotism | title = Patriotism}}&lt;/ref&gt;

==Education==
Currently, both the metric and English systems are taught in elementary school. Equal weight is given to both systems in textbooks. Since mostly English units are used by students at home, metric units are taught and then related to their equivalence in English units in school. Researchers believe that dual system usage causes students in the United States to perform poorly in math and science when compared to the rest of the world. &lt;ref name = &quot;Education System Benefits of U.S. Metric Conversion&quot;&gt;{{cite web |url=http://www.richardphelps.net/Eval-Rev-1996-Phelps-84-118.pdf | title = Education System Benefits of U.S. Metric Conversion}}&lt;/ref&gt; Computers are now reducing the need for fluency in both systems. With the click of a button, units can be converted back and forth. &lt;ref name = &quot;Solidworks Unit Conversions&quot;&gt;{{cite web |url=http://help.solidworks.com/2012/English/SolidWorks/sldworks/t_dual_dims_measure_tool.htm | title = Solidworks Dual Unit Conversions}}&lt;/ref&gt;&lt;ref name = &quot;Inventor Unit Conversions&quot;&gt;{{cite web |url=http://wikihelp.autodesk.com/Revit/enu/Community/Tips_and_Tricks/Project_Documentation/Tips | title = Inventor Dual Unit Conversions}}&lt;/ref&gt;

==Generalizable Lesson==
We've found that a system of measures is more than an a way to measure things. People have feelings and memories to their use of the customary system. Even its name embodies this link -- it is 'customary,' usual and friendly. Any program whose intent is to convert must take into account this emotional attachment, and navigate the waters carefully.

This is readily expandable into other social problems. In many situations, the technical community may agree on the benefits of some change, and want to implement it in society. To their chagrin, non-technical people oppose the shift. The technical community would do well to pay attention to the lesson that metrication can teach.

The United States is the last industrialized nation to convert to the metric system. Many communities in the United States have already converted to the metric system or are making attempts to do so as quickly as possible. Examples of such groups include scientists, the military and the medical community.  With the widespread use of computers it has become easy to quickly convert from one unit system to another using precise ratios established by scientific and governmental groups.  The United States government has tried to support conversion, but has not been successful for a variety of reasons.  There is a lack of popular support.  Rather than advertising the benefits of an international standard and the advantages to the average person, only tables of conversions were provided with groups willing to help conversion when assistance was requested.  A nation-wide conversion cannot occur without educating citizens and a strong government presence to oversee the process.

=References=
&lt;references /&gt;

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    <title>Structural Biochemistry/Organic Chemistry/Stereochemistry</title>
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      <text bytes="21535" sha1="mdxom5ax2cz7y90qoq39q254lu3wz5a" xml:space="preserve">Stereochemistry: By definition, stereochemistry is the arrangement of different atoms in space. Stereochemistry is a 3d representation of a carbon that is sp3 hybridized. There are many different types of stereoisomers. Let's first discuss a few basic concepts.

==Background==
Although most people are accustomed to thinking of organic chemistry as a bunch of drawings and structures, this &quot;paper chemistry&quot; is not really how these molecules act in real life. Of course, these molecules are really three-dimensional shapes, and not just 2d drawings. Stereochemistry aims to explain the natural phenomena of spatial arrangements of these organic molecules. 

[[File:Stereoisomers.png|thumb|Add caption here]]

'''Stereocenter''': Any atom in a molecule that is attached to 4 '''different''' atoms. Also known as ''chiral center''. A chiral molecule is special in that it is not identical to its mirror image, or in other words, the only criterion for chirality is that the object and its mirror image must be non-superimposable.&lt;ref name =&quot;Schore&quot; /&gt; For example, methane (CH4) is identical to its mirror image; therefore, this molecule is not chiral. In general, molecules with &quot;n&quot; chiral centers have 2^n stereoisomers. For example, a molecule with 3 stereocenters would give rise to a molecule with 8 stereoisomers. Stereoisomers are isomers of molecules with the same formula and connectivity, but with different arrangements of their atoms in space. Two stereoisomers have atoms linked together in the same order, but the two molecules do not have the same three-dimensional shapes. Wedges indicate bonds coming towards the viewer, while dashed lines indicate bonds going away from the viewer. 

A molecule classified as achiral either is not sp3 hybridized or the molecule is sp3 hybridized but has two or more substituents that are the same. These molecules are not chiral. An achiral molecule is super-imposable on itself.

==A Brief Introduction to Isomers==

[[File:Isomers danish.png|thumb|left|600px|This is a flow chart showing the different categories of isomers. Although this chart is in Danish, the spelling is very similar to English. Do not get the spelling confused.]]

Molecules that are isomers have the same molecular formulas but their structures are different from each other. There are two main categories of isomers: constitutional isomers and stereoisomers. Constitutional isomers, also known as structural isomers, are molecules that have the same molecular formulas but their connectivity is different. Oftentimes constitutional isomers have very different physical and chemical properties. Stereoisomers can be divided into two sub-categories: enantiomers and diastereomers. Enantiomers are molecules that are mirror images of one another and are not super-imposable. These molecules have the same connectivity but differ in their 3D arrangement of carbon substituents. Oftentimes enantiomers have very similar physical properties but they may differ in chemical properties. Diastereomers are molecules that are not related as mirror images but, like enantiomers, have the same connectivity but differ in their 3D arrangement of carbon substituents.

==Types of Stereochemistry==

==Enantiomers ==

Enantiomers are structures that are mirror images of one another that are ''non-superimposable.'' Non-superimposable means that no matter what way you rotate it, you will not be able to place it directly on top of the other facing the same way. You can easily demonstrate this with your hands because your hands are also non-superimposable, as you cannot place your hands on top of one another with your thumbs facing in the same direction, while your palms face the same way. Enantiomers have identical physical and chemical properties. 

[[File:RS isomer en.svg|thumb|left|500px|This is a depiction of both R and S enantiomers.]]

Enantiomers are generally classified with either an R or S configuration. To determine whether a stereocenter is labeled R or S, one must first rank its four substituents by molecular weight. The highest priority substituent (rank 1) will be the substituent atom with the largest molecular weight; conversely, the lowest priority substituent (rank 4) will be the substituent atom with the lowest molecular weight. For this reason, a hydrogen substituent is always given the lowest priority. Once the four substituents have been ranked by priority, a series of rules can be followed to determine the stereocenter's R or S configuration.&lt;ref name =&quot;Schore&quot;&gt;Schore, Neil E. (2011). Organic Chemistry Structure and Function 6th Edition. W. H. Freeman&lt;/ref&gt;

'''Rule 1:''' The lowest priority substituent is always placed as far away as possible, or in other words, placed into the paper or board.

'''Rule 2:''' Once the lowest priority substituent (generally a hydrogen) has been placed &quot;into the paper,&quot; there are only two possible arrangements of the remaining three substituents (R and S).  Looking down the lowest priority substituent bond with the carbon, if the remaining three substituents are increasing in the clockwise direction (1, 2, 3), the stereocenter is classified as R. If the three substituents are increasing in the counter-clockwise direction, the stereocenter is classified as S.

'''Rule 3:''' ''Point of difference rule''. This rule helps determine the priority of substituents that have the same rank when looking at the atoms directly attached to the stereocenter. A simple example is the difference between a methyl and an ethyl substituent.  Following the substituents beyond the initial carbon atom, a methyl is attached to three hydrogens, while the ethyl's carbon is attached to two hydrogens and another carbon.  This is the point of difference in the substituents. Since the carbon atom in the ethyl group is a higher priority than the 3rd hydrogen in the methyl group, the ethyl substituent is ranked higher in priority than the methyl group. &lt;ref name =&quot;Schore&quot; /&gt;
[[File:Rule 4.png|thumbnail|Rule 4: Double and triple bonds can be treated as single bonds when determining priority]]

'''Rule 4:''' Double and triple bonds can be viewed as if they were single bonds, but with the extra bonds added to each end of the multiple bond as single bonds. This can be seen on the figure to the right. The carbon is attached to a thiol group and is double bonded to an oxygen.  This double bond can be seen as the carbon attached to two single bonded oxygens, and the second oxygen is bonded a carbon (representing the double bond to the initial carbon).


Enantiomers are optically active, meaning they have the ability to rotate plane polarized light. A dextrorotatory (+) molecule rotates plane polarized light in a clockwise direction, while a levorotatory molecule rotates plane polarized light in a counterclockwise direction. Enantiomers differ in which direction each compound rotates plane polarized light. 

[[File:Enantiomers.png|thumb|left|200px|Enantiomers: As you can tell from this picture, the methyl is sticking out in one drawing, and sticking back in another. If you rotated one of these, you would NOT be able to superimpose it on the other. An example of Enantiomers.]]

As you can tell from the drawing, on the left, the methyl group is pointing out at us, while the hydroxyl group is pointing back. On the molecule in the right, this order is reversed. This is a prime example of enantiomers.

==Confusion with Enantiomers==

One might think that you can simply just rotate the molecule and they would be exactly the same, but this is not the case. If you rotated the molecule above, the hydroxyl and methyl will be imposed on one another, ''however'', the fluorine and hydrogen will now be on opposite sides. This is what it means to be '''non-superimposable'''

Oftentimes there are confusions with meso compounds. For example, say there are two molecules: one characterized with 1R, 2S configurations at its stereocenters (chiral center), the other molecule characterized with 1S, 2R configurations at its respective stereocenters. One might wonder why these molecules aren't enantiomers of each other—after all, they do have opposite configurations at all their respective stereocenters. However, these molecules aren't enantiomers because they are meso compounds. In this situation, each compound has a line of symmetry dividing its two stereocenters. Due to the plane of symmetry of each molecule, each compound is considered achiral (not chiral). 

==Racemization of Enantiomers==
Two molecules are considered to be '''racemic''' if they consist of the same amount of each enantiomer, a 50:50 ratio of the left-handed, and right handed enantiomer.

A racemic molecule is not shown to rotate plane-polarized light. Individually each enantiomer rotates plane-polarized light, but does so in equal and opposite directions, thus leading to a net rotation of 0.

Optical activity is directly proportional to the ratio of the two enantiomers of a compound.  When an equal ratio of the two enantiomers of a compound are present, the sample is considered optically inactive or racemic.  If only one of the enantiomers is present, then the sample is considered optically pure.  To describe a mixture that falls in-between these two extremes, one can use the enantiomer excess (ee) relation.  This is used for when one enantiomer is in excess of the other enantiomer in a mixture.&lt;ref name =&quot;Schore&quot; /&gt;  The enantiomer excess equation is:

Enantiomer excess (ee) = % of major enantiomer - % of minor enantiomer

The enantiomer excess equation tells us how much of one enantiomer is in excess of a racemic solution (since in a racemate the ratio of enantiomers is 1:1).  The enantiomer excess is also known as optical purity and can be found by the relation:

Enantiomer excess (ee) = optical purity = ([a]&lt;sub&gt;Mixture&lt;/sub&gt;/[a]&lt;sub&gt;Pure enantiomer&lt;/sub&gt;) X 100%

&lt;u&gt;'''Example'''&lt;/u&gt;

A solution of (+)-alanine from an artifact has a value of [a]&lt;sub&gt;Mixture&lt;/sub&gt; = 3.7; [a]&lt;sub&gt;pure enantiomer&lt;/sub&gt; = 8.5.  Find optical purity and actual enantiomer composition of the sample.&lt;ref name =&quot;Schore&quot; /&gt;

'''&lt;u&gt;Solution&lt;/u&gt;'''

Enantiomer excess = optical purity = (3.7/8.5) X 100% = 43.5 %
Therefore, 56.5% of the sample is racemic and 43.5% of the sample is the pure (+) isomer.  Thus, it can be concluded that the artifact under examination has an actual composition of 71.75%  (+) and 28.25% (-) alanine.

==Optical Activity==
If two enantiomers of a chiral molecule were isolated into two different containers in pure form, it would be extremely difficult to distinguish the two based on just their physical properties such as boiling points, melting points, and densities. Because only certain enantiomers of certain drugs are effective while the other enantiomers are ineffective or even detrimental to the body, it is very important to be able to distinguish enantiomers.

Fortunately chiral molecules possess a unique property that rotates the chiral molecules in a particular direction when a special kind of light called plane-polarized light is passed through a sample containing the enantiomers. This unique reaction that chiral molecules have with plane-polarized light is called optical activity which results in enantiomers often being referred to as optical isomers. The rotation of a particular chiral molecule as seen from the perspective of the viewer facing the light source can either be clockwise, dextrorotatory (“dexter” is Latin for “right”), or counterclockwise, levorotatory (“laevus” is Latin for “left”) [2]. Molecules that rotate clockwise are referred to as (+) enantiomers while molecules that rotate counterclockwise are referred to as (-) enantiomers [2]. The direction of rotation of any particular chiral molecule will always be the same and the enantiomer of that chiral molecule will have an opposite direction of rotation.

[[File:Wire-grid-polarizer.svg|thumb|left|400px|This is a simplified depiction of how polarizers filter out a beam of light leaving only the plane-polarized light to pass through.]]

Plane-polarized light result from passing a beam of ordinary light through a special material called a polarizer. A polarizer serves as a filter for the light waves that pass through it and filters out all but one light wave, the plane-polarized light wave [2]. Once the light has been filtered to just the plane-polarized light, the light wave travels through the chiral molecules. The electric field of the light wave interacts with the electrons around the chiral molecule and causes the molecule to rotate at a particular direction; this rotation is called optical rotation [2]. If there is optical rotation in the sample, then the sample is referred to as optically active. To determine if a sample is optically active, optical rotations have to be measured with a polarimeter.

Oftentimes reactions that yield chiral molecules are not stereospecific which may lead to products that have a 1:1 mixture of enantiomers. This reaction mixture is referred to as a racemic mixture because when a plane-polarized light passes through this sample, half of the chiral molecules would be rotating clockwise while the other half would be rotating counterclockwise. This results in a sample with no net optical activity. However, when the ratio of enantiomers is not 1:1, then there will be optical activity. One of the major goals of pharmaceutical companies is to perform reactions that yield optically pure products because this saves them very much time from the purification process of optically impure samples, not to mention money on the reactants used during production.

==Stereochemistry of Amino Acids==

[[File:Op isomer.svg|thumb|left|500px|This is a depiction of both L and D amino acids.]]

Amino acids are building blocks of protein. The general structure of an amino acid has an sp3 hybridized carbon atom linked to four other substituents. Three of the four substituents attached to the carbon of all amino acids are an amino group and a carboxylic acid group and a hydrogen atom. The last substituent is often represented with an R if the amino acid isn’t specified. R can represent one of the twenty different side chains the twenty different amino acids have.

The stereochemistry of amino acids are represented with an L,D system. The L,D system is essentially the same as the R,S system for absolute configuration with the exception that the L,D system is for amino acids. For the L,D system, L is the equivalent of the S absolute configuration leaving D to be the equivalent of the R absolute configuration.

Amino acids exist dominantly in the L absolute confirmation. There is no conclusive evidence for the dominance of the L isomer over the D isomer at the moment. Many speculate that the L isomer became the dominant amino acid configuration due to chance rather than any physical or chemical property. There have been experiments done trying to elucidate the reason behind the dominance of the L isomer. In the experiments, a protein would be artificially synthesized with only amino acids with the D absolute configuration. The conclusion was that the proteins made of only D amino acids were equally as active as the proteins made of L amino acids with the only difference being that the proteins made of D amino acids had reactions that were the reverse of the proteins made of L amino acids.

==Diastereomers==
Diastereomers are any stereoisomers that are not enantiomers. Thus, diastereomers are not mirror images of each other. Because diastereomers are stereoisomers that aren't mirror images of each other, they can be distinguished by different physical and chemical properties. Thus, it is possible to separate diastereomers by standard laboratory techniques. Diastereomers arise when stereoisomers of a molecule have different configurations at at least one of their stereocenters (but not all, for they would then be enantiomers). Each stereocenter possesses two configurations, so the number of stereoiosmers is increased by a factor of two. Cis/trans isomerism is a form of diastereomerism (cis-two substituents pointing in the same direction; trans-two substituents pointing in opposite directions). 

[[File:Cis-trans.svg|thumb|left|400px|This picture shows the cis and trans diastereomers of a butene molecule. For alkenes only, the cis and trans prefix is often replaced with Z or E.]]
[[File:Cis-trans-cyclo1.svg|thumb|right|400px|This picture shows the cis and trans diastereomers of a cyclic molecule.]]

While enantiomers have identical chemical properties, diastereomers do not. 

There are 3 stereocenters in the molecule below. The atoms have been switched in the first and third carbon. This is an example of diastereomers. If the second carbons atoms were switched also, then these two molecules would be enantiomers.

[[File:DiastereomersImageRH11.png]]

==Stereochemistry for Alkenes==
Cis and trans prefixes are essentially absent when naming alkenes with an E,Z system being used instead. The E,Z system sets priorities for carbon substituents the same way as the R,S system with the difference being that the double-bond having no bearing on the priority, only the two substituents that are single-bonded to the carbon. An E isomer (E comes from the German word “entgegen” which means “opposite”) would be where the highest priority substituents on each carbon on the double bond are opposite of each other [1]. A Z isomer (Z comes from the German word “zusammen” which means “together”) would be where the highest priority substituents on each carbon on the double bond are on the same side of each other [2].

==Epimers==
Epimers are diastereomers that differ at only ONE stereocenter.

There are 3 stereocenters in the molecule below. The atoms have been switched at the stereocenter in ONLY the third carbon(on right). This is an example of epimers. 
[[File:EpimersFinal.png]]

==Anomers==
&quot;Anomers are isomers that differ at a new asymmetric carbon atom formed on ring closure.&quot;&lt;ref name =&quot;Berg&quot;&gt;Berg, Jeremy Mark, John L. Tymoczko, and Lubert Stryer.Biochemistry. 6th. New York: W H Freeman &amp; Co, 2012. Print.&lt;/ref&gt;  An example of this is Alpha-D-Glucose and Beta-D-Glucose (shown below).
[[File:Alpha-D-glucose Haworth.svg|200px|thumbnail|left|Alpha-D-Glucose]][[File:Beta-D-Glucopyranose.svg|200px|thumbnail|center|Beta-D-Glucose]]

==Fischer Projections==

[[File:D-Glycerinaldehyd.png]] 
D-Glyceraldehyde 

In Fischer projections, bonds attaching substituents to the central carbon are shown using horizontal and vertical lines. Similar to the hashed-wedged line structures, horizontal lines are indicative of bonds coming towards the viewer, while vertical lines indicate bonds going away from the viewer. It is a simplified way of showing tetrahedral carbon atoms and their substituents in 2 dimensions. Converting hashed-wedged structures to Fischer projections is a simple process. Here is an example shown below. 

[[File:Fischer projection conversion.png]]
{{BookCat}}

==Shortcuts for Stereochemistry==
A simple method to differentiate between an enantiomer and a diastereomer is to look at the molecules being compared and, assuming they both are in fact stereoisomers, look at each stereocenter. If the carbon substituents of each stereocenter have been switched once, then the molecules are related as enantiomers. If not all of the carbon substituents of each stereocenter have been switched once, then the molecules are related as diastereomers. In this case, the switching of carbon substituents means, for example, that a dashed OH substituent and a wedged H substituent of the same carbon switched to a wedged OH substituent and a dashed H substituent while the other two carbon substituents remain in the same positions. By this logic, if the carbon substituents of the stereocenters have been switched twice, then the molecules are related as enantiomers, not diastereomers.


==Chiral Drugs==
Today, the majority of chiral medicines are sold as racemic mixtures. In two enantiomers, there would be one that is inactive. According to Vollhardt&lt;ref&gt;Vollhardt, Peter and Schore, Neil. (2009). Organic Chemistry 9th Edition. W.H. Freeman and Company. {{ISBN|978-1-4292-0494-1}}. &lt;/ref&gt;, the other enantiomer would stop the activity of the other enantiomer by becoming the blocker of the biological receptor site. As a result, the FDA started to produce individual enantiomers of the medicine. Consequently, they began to test the pure enantiomers to help increase the performance of the medicine and it may help to lengthen the patent of medicine. Also, they applied the &quot;chiral switch&quot; method to develop different ways of enantioselective synthesis. &quot;Chiral switch&quot; is the method of switching achiral reactant to a chiral product in a reaction that can be catalyzed by a chiral catalyst. Vollhardt states that such methods have been applied to the production of medicine such as antiarthritic, analgesic naproxen, and antihypertensive propranolol.

==References==
1. Schore, Neil E. (2011). Organic Chemistry Structure and Function 6th edition. W. H. Freeman.

2. Berg, Jeremy Mark, John L. Tymoczko, and Lubert Stryer.Biochemistry. 6th. New York: W H Freeman &amp; Co, 2012.

3. Schore, Neil E. (2007). Organic Chemistry Structure and Function 5th edition. W. H. Freeman.

3. Berg, Jeremy M. (2002). Biochemistry 5th edition. W. H. Freeman.

4. Vollhardt, Peter and Schore, Neil. (2009). Organic Chemistry 9th Edition. W.H. Freeman and Company. {{ISBN|978-1-4292-0494-1}}.</text>
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==Introduction==


# learn to think like a programmer 
# learn C C++
# learn how the AROS (Amiga) works


The first basically is about algorithm designs and things to avoid. Do you know what spaghetti code means? Structured programming? How about OO design? Big &quot;O&quot; notation? Recursion? Quick Sort? These are important concepts regardless of what language you use or platform you're on.


Where game engine systems lie regarding other systems
*1st - C, C++ base level 
*2nd - SDL with OpenGL framework, Raylib with OpenGL 
*3rd - Godot engines, Ren'Py, GameMaker, Scratch, Unity, Unreal ect (AROS does not have) 


=== Game design document GDD ===

 Top 100 games have over 90% market share 

Concept (marketing hook in order of difficulty) - '''gamestyle with gamestyle but ... ''' 
 arcade, catch, platformer, endless runner, mazes, tower defense, puzzle, city builder, rhythm, text, visual novel, adventure, party, shoot 'em up, metrovania, bullet hell, beat 'em up, walking sim on rails, card deck builder, cooking, racing, fighter, tactical turn based, real time strategy rts long term, relationships, survival, collectothon, fps, sandbox, management simulator, trading, sports, betting, rogue, roguelite, roguelike, dungeon crawler, rpg, horror, souls, soulslite, soulslike, battle royale, arena, 

Similar games - same ... look / feel / genre / scope / target avatar audience 

Measurement (Scale) - how the character / enemies interact, like reach, swing, grab, etc 

Prototype - ugly grey box version, mock ups, concepts, screenshots, color schemes, logo, fonts, etc 

10min demo - vertical slice for funding, getting initial impressions about good bits, poor ideas, missing parts, etc i.e playtesting 


Fun like overcoming obstacles challenges, progression (quests logs to do lists) and rewards, sense of wonder 

Design Patterns - finite state machines (behaviors through states entities transitions), event bus singleton (manage signals from objects), entity component patterns (building blocks) 

Game Play Loop -   , extraction, looter shooter, 


Mechanics
* Walls, Spikes, , etc 
* 

Story - Narrative, storyline, 

Shipping 
* game jam to limit scope and allow completion within 1 week with clean code 
* then expand to 3,4 and 6 month cycles later as you gain experience and knowledge, later 9 or 12 months max to gain experience and stay focused 


&lt;pre&gt;
design process 
- Empathize: Research user needs
- Define: State user needs and problems
- Refine: Challenge assumptions and create ideas
- Prototype: Start to create solutions
- Test: Try out solutions
&lt;/pre&gt;


==AROS (Amiga)==

Of course you need to learn programming on the Amiga. Lucky for you, the Amiga is a fun computer to program with a relatively small API. That is to say, you won't be swamped with OS calls, however, there are some ugly parts out there. If you want to just open a window and create some buttons, that's easy. Wanna write a web browser? That is gonna be hard and that is mostly because the OS doesn't provide a lot of the things you would need so you will end up writing it yourself. :-)

==2D==

* [http://lazyfoo.net/SDL_tutorials/index.php Setup] [http://www.metanetsoftware.com/technique/tutorialB.html Tiles], masking, [http://web.archive.org/web/20071018032333/http://jnrdev.72dpiarmy.com/ platform stuff], etc 
* Double buffering for moving objects sprites ([http://www.metanetsoftware.com/technique/tutorialA.html#section6 collisions]) or scrolling
* 2D frameworks which take some of the hard work of writing your own routines (like [http://lazyfoo.net/SDL_tutorials/ SDL1.2]) or raylib5 
* Make at least one type of game from each of the following board/grid, maze, card, etc for the experience

[http://www.youtube.com/watch?v=CN8m_8rZLRM SDL youtube video]


* [http://gamasutra.com/blogs/ChrisHildenbrand/20111015/8669/2D_Game_Art_For_Programmers__Part_1_updated.php Sprites Artwork] 
* [http://www.gamasutra.com/blogs/ChrisHildenbrand/20111114/8882/2D_Game_Art_For_Programmers__Part_5.php Animation Artwork]


*hitbox 
*rotation 
*gravity 
*resistance - spring damper mass model 
*regioning art assets 
*lighting / shadows using 2d normal maps 
*particle effects, fire, embers, smoke, spark, lantern, 
*collisions 
*


===Rotations===

Point Rotation
 x' = x*cos(a) - y*sin(a) 
 y' = x*sin(a) + y*cos(a) 
Mentioned briefly in the tutorial as being an expensive operation, here is the basic point rotation formula. As you can see, it's at least 2 table lookups, 4 multiplications, and two additions, but the sine and cosine values can be reused for each point. Rotating for the purpose of hills would mean rotating the Z and Y coordinates, not the X and Y coordinates. To find the derivation of this formula, look up Rotation of Axes.

&lt;syntaxhighlight lang=&quot;c&quot;&gt;
/* assuming width and height are integers with the image's dimensions */

for(int x = 0; x &lt; width; x++) {
    int hwidth = width / 2;
    int hheight = height / 2;

    double sinma = sin(-angle);
    double cosma = cos(-angle);

    for(int y = 0; y &lt; height; y++) {

        int xt = x - hwidth;
        int yt = y - hheight;

        int xs = (int)round((cosma * xt - sinma * yt) + hwidth);
        int ys = (int)round((sinma * xt + cosma * yt) + hheight);

        if(xs &gt;= 0 &amp;&amp; xs &lt; width &amp;&amp; ys &gt;= 0 &amp;&amp; ys &lt; height) {
             /* set target pixel (x,y) to color at (xs,ys) */
             } else {
             /* set target pixel (x,y) to some default background */
             }
        }
}
&lt;/syntaxhighlight&gt;


===2.5D Calculating Distance/Depth===

A perspective transform ([http://en.wikipedia.org/wiki/3D_projection perspective projection] == divide by Z) amounts to

 x = x*d/z+d
 y = y*d/z+d

where d is the distance from the viewpoint and x, y, z are (obviously ) your x, y, z coordinates in 3d space
If there is no &quot;divide by Z&quot;, the depth-wise movement will feel wrong and it will feel like your object is accelerating/braking at the wrong times.


3d Projection
 y_screen = (y_world / z) + (screen_height &gt;&gt; 1)
or:
 z = y_world / (y_screen - (screen_height &gt;&gt; 1))

This formula takes the x or y world coordinates of an object, the z of the object, and returns the x or y pixel location. Or, alternately, given the world and screen coordinates, returns the z location.


Fast Linear Interpolation
 o(x) = y1 + ((d * (y2-y1)) &gt;&gt; 16)
This assumes that all the numbers are in 16.16 fixed point. y1 and y2 are the two values to interpolate between, and d is the 16-bit fractional distance between the two points. For example, if d=$7fff, that would be halfway between the two values. This is useful for finding where between two segments a value is.

Fixed Point Arithmetic
Floating point is very expensive for old systems which did not have specialized math hardware. Instead, a system called fixed point was used. This reserved a certain number of bits for the fractional part of the number. For a test case, say you only reserve one bit for the fractional amount, leaving seven bits for the whole number amounts. That fraction bit would represent one half (because a half plus a half equals a whole). To obtain the whole number value stored in that byte, the number is shifted right once. This can be expanded to use any number of bits for the fractional and whole portions of the number.

Fixed point multiplication is trickier than addition. In this operation, you would multiply the two numbers and then shift right by however many bits are reserved for fractions. Due to overflow, sometimes you may need to shift before multiplication instead of after. See &quot;Fast Linear Interpolation&quot; for an example of fixed point multiplcation.


Avoid Division
Instead of dividing by the z of an object in the standard projection formulas, you can take advantage of some properties of the road to speed up calculations. Say you have a 3d segment z position and a y position, and you want to find which line of the screen it belongs on. First, read through the z-map until you get to the 3d segment's z position. Then, multiply the height of the segment by the corresponding scaling value. The result is the number of pixels above the road that the segment belongs.

Use Z as Scaling Value
Scaling routines work by slowing or speeding up the speed at which a draw routine reads through graphics data. For example, if you were to set the read speed to half, this would draw a sprite double the size. This is because for each time a pixel is drawn, the read position in the sprite data is only incremented by half, causing the read position to only increment by a whole number every two pixels.

Usually, a scaling routine has parameters like x, y, and scaling factor. But since a scaling factor is just 1/z, we can just reuse the Z value of that sprite! We will still need the scaling factor though to determine the boundaries of the sprite so that we can keep it centered as it scales.

Read more about 2.5D [http://eab.abime.net/showthread.php?t=26679 here]

====Maze Generation====

Binary Tree (simple but has limitations) 
Sidewinder (better Binary tree) 
Depth First Search (DFS) (good simple mazes - traces route and backtracks to fill in missing) 
Growing Tree

Recursive Subdivision (wall adding - fractal like)

Aldous-Broeder (inefficient uniform spanning tree based)
Wilson's (better spanning tree) 
Prim's (another spanning tree) 
Kruskal's (good but complex tree spanning)

Solving Algorithms

Dead-end retrace

a perfect maze, there is one and only one path from any point in the maze to any other point. 
That is, there are no inaccessible sections, no circular paths, and no open regions. 
A perfect maze can be generated easily with a computer using a depth first search algorithm.

A two dimensional maze can be represented as a rectangular array of square cells. 
Each cell has four walls. The state of each wall (north, south, east, and west) of 
each cell is maintained in a data structure consisting of an array of records. 
Each record stores a bit value that represents the state of each wall in a cell. 
To create a path between adjacent cells, the exit wall from the current cell and 
the entry wall to the next cell are removed. For example, if the next cell is to 
the right (east) of the current cell, remove the right (east) wall of the current 
cell and the left (west) wall of the next cell.

Depth-First Search - simplest maze generation algorithm

# Start at a random cell in the grid 
# Look for a random neighbor cell you haven't been to yet 
# If you find one, move there, knocking down the wall between the cells. If you don't find one, back up to the previous cell 
# Repeat steps 2 and 3 until you've been to every cell in the grid

PSEUDOCODE

&lt;pre&gt;
create a CellStack (LIFO) to hold a list of cell locations
set TotalCells = number of cells in grid
choose a cell at random and call it CurrentCell
set VisitedCells = 1
 
while VisitedCells &lt; TotalCells

      find all neighbors of CurrentCell with all walls intact
      if one or more found
            choose one at random
            knock down the wall between it and CurrentCell
            push CurrentCell location on the CellStack
            make the new cell CurrentCell
            add 1 to VisitedCells 
      else
            pop the most recent cell entry off the CellStack
            make it CurrentCell 
      endIf
 
endWhile
&lt;/pre&gt;

assumed u have a x b size of maze, u need a + 1 and b + 1 of walls, so size of array should be (size * 2 + 1)

&lt;syntaxhighlight lang=&quot;c&quot;&gt;
#include&lt;stdio.h&gt;
#include&lt;conio.h&gt;
#include&lt;stdlib.h&gt;
#include&lt;time.h&gt;
 
#define MAX 61  // 30 * 2 + 1
#define CELL 900  // 30 * 30
#define WALL 1
#define PATH 0
 
void init_maze(int maze[MAX][MAX]);
void maze_generator(int indeks, int maze[MAX][MAX], int backtrack_x[CELL], int bactrack_y[CELL], int x, int y, int n, int visited);
void print_maze(int maze[MAX][MAX], int maze_size);
int is_closed(int maze[MAX][MAX], int x, int y);
 
int main(void)
{
    srand((unsigned)time(NULL));
 
    int size;
    int indeks = 0;
    printf(&quot;MAZE CREATOR\n\n&quot;);
    printf(&quot;input  (0 ~ 30): &quot;);
    scanf(&quot;%d&quot;, &amp;size);
    printf(&quot;\n&quot;);
    int maze[MAX][MAX];
    int backtrack_x[CELL];
    int backtrack_y[CELL];
 
    init_maze(maze);
 
    backtrack_x[indeks] = 1;
    backtrack_y[indeks] = 1;
 
    maze_generator(indeks, maze, backtrack_x, backtrack_y, 1, 1, size, 1);
    print_maze(maze, size);

    getch();
    return 0;
}
 
void init_maze(int maze[MAX][MAX])
{
     for(int a = 0; a &lt; MAX; a++)
     {
         for(int b = 0; b &lt; MAX; b++)
         {
             if(a % 2 == 0 || b % 2 == 0)
                 maze[a][b] = 1;
             else
                 maze[a][b] = PATH;
         }
     }
}
 
void maze_generator(int indeks, int maze[MAX][MAX], int backtrack_x[CELL], int backtrack_y[CELL], int x, int y, int n, int visited)
{
    if(visited &lt; n * n)
    {
        int neighbour_valid = -1;
        int neighbour_x[4];
        int neighbour_y[4];
        int step[4];
 
        int x_next;
        int y_next;
 
        if(x - 2 &gt; 0 &amp;&amp; is_closed(maze, x - 2, y))  // upside
        {
            neighbour_valid++;
            neighbour_x[neighbour_valid]=x - 2;;
            neighbour_y[neighbour_valid]=y;
            step[neighbour_valid]=1;
        }
 
        if(y - 2 &gt; 0 &amp;&amp; is_closed(maze, x, y - 2))  // leftside
        {
            neighbour_valid++;
            neighbour_x[neighbour_valid]=x;
            neighbour_y[neighbour_valid]=y - 2;
            step[neighbour_valid]=2;
        }
 
        if(y + 2 &lt; n * 2 + 1 &amp;&amp; is_closed(maze, x, y + 2))  // rightside
        {
            neighbour_valid++;
            neighbour_x[neighbour_valid]=x;
            neighbour_y[neighbour_valid]=y + 2;
            step[neighbour_valid]=3;
 
        }

        if(x + 2 &lt; n * 2 + 1 &amp;&amp; is_closed(maze, x + 2, y))  // downside
        {
            neighbour_valid++;
            neighbour_x[neighbour_valid]=x+2;
            neighbour_y[neighbour_valid]=y;
            step[neighbour_valid]=4;
        }
 
        if(neighbour_valid == -1)
        {
            // backtrack
            x_next = backtrack_x[indeks];
            y_next = backtrack_y[indeks];
            indeks--;
        }

        if(neighbour_valid!=-1)
        {
            int randomization = neighbour_valid + 1;
            int random = rand()%randomization;
            x_next = neighbour_x[random];
            y_next = neighbour_y[random];
            indeks++;
            backtrack_x[indeks] = x_next;
            backtrack_y[indeks] = y_next;
 
            int rstep = step[random];
 
            if(rstep == 1)
                maze[x_next+1][y_next] = PATH;
            else if(rstep == 2)
                maze[x_next][y_next + 1] = PATH;
            else if(rstep == 3)
                maze[x_next][y_next - 1] = PATH;
            else if(rstep == 4)
                maze[x_next - 1][y_next] = PATH;
            visited++;
        }
 
        maze_generator(indeks, maze, backtrack_x, backtrack_y, x_next, y_next, n, visited);
    }
}
	 
void print_maze(int maze[MAX][MAX], int maze_size)
{
     for(int a = 0; a &lt; maze_size * 2 + 1; a++)
     {
         for(int b = 0; b &lt; maze_size * 2 + 1; b++)
         {
             if(maze[a][b] == WALL)
                 printf(&quot;#&quot;);
             else
                 printf(&quot; &quot;);
         }
         printf(&quot;\n&quot;);
     }
}
	 
int is_closed(int maze[MAX][MAX], int x, int y)
{
    if(maze[x - 1][y]  == WALL
       &amp;&amp; maze[x][y - 1] == WALL
       &amp;&amp; maze[x][y + 1] == WALL
       &amp;&amp; maze[x + 1][y] == WALL
    )
    return 1;
	 
    return 0;
}
&lt;/syntaxhighlight&gt;

Growing Tree

It starts by selecting a random cell and adding it to the list

&lt;pre&gt;
x, y = rand(width), rand(height)
cells &lt;&lt; [x, y]

The program them simply loops until the list is empty

until cells.empty?
  # ...
end
&lt;/pre&gt;

Within the loop, we first select the cell to operate on. I’m going to mask my own program’s complexity here behind a simple “choose_index” method; it takes a number and returns a number less than that.

&lt;pre&gt;
index = choose_index(cells.length)
x, y = cells[index]
&lt;/pre&gt;

Next, we iterate over a randomized list of directions, looking for an unvisited neighbor. If no such neighbor is found, we delete the given cell from the list before continuing.

&lt;pre&gt;
[N, S, E, W].shuffle.each do |dir|
  nx, ny = x + DX[dir], y + DY[dir]
  if nx &gt;= 0 &amp;&amp; ny &gt;= 0 &amp;&amp; nx &lt; width &amp;&amp; ny &lt; height &amp;&amp; grid[ny][nx] == 0
    # ...
  end
end

cells.delete_at(index) if index
&lt;/pre&gt;

When a valid, unvisited neighbor is located, we carve a passage between the current cell and that neighbor, add the neighbor to the list, set index to nil (to indicate that an unvisited neighbor was found), and then break out of the innermost loop.

&lt;pre&gt;
grid[y][x] |= dir
grid[ny][nx] |= OPPOSITE[dir]
cells &lt;&lt; [nx, ny]
index = nil
break
&lt;/pre&gt;

And that’s really all there is to it. Some possible implementations of the choose_index method might be:

&lt;pre&gt;
def choose_index(ceil)
  return ceil-1 if choose_newest?
  return 0 if choose_oldest?
  return rand(ceil) if choose_random?
  # or implement your own!
end
&lt;/pre&gt;


==3D==


* solo route (hard and time consuming) ?? and use OpenGL (Mesa) routines] or raylib
* 3d game engines like [http://www.arkham-development.com/ Antiryad Gx], [http://www.panda3d.org/ panda 3d], [Quake Doom id Tech 1 to 3 in C],  [id Tech 4 in C plus plus],  [http://code.google.com/p/urho3d/ Urho3d], [Terathon C4 (Commercial),

AROS currently does not have these 
* 3d graphics engine (need to add networking, physics, sound, scripting, level editor etc.) like [http://www.ogre3d.org/ Ogre3D C++], [http://www.crystalspace3d.org/main/Main_Page Crystal Space C++], [http://irrlicht.sourceforge.net/ irrlicht C++], [http://www.sea-of-memes.com/ 3D voxel etc], [http://cubeengine.com/ Cube C], [http://www.luxinia.de/ Luxinia in C BSD], [http://horde3d.org/ horde 3d], [],  [ torque 3d],


==2D texture onto 3D object==

You need to create a separate context per class object instance and keep it alive as long as instance is alive. The context is bound to executing task and you can only have one context bound at a time. Opening the library itself does nothing - all actions are executed in relation to the context.

To get access to the right mouse button in an subclass I've added this in the setup method :
 set(_win(obj), MUIA_Window_NoMenus, TRUE);

f the square you are drawing to is 2 dimensional and not rotated, you may be looking for glDrawPixels. It allows you to draw a rectangular region of pixels directly to the buffer without using any polygons.

glTexImage2D. This is the call that loads the image in OpenGL for a 2D Texture. glTexImage2D actually takes a raw pointer to the pixel data in the image. You can allocate memory yourself, and set the image data directly (however you want), then call this function to pass the image to OpenGL. Once you've created the texture, you can bind it to the state so that your triangles use that texture. If the texture needs to change, you will need to either re-do the glTexImage2D() call, or use something like glTexSubImage2D() to do a partial update.

just pass an array of GLubyte to the glTexImage2D function (as well as all the functions needed to bind the texture, etc). Haven't tried this exact snippet of code, but it should work fine. The array elements represent a serial version of the rows, columns and channels.


&lt;pre&gt;
int pixelIndex = 0;
GLubyte pixels[400];

for (int x = 0; x &lt; 10; x++)
{
    for (int y = 0; y &lt; 10; x++)
    {
         for (int channel = 0; channel &lt; 4; channel++)
         {
             // 0 = black, 255 = white
             pixels[pixelIndex++] = 255;
         }
    }
}

glTexImage2D(
    GL_TEXTURE_2D, 0, GL_RGBA, SIZE, SIZE, 0,
    GL_RGBA, GL_UNSIGNED_BYTE, pixels);
&lt;/pre&gt;

An OpenGL book you can use a 2D array for monochrome images, so assume you could use a 3D array also.


==Collision==


===Ray with Plane===

* about a sphere intersecting triangles, and something called the &quot;2PI summation method&quot; to determine if a ray strikes a triangle. 
* create a plane from your wall and do a ray to plane intersection test. A very simple test is a dot product test. If the result of the vector based scalar 'dot' product is &gt;0, or the vectors are pointing in somewhat the same direction, then you know you are on the near side of the wall. If its &lt;0, or the vectors are nearly pointing in the opposite directions then you are on the far side of the wall 
* test for pixel perfect, you must find the point at which your camera ray intersected the plane in question. Back the camera up to that point, compute the collision response, and move on 
* does a ray to triangle test and then computes the barycentric coords of the ray inside of the triangle. It will also bail on NAN's which can cause problems. It only uses dot products as well


The equation for intersection of ray and plane. This is easier to do in a maze because we know that all walls will form a plane so there is no need to do an expensive ray to triangle test.

&lt;pre&gt;
    Ray: p(t)=p0+td
    Plane: p dot n=d

    Equation: t=(d-p0 dot n)/(d dot n)

    If ray is parallel to plane or (d dot n)=0 then there is no intersection.
    If (d dot n)&lt;0 then intersection during interval.
&lt;/pre&gt;

Solve for p0 which is actually p0.x and p0.y in this equation as well. This has been solved for t to test for intersection in this time interval. The nice thing about this is the dot product test and point test are rolled up in one algorithm. If the first time interval test passes, you can then solve for p0.

==[http://en.wikipedia.org/wiki/Pathfinding Pathfinding]==

* grid-based 
*


* [http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm Djikstra's] with [http://www.gamedeveloperskills.com/?p=13 tutorial] and close cousin [http://en.wikipedia.org/wiki/A%2a_search_algorithm A* star] and [http://www.anotherearlymorning.com/2009/02/pathfinding-with-a-star/ pseudocode] 
* [http://www.gamasutra.com/features/20020405/smith_01.htm flooding algorithm] and then [http://www.gamasutra.com/features/gdcarchive/2002/chris_carollo.ppt sound propagation]. [http://sourceforge.net/projects/argorha/ placing a virtual version of my character at a given position (usually a spawn point). I do an initial ray cast to place him flush against the ground and then add that single occupy-able cell to my &quot;unprocessed list&quot;. I then check the four neighbours of that cell to see if a virtual player could step there with out being impeded/falling-climbing too much. I add the successful cells to the unprocessed list, and take my initial cell and put it in the processed list. repeat until the unprocessed list is empty. The cells in my algorithm also store a position of the ground, and this can go up or down depending on if stairs or a ramp were encountered. When a sector is created from a list of related cells, it uses a axis aligned bounding box that encompass all these points (+ the width and height of each cell)] 
* D* being a dynamic A* 
* 3D mesh navigation


===A star===

A* (A star) algorithm moves through the grid from the given starting point to the given destination. Each &quot;step&quot; it makes it stores in a node. In this node data structure is stored three values 
* the coordinates of the current block from the start 
* the distance to the finish position from this block (different methods manhattan, diagonal, euclidian (optional squared version) 
* and this block's parent block (for a later backtrack mode) 
As the algorithm moves towards the destination, it stores these nodes into two stacks: the open and the closed stack. Once the destination block is reached, we go back through the nodes (using the pointer to the parent nodes) and we have our path described

Youtube video [http://www.youtube.com/watch?v=Kw8AMmyc6vg here] and [http://www.youtube.com/watch?v=uvIIzj11ayM here]

&lt;pre&gt;

Insert the start point onto the open stack

while( nodes left on open stack ) {

    //Pop first (closest) node off of open stack

    node = openstack.pop

    if( node.bDestination == 1 ) {

        Loop upwards through data structure to generate path

        exit function

    }

    //If we get here, this node isn't the destination

    for( up, down, forward, backward, left, right in grid ) {

        //GetNewNode returns null if block is occupied or in //closed stack

        newnode = GetNewNode( currentDirection )

        if( newnode ) {

        //This InsertSorted function inserts the node //sorted based on the block's distance from the //destination

            openstack.InsertSorted(newnode);

        }

    }

    //Once a node is on the closedstack it is no longer used //(unless one of its children is the destination)

    closedstack.push( node );

}
&lt;/pre&gt;

In this algorithm the tricky part is actually in the InsertSorted() method of the openstack. You sort the nodes by their distance to the destination. This is the most important part of the algorithm, because the order in which the algorithm picks nodes to search is based on this sorting. Traditionally, (at least in the examples I've seen) you use the Manhattan Distance, which is the distance in grid blocks from the destination. I tweaked this distance function, and instead used the 3D distance of the centerpoint of the current block to the destination (using BlockDistance3D). For whatever reason this made the algorithm work better in my case...it consistently searched less nodes than using the manhattan distance.

; Read more [http://www.policyalmanac.org/games/aStarTutorial.htm here] and [http://aigamedev.com/open/tutorials/clearance-based-pathfinding/ here]


==Examples==

Minecraft clones [https://github.com/jrupac/minecraft-opengl OpenGL], [http://c55.me/minetest/ Minetest C55], [http://www.sea-of-memes.com/LetsCode1/LetsCode1.html Theory],


===Simpler Language Introductions===

* Javascript - [http://javascriptsource.com/games/ ], [http://www.webresourcesdepot.com/25-amazing-javascript-games-some-fun-and-inspiration/ ], [http://www.lutanho.net/stroke/online.html ], [http://www.queness.com/post/2584/12-amazing-and-creative-javascript-games ], [http://www.javascriptkit.com/script/cutindex22.shtml ], [http://plaza.harmonix.ne.jp/~jimmeans/ ], [http://php.resourceindex.com/Complete_Scripts/Games/ ], 
* Lua - Love2D, 
* [http://html5gameengines.com/ HTML5], [https://gist.github.com/768272 ], 
* PortablE


===C and company===

* C 
* C++ [http://www.youtube.com/watch?v=-B7Vqc9c_wk&amp;feature=related]

Learning C isn't that hard, learning how to use it properly can be. The syntax for C although terse, is very simple. You could describe the entire C language on a couple of sheets of paper, there's little too it. However, like my prof always told me, C gives you enough rope to hang yourself (and everyone around you actually). Basically, C trusts that you know what you are doing and if you tell it to trash memory it will happily do it for you. C will not hold your hand like Basic, but it gives you more power and flexibility then just about any other language. Anyways, my best advice to anyone who's new with C is to study your pointers. And when you think you [http://www.youtube.com/watch?v=Rxvv9krECNw understand pointers], [http://www.youtube.com/watch?v=m_q6FUdLuCI study] them [http://www.youtube.com/watch?v=7uiga0LzHZQ some more]. That's the one part of C that gives people major headaches.

Read more at [http://www.gametutorials.com/default.aspx?sp=tutorials C] or [http://www.cppgameprogramming.com/ C++]

&lt;pre&gt;
c value
&amp;c address of c 
*c pointed to by c 
&lt;/pre&gt;

&lt;pre&gt;
.c should contains functions 
.h usually contain #define #include typedef enum struct extern 
&lt;/pre&gt;


&lt;pre&gt;
C++ 
BASIC
types and variables
conditionals and loops
i/o
structures

MEDIUM
class
constructor
destructor
methods
instance variables
&lt;/pre&gt;

the underlying semantics of C++ ( e.g. when to use virtual, what a copy constructor does ) 
Even without data-hiding inheritance, references, polymorphism you have a powerful structure. 
Very handy features, such as function name overloading (used with care and sparingly), declarations as statements, references to name but a few...
Data and methods put together called a class. OO language is classes &amp; objects. 
OO means objects that have data and methods. Methods are sent from object to object and the object manipulates it's data.
It has its share of problems too: the syntax is complex (although not unbearably so), it has no garbage collection and still has you managing memory by yourself, and quite a number of others, which may not directly affect a programmer working on his own, but will when working in a team.

[http://javilop.com/game-development/tetris-tutorial-in-c-platform-independent-focused-in-game-logic-for-beginners/ Tetris], , etc


===Algorithms===

 [http://www-cs-students.stanford.edu/~amitp/gameprog.html Algorithms] with [http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/video-lectures/ youtube videos] but start with small WB games first.

Read more [http://www.mangatutorials.com/forum/showthread.php?742-The-Ultimate-Indie-Game-Developer-Resource-List here]


Algorithm theory involves thinking about the growth rates (Space and Time) and providing a breakdown of the issue into pseudo code and [http://en.wikipedia.org/wiki/Big_O_notation big O notation] which taught what items to think about when choosing and optimizing algorithms

* search trees like: B tree, B+ tree, Red-black tree 
* sorting algorithms like quicksort, [http://eab.abime.net/showthread.php?t=40645 natural merge sort], bucket bin sort, 
* Breadth-first search (http://en.wikipedia.org/wiki/Graph_traversal), Depth-first search (mazes), 
* path finding like Hill Climb, A-star, 
* [http://www.youtube.com/watch?v=sDDRZv5QM80 Vectors],


====Quicksort====

&lt;pre&gt;
# A is the array, p is the start position and r the end position 
# i is the pivot value, p the start position value and r the end position value 
#
# Randomised-Partition(A,p,r)
# i &lt;- Random(p,r)
# exchange A(r) with A(i)
# return Partition(A,p,r)
#
# Randomised-QuickSort(A,p,r)
# if p&lt; r then
#     q &lt;- Randomised-Partition(A,p,r)
#     Randomised-QuickSort(A,p,q)
#     Randomised-QuickSort(A,q+1,r)
#
#
void quickSort(int numbers[], int array_size)
{
  q_sort(numbers, 0, array_size - 1);
}
 
# choose (random?) pivot number and partition numbers into lower and greater around pivot 
void q_sort(int numbers[], int left, int right)
{
  int pivot, l_hold, r_hold;
 
  l_hold = left;
  r_hold = right;
  pivot = numbers[left];
  while (left &lt; right)
  {
    while ((numbers[right] &gt;= pivot) &amp;&amp; (left &lt; right))
      right--;
    if (left != right)
    {
      numbers[left] = numbers[right];
      left++;
    }
    while ((numbers[left] &lt;= pivot) &amp;&amp; (left &lt; right))
      left++;
    if (left != right)
    {
      numbers[right] = numbers[left];
      right--;
    }
  }
  numbers[left] = pivot;
  pivot = left;
  left = l_hold;
  right = r_hold;
  if (left &lt; pivot)
    q_sort(numbers, left, pivot-1);
  if (right &gt; pivot)
    q_sort(numbers, pivot+1, right);
}
&lt;/pre&gt;

Youtube video [http://www.youtube.com/watch?v=Fju_NstEy3M here] and


====Merge Sort====

&lt;pre&gt;
void mergeSort(int numbers[], int temp[], int array_size)
{
  m_sort(numbers, temp, 0, array_size - 1);
}
 
 
void m_sort(int numbers[], int temp[], int left, int right)
{
  int mid;
 
  if (right &gt; left)
  {
    mid = (right + left) / 2;
    m_sort(numbers, temp, left, mid);
    m_sort(numbers, temp, mid+1, right);
 
    merge(numbers, temp, left, mid+1, right);
  }
}
 
void merge(int numbers[], int temp[], int left, int mid, int right)
{
  int i, left_end, num_elements, tmp_pos;
 
  left_end = mid - 1;
  tmp_pos = left;
  num_elements = right - left + 1;
 
  while ((left &lt;= left_end) &amp;&amp; (mid &lt;= right))
  {
    if (numbers[left] &lt;= numbers[mid])
    {
      temp[tmp_pos] = numbers[left];
      tmp_pos = tmp_pos + 1;
      left = left +1;
    }
    else
    {
      temp[tmp_pos] = numbers[mid];
      tmp_pos = tmp_pos + 1;
      mid = mid + 1;
    }
  }
 
  while (left &lt;= left_end)
  {
    temp[tmp_pos] = numbers[left];
    left = left + 1;
    tmp_pos = tmp_pos + 1;
  }
  while (mid &lt;= right)
  {
    temp[tmp_pos] = numbers[mid];
    mid = mid + 1;
    tmp_pos = tmp_pos + 1;
  }
 
  for (i=0; i &lt;= num_elements; i++)
  {
    numbers[right] = temp[right];
    right = right - 1;
  }
}
&lt;/pre&gt;

Youtube video [http://www.youtube.com/watch?v=GCae1WNvnZM here]

[], [],


===Uses===</text>
      <sha1>i1nk3xant6x84vc0sfg2y54ntju9dp8</sha1>
    </revision>
  </page>
  <page>
    <title>Descriptive Geometry/Complex Solid Intersections</title>
    <ns>0</ns>
    <id>290561</id>
    <revision>
      <id>4631270</id>
      <parentid>3175832</parentid>
      <timestamp>2026-04-18T21:21:53Z</timestamp>
      <contributor>
        <username>~2026-23742-78</username>
        <id>3577508</id>
      </contributor>
      <comment>{\fVerdana|b0|i0|c0|p34;\H0.75x;Show the intersection (Curve/Line) of the intersection of these 2 object in the front view\H1.33333x; }</comment>
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      <text bytes="10179" sha1="2obi390aywdoadigtmgemslae7jhqxg" xml:space="preserve">&lt;hiero&gt;
{\fVerdana|b0|i0|c0|p34;\H0.75x;Show the intersection (Curve/Line) of the intersection of these 2 object in the front view\H1.33333x; }

&lt;/hiero&gt;{\fVerdana|b0|i0|c0|p34;\H0.75x;Show the intersection (Curve/Line) of the intersection of these 2 object in the front view\H1.33333x; }


==Cylinders==
Line Intersecting a solid
Find piercing points by using sectional views.

1.	Think of the given line in one view to be the edge view of a section of the given solid.
2.	The points where the line crosses the base of the cylinder in this view are the piercing points.
3.	Find the piercing points in the other view by projecting these points down until they cross the given line.

Line Intersecting a cylinder

Use a vertical section plane.

1.	In the top view, consider the given line to be the edge view of a vertical section plane.
2.	Locate the points where the line crosses the base. Project theses piercing points down to the front view.
3.	The piercing points are where theses points intersect the line in the front view. 

Line Intersecting an inclined cylinder

Create a section plane parallel to the axis of the cylinder.

1.	In front view, select two points on the given line and draw two lines that are parallel to the axis through these points.
2.	Mark the points at which these two lines intersect the edge view of the base of the cylinder. Project these points to the top view until they hit the base.
3.	Draw lines parallel to the axis, to the other base of the cylinder.
4.	Where these two lines intersect the given line are the two piercing points.

Plane Intersecting a Cylinder

A cylinder is made of infinite number of lines that are parallel to the axis, along the base. Create a convenient cutting plane in one view, that contains a piercing point and project it to find the trace and to locate the piercing point in the other view.

1.	In top view, locate the apparent piercing points, where each side of the given plane crosses the base of the cylinder.
2.	Project these piercing points to the front view and mark them where they meet the same side of the plane.
3.	To find more points that make up the plane of intersection, use cutting planes and traces:
a.	Pick a point on the base in top view (should lie inside the given plane and along the base of the cylinder).
b.	Draw a line (represents the edge view of the cutting plane) that contains that point, across the given plane.
c.	Project the line and the piercing point down to the front view. The piercing point (in the front view) is located on that line.
d.	Connect these piercing points (in the front view), taking visibility into consideration, to find the plane of intersection.

For example, questions, see files &quot;Problem 1 Line Intersecting Cylinder&quot; and &quot;Problem 2 Plane Intersecting Cylinder.&quot;

&lt;gallery&gt;
File:Problem1 Line Intersecting Cylinder.pdf|Problem1 Line Intersecting Cylinder
File:Problem1 Solution.pdf|Problem 1 Solution
&lt;/gallery&gt;

&lt;gallery&gt;
File:Problem2 Plane Intersecting Cylinder.jpg|Problem2 Plane Intersecting Cylinder
File:Problem2 Solution.jpg|Problem 2 Solution
&lt;/gallery&gt;

==Cones==

''Intersection between plane and cone (given two views)''

Draw line parallel to folding line in one view through one point of intersecting plane. Transfer line to other given view with points on respective lines to get line in true length.  Draw folding line perpendicular to true length line and transfer points of cone to auxiliary view.  Transfer intersecting plane to auxiliary view to show plane in edge view and treat it as a cutting plane.  Draw lines from edge view of base plane to the vertex of the cone.  Transfer lines to both other views. Find intersections between cutting plane and cone line in auxiliary view and transfer points back to other two views to find intersections.  Determine visibility.

&lt;gallery&gt;
File:Problem1.1.jpg|Problem 1
File:Problem1.2.jpg|Problem 1 - Select arbitrary points on intersecting line
File:Problem1.3.jpg|Problem 1 - Extend lines from vertex
File:Problem1.4.jpg|Problem 1 - Find piercing points
&lt;/gallery&gt;

''Intersection between line and cone (given two views)''

Choose two arbitrary points on the intersecting line in front view.  Draw lines through these two points from the vertex to intersect the edge view of the base plane of the cone, label these intersections.  Project two arbitrary points on line vertically to top view onto intersecting line and draw lines through these points from the vertex in top view.  Project the two labelled intersections to top view and mark where it intersects the previously drawn lines from the vertex through the two projected arbitrary points.  Connect these two points to create the cutting plane of the base plane.  Connect where this line intersects the base of the cone in top view to the vertex,  where these two lines crosses the intersecting line are the piercing points of the line.  Carry the piercing points back to front view.

&lt;gallery&gt;
File:Problem 2.1.jpg|Problem 2
File:Problem 2.2.jpg|Problem 2 - Find one line in true length
File:Problem 2.3.jpg|Problem 2 - Create edge view of intersecting plane from true length line
File:Problem 2.4.jpg|Problem 2 - Extend lines from vertex of cone
File:Problem 2.5.jpg|Problem 2 - Find intersections
&lt;/gallery&gt;


[[File:DescGeoSphere.pptx|thumbnail]]
[[File:Fig1.1|thumbnail]]

==Spheres==
Section of a sphere:

Given two views of a sphere and its section. (Fig1.1)
[[File:Fig1.1.png|thumb|Figure1.1]]
To find the true section of the sphere, we draw an auxiliary view.

The folding line for this view must be parallel to the line (edge view of section shown in other view).(fig1.2)[[File:Fig1.2.png|thumb|Figure1.2]]
This will give us the true size of the section of the sphere.
The auxiliary view is constructed by simply transferring distances. (Fig1.3)
[[File:Fig1.3.png|thumb|Figure1.3]]

Intersection of a line and a sphere:
(Fig2.1)
[[File:Fig.2.1.png|thumb|Figure2.1]]
We can use the above construction to figure out the piercing points of a line and a sphere. To do so, we assume that the line in one of the two views is the edge view of the cutting plane (section plane). (Fig2.2)
[[File:Fig.2.2.png|thumb|Figure2.2]]
Using this information, we can construct the corresponding section of the sphere in the other view. 
The points where the line intersects this section can be transferred back onto the line to find the piercing points of the line and the sphere.
We can project these back to the first view.(Fig2.3)
[[File:Fig.2.3.png|thumb|Figure2.3]]

Practice problems: (with solutions)
(Refer to pictures named: 
Problem 1. Find (a)true section of the sphere and (b) piercing points
Solution 1(a)
Solution 1(b)
Problem 2
Solution 2 (a&amp;b)
[[File:Problem 1. Find (a)true section of the sphere and (b) piercing points.jpg|thumb|Practice Problem 1]]
[[File:Solution 1(a).jpg|thumb|True section]]
[[File:Solution 1(b).jpg|thumb|Piercing points]]
[[File:Problem 2. Find (a)true section of the sphere and (b) piercing points of the plane.jpg|thumb|Practice problem]]
[[File:Solution 2 (a&amp;b).jpg|thumb|True section and piercing points]]

==Intersections between Complex Solids==

Intersection between Complex Solids with Complex Solids
Cylinder and Cylinder
Cone and Cylinder
First determine if the two complex solids have axis that are parallel to each other.
If not, the first step is to construct a plane that contains a line that is parallel to the axis of both the complex solids which intersects each solid in top view.
If the faces of the shapes are not coplanar, extend one in front view until they have a common base plane.
Construct cutting planes that intersect the bases of the two solids parallel to the previously constructed plane to determine the intersections between the planes and the solids as shown in figure
Project the intersecting points obtained in the second step onto the corresponding solid in the other view.
Using those points of intersection, construct lines parallel to the axis of that solid so it intersects the solids in the same view that you used for the step above.
Mark the intersections that are generated from the same cutting plane. The two complex solids intersect at these points.
Join the intersecting points to get the shape where the two solids intersect each other.
Project the above obtained points corresponding to the solids in the other view to obtain the shape of the intersecting plane.
Note: You should use a good number of intersecting planes and follow the steps above to obtain an accurate shape of intersection.

To understand this topic better do the practice problems assigned below.
&lt;gallery&gt;
File:Problem2dg.jpeg|thumb|Descriptive Geometry
File:Solution2dg.jpeg|thumb|Descriptive Geometry
File:Problemdg.jpeg|thumb|Descriptive Geometry
File:Solutiondg.jpeg|thumb|Descriptive Geometry
File:PracticeProblem1Question.jpg|Image for Question 1 on Intersection of Complex Solids
File:PracticeProblem1Solution.jpg|Answer for Question 1 on Intersection of Complex Solids
File:PracticeProblem2Question.jpg|Image for Question 2 on Intersection of Complex Solids
File:PracticeProblem2Solution.jpg|Answer for Question 2 on Intersection of Complex Solids
&lt;/gallery&gt;

==Intersections with Complex and Planar Solids==

Break the planar solid into individual surfaces and address each surface individually.
For each surface on the planar solid, find intersections between the surface and the complex solid.  (Top View)
Using piercing points project these intersections into front view. These will be the points where the two solids intersect. (Top/Front View)
If more points are needed to complete the shape of the intersection, create lines on each surface and using the same method as before find more points of intersection.
Repeat with the other surfaces on the planar solid until a complete shape of intersection is found.
&lt;gallery&gt;
File:Prob1.jpg|Example Problem 1
File:Solution 1.jpg|Example Problem 1 Solution
File:Prob 2.jpg|Example Problem 2
File:Solution 2.jpg|Example Problem 2 Solution
File:Problem 3 Setup.JPG|thumb|Setup
File:Problem 3 Solution.JPG|thumb|Solution
File:Problem 4 Setup.JPG|thumb|Setup
File:Problem 4 Solution.JPG|thumb|Solution
&lt;/gallery&gt;

{{BookCat}}</text>
      <sha1>2obi390aywdoadigtmgemslae7jhqxg</sha1>
    </revision>
  </page>
  <page>
    <title>Flora of New York/Ericales</title>
    <ns>0</ns>
    <id>291752</id>
    <revision>
      <id>4631230</id>
      <parentid>4629983</parentid>
      <timestamp>2026-04-18T13:56:26Z</timestamp>
      <contributor>
        <username>Nonenmac</username>
        <id>6290</id>
      </contributor>
      <comment>/* Primula subg. Auriculastrum */</comment>
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      <text bytes="169666" sha1="lpfb9w9gce89n5xit8kbsh7f4bmsfvj" xml:space="preserve">{{../header
| this      = Ericales
| prev-link = Cornales
| next-link = Gentianales
}}

==Ericales introduction==

{{:Flora of New York/Clade1 Ericales}}
{{:Flora of New York/TOC-Ericales}}

==Family Balsaminaceae==

{{../txt|The {{../w|Balsaminaceae}} ('''balsam family''') contains the two genera: ''Impatiens'' with about 1000 species and ''Hydrocera'' with a single species. Of these, four ''Impatiens'' species have been reported growing wild in New York.&lt;ref&gt;{{../nyfa-fam|Balsaminaceae}}&lt;/ref&gt;&lt;ref&gt;{{../usda-fam|Balsaminaceae}}&lt;/ref&gt;&lt;ref&gt;[http://www.mobot.org/mobot/research/apweb/orders/ericalesweb.htm#Balsaminaceae P.F. Stevens, P. F. (2001-2015). ''Angiosperm Phylogeny Website''. Version 14, University of Missouri, St Louis, and Missouri Botanical Garden, April 2015.]&lt;/ref&gt;}}

===''Impatiens''===

{{../txt|[[File:Impatiens capensis and-or pallida SCA-4330.jpg|thumb|right|Jewelweed with bright white water droplets at the edges of the leaves.]]
Of the four ''Impatiens'' species found outside of cultivation in New York, all are annual herbaceous plants The two native species, '''spotted jewelweed''' (''Impatiens capensis'') and '''pale jewelweed''' (''Impatiens pallida''), are common in wet wooded areas throughout much of the state.

Two non-native ''Impatiens'' species may also be found, but are less common. '''Himalayan balsam''' (''Impatiens glandulifera'') is considered to be moderately invasive in New York.  '''Garden balsam''' (''Impatiens balsamina'' has been reported, but whether it has truly naturalized in the state is in question.}}

{{../table|order=Ericales|Balsaminaceae||||Impatiens|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Impatiens|Touch-me-not|90|4|IMPAT|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens capensis
| au-abbr = Meerb.
| au-full = Nicolaas Meerburgh
| au-pub  = Afb. Zeldz. Gew.: 8 (1775)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1775|Impatiens capensis    |Meerb.}}
{{../sp-1|1788|Impatiens biflora     |Walter}}
{{../sp-1|1813|Impatiens maculata    |Muhl. (not validly publ.)}}
{{../sp-1|1818|Impatiens fulva       |Nutt.}}
{{../sp-1|1849|Balsamina fulva       |(Nutt.) Ser.}}
{{../sp-1|1910|Impatiens nortonii    |Rydb.}}
{{../sp-1|1916|Chrysaea biflora      |(Walter) Nieuwl. &amp; Lunell}}
{{../ssp1|1967|Impatiens noli-tangere|L. |biflora |(Walter) Hultén}}
&lt;!-- -- --&gt;
| en1     = Spotted jewelweed
| en2     = Orange touch-me-not
| en3     = Spotted touch-me-not
| en4     = Spotted snapweed
| ---
| fr1     = Impatiente du Cap
| fr2     = Chou sauvage
| fr3     = Impatiente biflore
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 3
| wetland = FACW
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = S5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6618|5|Impatiens capensis Meerb. - spotted jewelweed, touch-me-not, snapweed - Floodplain forests, wet forests, seepage areas, swamps, marshes, fens, stream banks, thickets, disturbed areas, shaded roadsides and trail edges, and ditches. A very common annual of wet to mesic soils, preferring wetter sites. It can form dense large stands.}}
| usda    = {{../usda|IMCA|NN|}}
| vascan  = 3655
| gobot   = impatiens/capensis
| its-id  = 29182
| ars-id  = 19813
| fna-id  = 
| tro-id  = 3100006
| map     = Impatiens capensis NY-dist-map.png
| image1  = Jewelweed - Impatiens capensis, Julie Metz Wetlands, Woodbridge, Virginia - 27275915403.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens pallida
| author  = Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Impatiens noli-tangere |Michx. (sensu auct.)}}
{{../sp-1|1813|Impatiens aurea        |Muhl.}}
{{../sp-1|1818|Impatiens pallida      |Nutt.}}
{{../var1|1904|Impatiens pallida      |Nutt. |alba |Clute}}
{{../sp-1|1916|Chrysaea aurea         |(Muhl.) Nieuwl. &amp; Lunell}}
&lt;!-- -- --&gt;
| en1     = Pale jewel-weed
| en2     = Pale snapweed
| en3     = Pale touch-me-not
| en4     = Yellow touch-me-not
| ---
| fr1     = Impatiente pâle
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| wetland = FACW
| c-rank  = 3
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = S4, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|522|4|Impatiens pallida Nutt. - pale jewel-weed - Floodplain forests, wet forests, seepage areas, swamps, marshes, fens, stream banks, thickets, disturbed areas, shaded roadsides and trail edges, and ditches. Similar habitat to I. capensis and sometimes occurring together but not as common in at least parts of NY. It perhaps prefers more calcareous soils. Like I. capensis it can form dense large patches.}}
| usda    = {{../usda|IMPA|NN|}}
| vascan  = 3659
| gobot   = impatiens/pallida
| its-id  = 29189
| ars-id  = 417739
| fna-id  = 
| tro-id  = 3100027
| map     = 
| image1  = Impatiens pallida SCA-5938.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens glandulifera
| author  = Royle
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1835|Impatiens glandulifera |Royle}}
{{../sp-1|1840|Impatiens glanduligera |Lindl.}}
{{../sp-1|1842|Impatiens roylei       |Walp. (nom. superfl.)}}
{{../sp-1|1849|Balsamina roylei       |(Walp.) Ser.}}
{{../var1|2021|Impatiens sulcata      |Wall. |glandulifera|(Royle) R.Kr.Singh &amp; D.Borah}}
&lt;!-- -- --&gt;
| en1     = Ornamental jewelweed
| en2     = Policeman's helmet
| en3     = Himalayan balsam
| en4     = Himalayan touch-me-not
| ---
| fr1     = Balsamie de l'Himmalaya
| fr2     = Impatiente glanduleuse
| fr3     = Balsamine géante
&lt;!-- -- --&gt;
| status  = Introduced
| from    = India, Nepal
| from1   = Pakistan
| status1 = Moderately invasive
| c-rank  = X
| habit0  = Annual
| habit1  = Herb-forb
| nyis    = 67%&lt;ref&gt;{{../nyis-ref|Impatiens glandulifera|Moderate|67|first}}&lt;/ref&gt;
| ny-rank = SNA, GNR
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|521|X|Impatiens glandulifera Royle - policeman's helmet}}
| usda    = {{../usda|IMGL|XX|Impatiens glandulifera Royle - ornamental jewelweed }}
| vascan  = 3657
| gobot   = impatiens/glandulifera
| its-id  = 29187
| ars-id  = 19826
| fna-id  = 
| tro-id  = 3100049
| map     = 
| cos     = Columbia, Essex, Jefferson, Lewis, Madison, Schuyler, Sullivan
| image1  = Impatiens glandulifera on way from Gangria to Valley of Flowers National Park - during LGFC - VOF 2019 (12).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens balsamina
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Impatiens balsamina |L.}}
{{../sp-1|1790|Balsamina foemina   |Gaertn.}}
{{../sp-1|1922|Impatiens giorgii   |De Wild.}}
{{../sp-1|1962|Impatiens eriocarpa |Launert}}
&lt;!-- -- --&gt;
| en1     = Garden balsam
| en2     = Garden touch-me-not
| en3     = Spotted snapweed
| en4     = Rose balsam
&lt;!-- -- --&gt;
| status  = Introduced
| from    = India, Myanmar
| c-rank  = 
| wetland = UPL
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = SNA, GNR
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|520|X|Impatiens balsamina L. - garden touch-me-not, garden balsam, rose balsam}}
| usda    = {{../usda|IMBA|XX|}}
| vascan  = 
| gobot   = 
| its-id  = 29185
| ars-id  = 19810
| fna-id  = 
| tro-id  = 3100102
| map     = 
| cos     = Erie, Kings, Nassau, Suffolk
| image1  = Balsam Malaysia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens balfourii
| au-abbr = Hook. f.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1903|Impatiens balfourii |Hook. f.}}
{{../sp-1|1928|Impatiens mathildae |Chiov.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Balfour’s touch-me-not |2022 New York Flora Atlas}}
{{../vn1|Kashmir Balsam         |2023 iNaturalist}}
{{../vn1|Poor Man's Orchid      |2023 iNaturalist}}
| fr-vns  = 
{{../vn1|Impatience de Balfour  |2023 iNaturalist}}
{{../vn1|Impatiente des jardins |2023 iNaturalist}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Potentially invasive
| status2 = 
| imap    =
| ipa-us  = 
| griisus = 
| ny-tier =
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7807|Xu|}}
| usda    = {{../usda|||}}
| gbif    =
| pow-id  = 373976-1
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| fna-i1  = 
| fna-i2  = 
| tro-id  =
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;New York (2015)&quot;&gt;New York (2015)&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Nassau (2020), New York (2017-21)&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Impatiens balfourii Niecierpek Balfoura 2007-08-11 01.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

==Family Polemoniaceae==

The {{../w|Polemoniaceae}} (phlox family).&lt;ref&gt;{{../nyfa-fam|Polemoniaceae}}&lt;br&gt;{{../usda-fam|Polemoniaceae}}&lt;/ref&gt;

===Subamily Polemonioideae===

====Tribe Polemonieae====

=====''Polemonium''=====

{{../txt
|img=Jacob's ladder (24735231220).jpg
|cap=''Polemonium reptans''
|Although the New York Flora Atlas does not include ''Polemonium caeruleum'', Go Botany reports that it has escaped cultivation in New England.}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Polemonieae|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Polemonium|Jacob's ladder|474|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium reptans
| author  = L.
| var0    = reptans
&lt;!-- -- --&gt;
| en1     = Common Jacob's-ladder
| en2     = Creeping Jacob's-ladder
| en3     = Spreading Jacob's-ladder
| en4     = Greek valerian
| en5     = Charity
| fr1     = Polémoine rampante
| fr2     = Valériane grecque
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = FAC
| nwi2    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| habit2  = Subshrub
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6894|4|}}
| usda    = {{../usda|PORER|NN|}}
| vascan  = 8087               &lt;!-- Creeping Jacob's-ladder --&gt;
| gobot   = polemonium/reptans &lt;!-- Spreading Jacob's-ladder --&gt;
| its-id  = 529739             &lt;!-- Greek valerian --&gt;
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20reptans
| map     = Polemonium reptans var reptans nymap.svg
| image1  = Jacob's ladder (25910631796).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium vanbruntiae
| author  = Britton 
&lt;!-- -- --&gt;
| en1     = Vanbrunt's Jacob's ladder
| en2     = Vanbrunt's polemonium
| en3     = Bog Jacob's ladder
| fr1     = Polémoine de Van-Brunt
&lt;!-- -- --&gt;
| status  = Native
| status1 = Rare
| c-rank  = 7
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2418|3|}}
| usda    = {{../usda|POVA5|NN|}}
| vascan  = 8088
| gobot   = polemonium/vanbruntiae
| its-id  = 504486
| ars-id  = 310258
| fna-id  = 
| tro-id  = 25800477
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20vanbruntiae
| map     = Polemonium vanbruntiae nymap.svg
| image1  = Polemonium vanbruntiae FWS-2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Polemonium|Jacob's ladder|474|X|txt=(unlisted&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium caeruleum
| author  = L.
&lt;!-- -- --&gt;
| en1     = Blue Jacob's-ladder
| en2     = Charity
| en3     = Greek-valerian
| fr1     = Polémoine bleue
| fr2     = Valériane grecque
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| status1 = Cultivated
| c-rank  = 
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|POCA2||}}
| vascan  = 8080
| gobot   = polemonium/caeruleum
| its-id  = 31005 
| ars-id  = 29176
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20caeruleum
| map     = Excluded nymap.svg
| nyfa-co = &lt;abbr title=&quot; &quot;&gt;Not listed&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot; &quot;&gt;Jefferson County 2017&lt;/abbr&gt;
| image1  = Polemonium caeruleum BotGardBln 20170610 I.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Phlocideae====

=====''Phlox''=====

{{:Flora of New York/txt
|The sectional classification of the genus ''Phlox'' used here (Wherry 1955, Grant 1959)&lt;ref&gt;[http://w3.biosci.utexas.edu/prc/pdfs/Grant02_Lundellia04.pdf Verne Grant (2001). &quot;Nomenclature of the Main Subdivisions of ''Phlox'' (Polemoniaceae).&quot; ''Lundellia'' '''4:25''', 2001.]&lt;/ref&gt; is:
* Sect. '''''Phlox''''' (''alpha-Phlox''). Perennial herbs with soft deciduous leaves and long styles (''P. glaberrima, P. paniculata, P. subulata''...)
* Sect. '''''Divaricatae''''' (''Protophlox''). Perennial or annual herbs with soft deciduous leaves and short styles (''P. divaricata, P. drummondii, P. nivalis, P. pilosa''...)
* Sect. '''''Occidentales''''' (''Microphlox''). Cespitose or cushion-like subshrubs with stiff evergreen leaves and short styles. (''P. caespitosa, P. hoodii, P. sibirica''...)
Of these sections, only the herbaceous North American sections ''Phlox'' and ''Divaricatae'' contain native or naturalized species reported in New York.
}}

======''Phlox'' sect. ''Divaricatae''======

{{../txt
|img=Phlox divaricata 3.jpg
|cap=''Phlox divaricata''
|Sect. ''Divaricatae'' contains perennial or annual herbs with soft deciduous leaves and short styles.
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Phlocideae||Phlox||Divaricatae|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Phlox|Phlox|297|6|sect=Divaricatae|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox divaricata
| author  = L.
| ssp0    = divaricata
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Timber phlox|2021 New York Flora Atlas}}
{{../vn1|Wild blue phlox|2021 New York Flora Atlas}}
{{../vn1|Woodland phlox|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 8
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6891|4|Phlox divaricata L. ssp. divaricata - wild blue phlox - Native. Apparently secure in New York State. - Rich mesic to dry-mesic forests, floodplain forests, and forest edges.}}
| usda    = {{../usda|PHDID3|NN|Endangered in New Jersey}}
| vascan  = 
| gobot   = Phlox/divaricata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Phlox divaricata - Wild Blue Phlox 2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox pilosa
| author  = L.
| ssp0    = pilosa
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox pilosa|L.}}
{{../sp-1|1911|Phlox argillacea|Clute &amp; Ferriss}}
{{../var1|1931|Phlox pilosa|L.|amplexicaulis|(Raf.) Wherry}}
{{../var1|1931|Phlox pilosa|L.|virens|(Michx.) Wherry}}
&lt;!-- -- --&gt;
| en1     = Downy phlox
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| status2 = Impersistent
| c-rank  = 10
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = SH
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6892|1z|Phlox pilosa L. ssp. pilosa - downy phlox - Historically known from New York State, but not seen in the past 15 years.}}
| usda    = {{../usda|PHPIP2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 30975
| ars-id  = 432155
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = Phlox+pilosa
| bna-id  = Phlox%20pilosa
| map     =  nymap.svg
| cos     = Niagara (1838)
| image1  = Phlox pilosa2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

======''Phlox'' sect. ''Phlox''======

{{../txt
|img=2021-04-17 12 22 06 Creeping Phlox along Old Dairy Road in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg
|cap=''Phlox subulata''
|Section '''''Phlox''''' contains perennial herbs with soft deciduous leaves and long styles, including native '''moss phlox''' (''Phlox subulata''), endangered '''wild sweet William''' (''Phlox maculata''), as well as common introduced '''garden phlox''' (''Phlox paniculata'').
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Phlocideae||Phlox||Phlox|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Phlox|Phlox|297|6|sect=Phlox|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox subulata
| author  = L.
| ssp0    = subulata
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox subulata   |L.}}
{{../sp-1|1823|Phlox aristata   |Lodd.}}
{{../sp-1|1891|Armeria subulata |(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Moss phlox|2021 New York Flora Atlas - 2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening }}
{{../vn1|Creeping phlox|2021 ARS-GRIN: Locklear, J. H. 2011. Phlox: a natural history and gardener's guide Timber Press, Portland, Oregon. 248-253. Note: recognizes subsp. setacea (Linnaeus) Locklear and brittonii (Small) Wherry as well }}
{{../vn1|Moss pink|2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Ground pink|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Mountain phlox|2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening }}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S4
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6893|4|}}
| usda    = {{../usda|PHSUS2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 314438
| fna-id  = 
| tro-id  = 25800441
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Mauve Flowers 2 (4540691499).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox maculata
| author  = L.
| ssp0    = maculata
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox maculata|L.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Wild sweet-william|2021 New York Flora Atlas - 2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening. - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Wild sweetwilliam|2021 USDA NRCS National Plant Data Team}}
{{../vn1|Meadow phlox|2021 ARS-GRIN: Gleason, H. A. &amp; A. Cronquist. 1991. Manual of vascular plants of northeastern United States and adjacent Canada, ed. 2. - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Spotted phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967-. The vascular flora of Ohio.}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 7
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S2, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2416|1|}}
| usda    = {{../usda|PHMAM|NX|}}
| gbif    =
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 524492
| ars-id  = 405814
| fna-id  =
| tro-id  = 25800411
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = Phlox+maculata
| bna-id  = Phlox%20maculata
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = Phlox maculata ssp maculata NY-dist-map.png
| image1  = Phlox maculata NRCS-1 (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox paniculata
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox paniculata|L.}}
{{../sp-1|1813|Phlox acuminata|Pursh}}
{{../sp-1|1813|Phlox decussata|Lyon ex Pursh }}
&lt;!-- -- --&gt;
| en1     = Fall phlox
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = 
| c-rank  = 
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Sun
| chro-no = 
| ny-rank = SNA, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2414|X|}}
| usda    = {{../usda|PHPA9|NX|}}
| gobot   = Phlox/paniculata
| its-id  =
| ars-id  = 401812
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = PHPA9
| nse-id  = Phlox+paniculata
| bna-id  = Phlox%20paniculata
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  =  
| map     =
| image1  = Phlox paniculata Tatjana20140704 022.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox stolonifera
| author  = Sims
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1892|Phlox stolonifera|Sims}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Creeping phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967. The vascular flora of Ohio. - 2021 New York Flora Atlas.}}
{{../vn1|Cherokee phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967. The vascular flora of Ohio.}}
| en1     = Creeping phlox
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Not naturalized
| c-rank  = X
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = SNA, G4-5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2417|X|}}
| usda    = {{../usda|PHST3||}}
| vascan  = 
| gobot   = Phlox/stolonifera
| its-id  = 
| ars-id  = 405816
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = Phlox+stolonifera
| bna-id  = Phlox%20stolonifera
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| cos     = Fulton, Rensselaer (1947)
| image1  = Creeping Phlox Phlox stolonifera Flowers 3008px (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====Tribe Gilieae====

=====''Collomia''=====

{{:Flora of New York/txt
|img=Collomia linearis - Flickr - aspidoscelis (1).jpg
|cap=''Collomia linearis''
|''Collomia'' species, commonly known as '''trumpets''' or '''mountain trumpets''', are native to western North America. Of these the only species reported in the northeast is the '''tiny trumpet''' (''Collomia linearis'').
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Gilieae||Collomia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Collomia|Trumpet|86|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Collomia linearis
| author  = (Cav.) Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1800|Phlox linearis|Cav.}}
{{../sp-1|1818|Collomia linearis|(Cav.) Nutt.}}
{{../sp-1|1882|Gilia linearis (Nutt.)|A. Gray}}
{{../sp-1|1891|Navarretia linearis|(Nutt.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Tiny trumpet|2021 New York Flora Atlas - 2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
{{../vn1|Narrow-leaved collomia|2021 Tropicos: Canada Weed Committee. 1969. Comm. bot. nam. weeds Canada 1–67.}}
{{../vn1|Narrow leaved collomia|2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
{{../vn1|Narrow-leaved mountain-trumpet|2021 Native Plant Trust, Go Botany}}
{{../vn1|Slenderleaf collomia|2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = western N. America
| status1 = N. America native
| status2 = Impersistent
| status3 = Not naturalized
| c-rank  = 
| nwi1    = FACU
| nwi2    = UPL
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| ny-rank = SNA, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2413|Xm|}}
| usda    = {{../usda|COLI2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 316631
| fna-id  = 
| tro-id  = 25800001
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Collomia%20linearis
| map     = 
| cos     = Cattaraugus (1899)
| image1  = Collomia linearis (1).jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Gilia''=====

{{:Flora of New York/txt
|img=Gilia achilleifolia-2.jpg
|cap=''Gilia achilleifolia''
|About 35 ''Gilia'' species are native to western North America.  Of these, only a few have been reported in the northeast. The New York Flora Atlas shows only ''Gilia achilleifolia'' ('''California gilia'''), collected in Brooklyn in 1880. 

Although excluded by the New York Flora Atlas, &quot;reaserch-grade&quot; iNaturalist observations of ''Gilia capitata'' ('''bluehead gilia''') were made in 2020 in Nassau County&lt;ref&gt;[https://www.inaturalist.org/observations/49079716 ''Gilia capitata'' (Bluehead Gilia) iNaturalist observation, West Hempstead, NY, Jun 9, 2020]&lt;/ref&gt; and in 2021 in Ulster County.&lt;ref&gt;[https://www.inaturalist.org/observations/82928314 ''Gilia capitata'' (Bluehead Gilia) iNaturalist observation, Kingson, NY, Jun 13, 2021]&lt;/ref&gt;
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Gilieae||Gilia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gilia|Gilia|1025|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gilia achilleifolia
| author  = Benth.
| ssp     = achilleifolia
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1833|Gilia achilleifolia|Benth.}}
&lt;!-- -- --&gt;
| en1     = California gilia
| en2     = Blue gilia
&lt;!-- -- --&gt;
| status  = Introduced
| from    = California
| status1 = N. America native
| status2 = Impersistent
| c-rank  = 
| nwi1    = 
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| ny-rank = SNA
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7195|Xm|}}
| usda    = {{../usda|GIACA|X0|}}
| vascan  = 
| gobot   = Gilia/achilleifolia
| its-id  = 31096
| ars-id  = 
| fna-id  = 
| tro-id  = 25800683
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Gilia%20achilleifolia
| map     =  nymap.svg
| cos     = Kings (1880)
| image1  = Gilia achilleifolia 042.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gilia|Gilia|1025|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gilia capitata
| au-abbr = Sims
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1826|Gilia capitata|Sims}}
{{../sp-1|1891|Navarretia capitata|(Sims) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Bluehead gilia|2021 iNaturalist}}
{{../vn1|Gillyflower|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Field gilia|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
| fr-vns  = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = N. America native
| status2 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|799|EXCLUDED}}
| usda    = {{../usda|GICA5|NX|}}
| gbif    =
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 25800126
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Gilia%20capitata
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = Excluded nymap.svg
| cos     = Nassau (2020),&lt;br&gt;Ulster (2021)
| image1  = Gilia capitata (468859352).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Loeselieae====

======''Ipomopsis''======

{{:Flora of New York/txt
|img=Ipomopsis rubra - Standing Cypress Gilia rubra (8470988667).jpg
|cap=''Ipomopsis rubra''
|
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Loeselieae||Ipomopsis|
}}
{{../genus|Ipomopsis|Ipomopsis|479|X|txt=(excluded&amp;nbsp;taxa)|
}}
{{../taxon
| species = Ipomopsis rubra
| author  = (L.) Wherry
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Polemonium rubrum|L.}}
{{../sp-1|1891|Navarretia rubra|(L.) Kuntze}}
{{../sp-1|1895|Gilia rubra|(L.) Heller}}
{{../sp-1|1936|Ipomopsis rubra|(L.) Wherry}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Red standing-cypress|}}
{{../vn1|Standing-cypress|}}
&lt;!-- -- --&gt;
| status  = N. America native
| of      = southern U.S.
| status1 = Excluded
| nyfa    = {{../nyfa|X|846|}}
| usda    = {{../usda|IPRU2|NX|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 316899
| fna-id  = 
| tro-id  = 25800231
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = Excluded nymap.svg
| cos     = excluded
| image1  = Ipomopsis rubra.jpg
}}
{{../end table|Polemoniaceae|Polemonioideae|Loeselieae|
}}

==Family Ebenaceae==

{{../txt|The '''{{../w|Ebenaceae}}''' (ebony family) is primarily a tropical and warmer-temperate family of trees and shrubs. Its only species reported outside of cultivation in New York is the American persimmon tree, which consists mainly of escapes from cultivation. It is believed that the only native persimmon trees in the state are on Staten Island (Richmond County) and Long Island (Nassau and Suffolk Counties).&lt;ref&gt;{{../nyfa-fam|Ebenaceae}}&lt;/ref&gt;&lt;ref&gt;{{../usda-fam|Ebenaceae}}&lt;/ref&gt;}}

===Subfamily Ebenoideae===

====''Diospyros''====

{{../txt
|img=Persimmon American3 Asit.jpg
|cap=American Persimmons (''Diospyros virginiana'') in Tampa, Florida
|''Diospyros'' is primarily a tropical genus of trees, known generally as '''ebony''', but ''Diospyros virginiana'' ('''persimmon''') is native to the central and eastern United States, including New York.

In New York State, the only known native persimmon populations are in New York City and Long Island. Most of the naturalized trees in the state are assumed to be garden escapes. &lt;ref name=nynhp&gt;{{../nynhp-ref|9001|Diospyros virginiana|Threatened|S2|G5}}&lt;/ref&gt;

The genus name ''Diospyros'' was derived from the Greek ''dios'' (divine) and ''pyros'' (wheat or grain) in reference to the tree's &quot;divine fruit.&quot;

Persimmons are dioecious, requiring the presence of both male and female trees to produce fruit.
}}

{{../table|order=Ericales|Ebenaceae|Ebenoideae|||Diospyros}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Diospyros|Persimmon|673|1}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Diospyros virginiana
| au-abbr = L.
| au-full = Carl von Linnaeus (1707-1778).
| au-pub  = Species Plantarum 2: 1057–1058. 1 May 1753.
&lt;!-- -- --&gt;
| syns    = &lt;poem&gt;
1753.'' '''Diospyros virginiana'''   ''L.              in Sp. Pl.:1057
1817.''    Diospyros caroliniana  ''Muhl. ex Raf.   in Fl. Ludov.:139
1834.''    Diospyros persimon     ''Wikstr.         in Jahresber. Königl. Schwed. Akad. Wiss. Fortschr. Bot.
1838.''    Persimon virginiana    ''(L.) Raf.       in Sylva Tellur.:164 
1840.''    Diospyros angustifolia ''Audib. ex Spach in Hist. Nat. Vég. 9:405
1921.''    Diospyros mosieri      ''Small           in J. New York Bot. Gard. 22:33
1921.''    Diospyros virginiana   ''var. ''mosieri ''(Small) Sarg. in J. Arnold Arbor. 2(3): 170.
&lt;/poem&gt;
&lt;!-- -- --&gt;
| en1     = Persimmon
| en2     = Common persimmon
| en3     = American persimmon
| fr1     = Plaqueminier d'Amérique
&lt;!-- ========= --&gt;
| status  = Native
| status1 = Threatened
| nynhp   = '''S2'''&lt;ref&gt;[https://guides.nynhp.org/persimmon/ New York Natural Heritage Program. 2024. Online Conservation Guide for Diospyros virginiana.]&lt;/ref&gt;
| c-rank  = 8
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Tree
| light   = Heliophily: 6 &lt;br&gt; Part shade 
| chro-no = 2n = 60, 90
| ny-rank = S2, G5
| ns-rank = 
&lt;!-- ========= --&gt;
| nyfa    = {{../nyfa|1287|8 counties|Bronx, Nassau, New York, Richmond, Rockland, Suffolk, Westchester}}
| inatc   = {{../inat|83435-Diospyros-virginiana|13 counties|13 counties: Bronx, Dutchess, Kings, Monroe, Nassau, New York, Putnam, Queens, Richmond, Rockland, Suffolk, Ulster, Westchester}}
| gbifc   = {{../gbif||present in NY|present in New York State}}
| usda    = {{../usda|DIVI5|N0|NY is at the northern edge of this tree's US range}}
&lt;!-- ========= --&gt;
| col-id  = 
| pow-id  = 
| wfo-id  = 
| fsus    = 671
| vascan  = 
| gobot   = Diospyros/virginiana
| its-id  = 
| ars-id  = 14329
| fna-i1  = 242416451 | fna-i2 = Diospyros_virginiana
| tro-id  = 11500206
| lbj-id  = DIVI5
| ipn-id  = 
| nse-id  = Diospyros+virginiana
| bna-id  = diospyros%20virginiana
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| red-id  =
| mbg-id  = h740
| map     = nymap.svg
| image1  = Diospyros virginia (Ebenaceae) (leaves).JPG|width1=96
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Primulaceae==

{{../txt|The '''{{../w|Primulaceae}}''' {{../au|Batsch ex Borkh.}} (primrose family).&lt;ref&gt;{{../nyfa-fam|Primulaceae}}&lt;br&gt;{{../usda-fam|Primulaceae}}&lt;/ref&gt;  Note that this family does not include evening primroses (''Oenothera''), which are in the Onagraceae (evening primrose or willowherb family) in the order Myrtales.

The classification used here for Primulaceae is based primariliy on [http://www.mobot.org/MOBOT/research/APweb/orders/ericalesweb.htm#Ericales P. F. Stevens (2001-2015). ''Angiosperm Phylogeny Website''. Version 14, April 2015.]
}}

===Subfamily Theophrastoideae===

The Theophrastoideae (brookweeds &amp;&amp;) ...

====Tribe Samoleae====

The {{../w|Samoleae}} (brookweeds) ...

=====''Samolus''=====

{{../txt
|img=SamolusValerandiFlowers.jpg
|cap=''Samolus valerandi'' flowers
|The native ''Samolus valerandi'' ('''Water pimpernel''') is the only species of the genus ''Samolus'' reported in the wild in New York.
}}

{{../table|order=Ericales|Primulaceae|Theophrastoideae|Samoleae||Samolus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Samolus|Brookweed|816|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Samolus valerandi
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Samolus valerandi|L.}}
{{../sp-1|1818-1|S. parviflorus|Raf.}}
{{../sp-1|1818-2|S. floribundus|Kunth}}
{{../sp-1|1825|S. americanus|Spreng.}}
{{../ssp1|1971|S. valerandi|L.|parviflorus|Hultén}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Water pimpernel|}}
{{../vn1|Brookweed|}}
{{../vn1|Seaside brookweed|Go Botany}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|2539|4|as Samolus valerandi}}
| usda    = {{../usda|SAVAP|NN|Samolus valerandi ssp parviflorus}}
| vascan  = 21425       &lt;!-- Samolus parviflorus --&gt;
| gobot   = samolus/valerandi
| its-id  = 504999     &lt;!-- Samolus valerandi --&gt;
| ars-id  = 459207
| fna-id  = 242417215  &lt;!-- Samolus parviflorus --&gt;
| tro-id  = 26400042
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Samolus%20parviflorus &lt;!-- Samolus parviflorus --&gt;
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =  nymap.svg
| image1  = Samolus valerandi carex pre-a-pion-longpre-les-corps-saints 80 03062008 2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Primuloideae===

====''Primula''====

=====''Primula'' subg. ''Aleuritia''=====

{{../txt|[[File:Primula mistassinica.jpg|thumb|right|Lake Mistassini primrose, photographed in Wisconsin]] The only native species of ''Primula'' subg. ''Aleuritia'' is '''Lake Mistassini primrose''' or '''bird's eye primrose'''. {{../w|Lake Mistassini}} is the largest natural lake in Quebec.  Mistassini primrose is rare this far south. It is considered threatened in New York, where it is found only on cool wet cliff faces.&lt;ref&gt;{{../nynhp-ref|9257|Primula mistassinica|Threatened|S2|G5}}&lt;/ref&gt; 

The only non-native member of subg. ''Aleuritia'' reported to have naturalized in New York is '''Japanese primrose''', which has been found in Onondaga and Otsego counties.}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Aleuritia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|subg=Aleuritia|sect=Aleuritia|Primrose|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula mistassinica
| author  = Michx.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Primula mistassinica|Michx.}}
{{../var1|1888|Primula farinosa|L.|mistassinica|(Michx.) Pax}}
{{../var1|1894|Primula sibirica|Wulfen|mistassinica|(Michx.) Kurtz}}
{{../ssp1|1905|Primula farinosa|L.|mistassinica|(Michx.) Pax}}
{{../sp-1|1928|Primula intercedens|Fernald}}
{{../var1|1928|Primula mistassinica|Michx.|intercedens|Fernald}}
{{../var1|1966|Primula mistassinica|Michx.|intercedens|B.Boivin}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Lake Mistassini primrose|NYFA-1, Flora Novae Angliae, VASCAN-S}}
{{../vn1|Bird's-eye primrose|NYFA-2, NYNHP, VASCAN-S}}
{{../vn1|Mistassini primrose|VASCAN-A}}
{{../vn1|Dwarf Canadian primrose|VASCAN-S}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Primevère du lac Mistassini|VASCAN-A}}
{{../vn1|Primevère de Mistassini|VASCAN-S}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| nynhp   = 2&lt;ref&gt;{{../nynhp-ref|9257|Primula mistassinica|Threatened|S2|G5}}&lt;/ref&gt;
| c-rank  = 10
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2540|2|Cool calcareous perennially seepy north facing cliffs in relatively small isolated disjunct populations. Often found with Pinguicula vulgaris and Saxifraga aizoides.}}
| usda    = {{../usda|PRMI|NN|}}
| vascan  = 8375
| gobot   = primula/mistassinica
| its-id  = 24028
| ars-id  = 
| fna-id  = 250092230
| tro-id  = 26400185
| ipn-id  = 
| lbj-id  = PRMI
| nse-id  = 
| bna-id  = Primula%20mistassinica
| map     = 
| image1  = Primula mistassinica WFNY-159B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|subg=Aleuritia|sect=Proliferae|Primrose|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula japonica
| author  = A.Gray
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1858|Primula japonica|A.Gray}}
{{../sp-1|1980|Aleuritia japonica|(A.Gray) Soják}}
&lt;!-- -- --&gt;
| en1     = Japanese primrose
| fr1     = Primevère du Japon
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Japan
| status1 = Highly invasive
| status2 = Naturalized
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2527|X|Onondaga only}}
| usda    = {{../usda|PRJA2|X0|NY only}}
| gbif    = 5414324
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 29640
| fna-id  = 
| tro-id  = 26400450
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Onondaga, Otsego (2010)&quot;&gt;2 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Madison (2021), Nassau (2019), Oneida (2020), Tompkins (2017), Ulster (2019), Westchester (2020)&quot;&gt;6 counties&lt;/abbr&gt;
| image1  = Primula japonica 1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Primula'' subg. ''Primula''=====

{{../txt
|img=Cowslip (Primula veris) in Great Ashby District Park, Stevenage.jpg
|cap=''Primula veris'' - cowslip
|
}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Primula|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Primrose|subg=Primula|sect=Primula|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula veris
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Primula veris|L.}}
{{../sp-1|1765|Primula officinalis|(L.) Hill}}
&lt;!-- -- --&gt;
| en1     = Cowslip
| en2     = Cowslip primrose
| ---
| fr1     = Primevère officinale
| status  = Introduced
| from    = Europe
| from1   = temperate w. Asia
| status1 = Naturalized
| nyfa    = {{../nyfa|2543|X|}}
| usda    = {{../usda|PRVE2|XW|}}
| vascan  = 8378
| gobot   = 
| its-id  = 
| ars-id  = 400303
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Primula veris ENBLA03.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Primrose|subg=Primula|sect=Primula|255|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula × polyantha
| author  = Mill.
&lt;!-- -- --&gt;
| hp1     = Primula veris
| hp2     = Primula vulgaris
&lt;!-- -- --&gt;
| en1     = Elatior hybrid primrose
| en2     = False oxlip
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Excluded
| status2 = Only cultivated
| nyfa    = {{../nyfa|X|804|EXCLUDED}}
| usda    = {{../usda|PRPO2|XX|}}
| vascan  = 8379
| gobot   = 
| its-id  = 
| ars-id  = 29671 &lt;!-- as Primula polyantha Mill.  --&gt;
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = Excluded nymap.svg
| image1  = Primula denticulata.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Primula'' subg. ''Auriculastrum''=====

{{../txt
|img=Dodecatheon meadia 01.JPG
|cap=''Primula meadia'' &lt;small&gt;(L.) A.R.Mast &amp; Reveal&lt;/small&gt;&lt;br&gt;sooting star
|Based on molecular evidence, A. R. Mast &amp; Reveal, in 2006, transferred the 17 species of genus ''Dodecatheon'' (the '''sooting stars''') to a new sect. ''Dodecatheon'' in ''Primula'' subg. ''Auriculastrum''.&lt;ref&gt;[http://www.botany.wisc.edu/courses/botany_940/11JClub/MastReveal2007.pdf Austin R. Mast and James L. Reveal (2007). &quot;Transfer of ''Dodecatheon'' to ''Primula'' (Primulaceae).&quot; ''Brittonia'', The New York Botanical Garden. 59(1):79-82. 2007.]&lt;/ref&gt; The transfer included the only (and impersistent) New York native, ''Primula meadia'' ('''Eastern shooting star'''). The shooting stars are still listed in genus ''Dodecatheon'' by many sources.

The New York Natural Heritage Program Rare Plant Status List - Active Inventory List listed
*in 2004: ''Dodecatheon meadia'' ssp. ''meadia'' (Shooting-star) STEU-X G5T5 SX U&lt;ref&gt;[https://www.dot.ny.gov/divisions/engineering/environmental-analysis/manuals-and-guidance/epm/repository/4-1-h.pdf New York Natural Heritage Program 2004 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2022: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/5/rare-plant-status-lists-2022.pdf New York Natural Heritage Program 2022 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2023: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/5/rare-plant-status-lists-2023.pdf New York Natural Heritage Program 2023 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2025: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/508/rare-plant-status-list-2025.pdf New York Natural Heritage Program 2023 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Auriculastrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Shooting-star|subg=Auriculastrum|sect=Dodecatheon|255|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula meadia
| au-abbr = (L.) A.R.Mast &amp; Reveal
| au-full = 
| au-pub  = Brittonia 59: 81 (2007)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Dodecatheon meadia        |L.}}
{{../sp-1|1768|Meadia dodecatheon        |Mill.}}
{{../sp-1|1818|Dodecatheon angustifolium |Raf. }}
{{../sp-1|1903|Dodecatheon hugeri        |Small}}
{{../sp-1|1903|Dodecatheon brachycarpum  |Small}}
{{../sp-1|2007|'''Primula meadia'''      |A.R.Mast &amp; Reveal  In: Brittonia 59(1):81}}
&lt;!-- -- --&gt;
| en1     = Eastern shooting star
| en2     = Shooting star
| en3     = Pride of Ohio
| en4     = American cowslip
| en5     = Mead's shootingstar
| fr1     = Gyroselle de Virginie
&lt;!-- -- --&gt;
| status  = Native
| status1 = Extirpated
| c-rank  = 10
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Heliophily: 5
| chro-no = 
| ny-rank = SX, G5
&lt;!-- ========= --&gt;
| nyfa    = {{../nyfa|2544|Steuben |Steuben (no date)}}
| inatc   = {{../inat|549981-Primula-meadia|7 counties|7 counties}}
| gbifc   = {{../gbif|5640540|present in NY|present in New York State}}
| usda    = {{../usda|DOMEM3|NN|Steuben county only}} 
&lt;!-- ========= --&gt;
| gbif    =
| pow-id  = 60447606-2
| wfo-id  = 
| fsus    = 676
| vascan  = 8359      | Primula meadia (L.) A.R.Mast &amp; Reveal (2007)
| gobot   = primula/meadia
| its-id  = 836161    | Primula meadia (L.) A.R.Mast &amp; Reveal
| ars-id  = 459448    | Primula meadia (L.) A.R.Mast &amp; Reveal
| fna-id  = 220004300 | Dodecatheon meadia L.
| tro-id  = 50308078  | Primula meadia (L.) A.R.Mast &amp; Reveal
| ipn-id  = 
| nse-id  = Dodecatheon+meadia
| bna-id  = Dodecatheon%20meadia
| map     = nymap.svg
| image1  = Dodecatheon meadia habit.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====''Hottonia''====

{{../table|order=Ericales|Primulaceae|Primuloideae|||Hottonia|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Hottonia|Hottonia|494|1|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Hottonia inflata
| author  = Elliott 1817
| ---
| en1     = American featherfoil
| en2     = Featherfoil
| en3     = Hottonie enflée
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|2541|2|}}
| usda    = {{../usda|HOIN|NN|}}
| vascan  = 
| gobot   = hottonia/inflata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Hottonia inflata NRCS-2.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

====''Androsace''====

{{../table|order=Ericales|Primulaceae|Primuloideae|||Androsace|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Androsace|Rock jasmine|218|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Androsace maxima
| author  = L.
| ---
| en1     = Androsace
| en2     = Greater rock-jasmine
| en3     = Pussy-toes
| ---
| status  = Introduced
| status1 = 
| nyfa    = {{../nyfa|6482|X|}}
| usda    = {{../usda|ANMA14|X0|NY is only location shown}}
| image1  = Androsace maxima sl1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Myrsinoideae===

{{:Flora of New York/txt
|img=
|cap=
|The {{../w|Myrsinoideae}} subfamily has previously been grouped into its own family (Myrsinaceae). It contains about 35 genera worldwide, but of those, only '''''Lysimachia''''' is known to be native or naturlized in New York. The genera '''''Anagallis''''' and '''''Trientalis''''' are now treated as part of ''Lysimachia''.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51.&lt;/ref&gt;
}}

====''Lysimachia''====

{{../txt
|The grouping used here for genus ''Lysimachia'' is based on Anderberg (2007).&lt;ref&gt;[http://www.bgbm.org/sites/default/files/documents/wi37-2Anderberg%2Bal.pdf A. A. Anderberg, U. Manns, and M. Källersjö (2007). &quot;Phylogeny and floral evolution of the Lysimachieae (Ericales, Myrsinaceae): evidence from ''ndhF'' sequence data.&quot; ''Willdenowia'' '''37:'''407-421.]&lt;/ref&gt;}}

=====''Lysimachia'' subg. ''Trientalis''=====

{{:Flora of New York/txt
|img=Trientalis borealis 10137.JPG
|cap=''Lysimachia borealis''
|Based on phylogenetic research, members of the genus ''Trientalis'' ('''starflowers''') were transferred to this subgenus in 2009 by U. Manns &amp; A. A. Anderberg.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51. [''Trientalis borealis'' Raf. = ''Lysimachia borealis'' (Raf.) U.Manns &amp; Anderb., comb. nov.].&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Trientalis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Starflower|133|1|subg=Trientalis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia borealis
| au-abbr = (Raf.) U.Manns &amp; Anderb.
| au-full = 
| au-pub  = Willdenowia 39: 51 (2009)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Trientalis europaea   |Michx non L. 1753}}
{{../var1|1805|Trientalis europaea   |L.    |americana |Pers.}}
{{../sp-1|1808|Trientalis borealis   |Raf.}}
{{../sp-1|1813|Trientalis americana  |Pursh}}
{{../var1|1872|Lysimachia trientalis |Klatt |americana  |(Pursh) Klatt}}
{{../var1|1924|Trientalis borealis   |Raf.  |tenuifolia |House}}
{{../sp-1|2009|Lysimachia borealis   |(Raf.) U.Manns &amp; Anderb.}}
&lt;!-- -- --&gt;
| en1     = Northern starflower
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = FAC
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = S5 G5
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2536|49 counties|49 counties}}
| usda    = {{../usda|TRBOB|NN|}}
| gbif    =
| pow-id  = 77100389-1
| wfo-id  = 
| vascan  = 
| gobot   = lysimachia/borealis
| inat    = 204174-Lysimachia-borealis
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Trientalis borealis 1177.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group A&lt;/small&gt;=====

{{../txt
|img=Lysimachia hybrida iNat 140267462 (3x4).jpg
|cap=''Lysimachia hybrida''&lt;br&gt;at Alley Park in Queens&lt;br&gt;photographed by [https://www.inaturalist.org/observations/85269469 Zihao Wang]
|''Lysimachia'' subg. ''Lysimachia'' '''group A''' contains the two New-World sections: &lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51.&lt;/ref&gt;
*Sect. '''''Seleucia''''' (''Lysimachia ciliata'', ''L. quadriflora'' and ''L. hybrida'') 
*Sect. '''''Theopyxis''''' (''Lysimachia andina'', a South American species). 
The New York species of group A are in sect. ''Seleucia''.
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Seleucia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia ciliata
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|'''Lysimachia ciliata'''|L.}}
{{../sp-1|1843|Steironema ciliatum     |(L.) Baudo}}
{{../sp-1|1891|Nummularia ciliata      |(L.) Kuntze}}
&lt;!-- -- --&gt;
| en1     = Fringed loosestrife
| en2     = Fringed yellow loosestrife
| en3     = Ciliate loosestrife
| fr1     = Lysimaque ciliée
| fr2     = Lysimaque fimbriée
| fr3     = Stéironéma cilié
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 4
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6483|5|Fens, swamps, marshes, ditches, and wet thickets.}}
| usda    = {{../usda|LYCI|N|}}
| gbif    =
| pow-id  = 
| wfo-id  = 
| vascan  = 6685
| gobot   = lysimachia/ciliata
| inat    = 
| its-id  = 
| ars-id  = 468325
| tro-id  = 50297982
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Lysimachia ciliata WFNY-162.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia hybrida
| au-abbr = Michx.
| au-full = 
| au-pub  = Fl. Bor.-Amer. 1: 126 (1803)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1788|Lysimachia ciliata      |Walter non L. (nom. illeg)}}
{{../sp-1|1803|'''Lysimachia hybrida'''|Michx.}}
{{../var1|1860|Lysimachia ciliata      |L.              |hybrida  |(Michx.) Chapm.}}
{{../var1|1876|Lysimachia lanceolata   |Walter          |hybrida  |(Michx.) A.Gray}}
{{../sp-1|1877|Steironema lanceolatum  |(Walter) A.Gray (misapplied)}}
{{../var1|1877|Steironema lanceolatum  |(Walter) A.Gray |hybridum |(Michx.) A.Gray}}
{{../sp-1|1901|Steironema laevigatum   |Howell}}
{{../sp-1|1903|Steironema hybridum     |(Michx.) Raf. ex Small}}
{{../sp-1|1913|Steironema validulum    |Greene ex Wooton &amp; Standl.}}
{{../var1|1939|Lysimachia ciliata      |L.              |validula |Kearney &amp; Peebles}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Lowland loosestrife|2023 iNaturalist, 2023 New York Flora Atlas}}
{{../vn1|Lance-leaved loosestrife|}}
{{../vn1|Lowland yellow loosestrife|}}
{{../vn1|Mississippi loosestrife|}}
{{../vn1|Swamp candles|}}
| fr-vns  = 
{{../vn1|Lysimaque hybride |}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S1, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2530|1|}}
| usda    = {{../usda|LYHY|NN|}}
| gbif    =
| pow-id  = 148125-2
| wfo-id  = 
| vascan  = 
| gobot   = Lysimachia/hybrida
| inat    = 165002-Lysimachia-hybrida
| its-id  = 23990
| ars-id  = 
| fna-id  = 242416811 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =
| nyfa-co = &lt;abbr title=&quot;Allegany, Clinton, Kings, Nassau, Oneida, Orange, Queens, Richmond, Rockland, Suffolk, Washington&quot;&gt;11 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Queens and Nassau Counties only&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Lysimachia hybrida iNat 140267385.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia quadriflora
| au-abbr = Sims
| au-full = 
| au-pub  = Bot. Mag. 18: t. 660 (1803)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|'''Lysimachia quadriflora'''|Sims}}
{{../sp-1|1813|Lysimachia longifolia       |Pursh}}
{{../sp-1|1892|Steironema quadriflorum     |(Sims) C.L.Hitchc.}}
{{../sp-1|1913|Nummularia quadriflora      |(Sims) Farw.}}
&lt;!-- -- --&gt;
| en1     = Linear-leaved loosestrife
| en2     = Linear-leaf loosestrife
| en3     = Four-flowered loosestrife
| en4     = Fourflower yellow loosestrife
| en5     = Linear-leaf loosestrife
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = OBL
| nwi2    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S1, G5?
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2529|1|}}
| usda    = {{../usda|LYQU|NN|}}
| gbif    =
| pow-id  = 148140-2
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 165013-Lysimachia-quadriflora
| its-id  = 23996
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =
| nyfa-co = &lt;abbr title=&quot;Erie and Niagara Counties only as of Dec. 2023&quot;&gt;Erie, Niagara&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;No New York State observations as of Dec. 2023&quot;&gt;No observations&lt;/abbr&gt;
| image1  = Lysimachia quadriflora.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group B&lt;/small&gt;=====

{{../txt
|img=Lysimachia nummularia 5026.jpg
|cap=''Lysimachia nummularia''  L. - moneywort or creeping Jenny
|''Lysimachia'' subg. ''Lysimachia'' '''group B''' contains two members of sect. ''Nummularia'', which are introduced species that are listed as moderately invasive in New York: ''Lysimachia nummularia'' ('''moneywort''' or '''creeping Jenny'''), and ''Lysimachia punctata'' ('''spotted loosestrife'''). 
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Nummularia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia nummularia
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Lysimachia nummularia|L.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Moneywort|}}
{{../vn1|Creeping Jenny|}}
{{../vn1|Creeping loosestrife|}}
{{../vn1|Creeping yellow loosestrife|}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Lysimaque nummulaire|Darbyshire et al., 2000}}
{{../vn1|Herbe-aux-écus|Centre du réseau suisse de floristique, 2010}}
{{../vn1|Monnayère|Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Eurasia
| status1 = Moderately invasive
| habit0  = Perennial
| habit1  = Herb-forb
| nyis    = 64%&lt;ref&gt;{{../nyis-ref|Lysimachia nummularia|Moderate|64}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2535|X|Lysimachia nummularia L. - Thickets, successional forests, low forests, and swamps generally in wet to wet-mesic soils in shaded to partly shaded habitats. It does particularly well in compacted and high pH soils.}}
| usda    = {{../usda|LYNU|XX|}}
| vascan  = 6689
| gobot   = Lysimachia/nummularia
| its-id  = 
| ars-id  = 316741
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Lysimachia nummularia LJM040629-5447866.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia punctata
| author  = L.
| sn0     = {{../sn|1753|Lysimachia punctata|L.}}
| sn1     = {{../sn||L. punctata|var=verticillata|Klatt}}
| ---
| en1     = Spotted loosestrife
| en2     = Large yellow loosestrife
| ---
| status  = Introduced
| from    = temperate Eurasia
| status1 = Moderately invasive
| nyis    = 57%&lt;ref&gt;{{../nyis-ref|Lysimachia punctata|Moderate|57}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2537|X|}}
| usda    = {{../usda|LYPU2|XX|}}
| vascan  = 
| gobot   = Lysimachia/punctata
| its-id  = 23995 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Lysimachia punctata-2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group C&lt;/small&gt;=====

{{../txt
|img=Lysimachia arvensis - EMPALOUS - IB-512 (Morró).jpg
|cap=Blue form of ''Lysimachia arvensis''
|Based on phylogenetic research, ''Anagallis arvensis'' ('''scarlet pimpernel''') was transferred to this subgenus in 2009 by U. Manns &amp; A. A. Anderberg.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51. [''Anagallis arvensis'' L. = ''Lysimachia arvensis'' (Raf.) U.Manns &amp; Anderb., comb. nov.].&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia |Pimpernel|292|2|subg=Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia arvensis
| author  = (L.) U.Manns &amp; Anderb.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Anagallis arvensis        |L.}}
{{../sp-1|1759|Anagallis caerulea        |L.}}
{{../var1|1764|Anagallis arvensis        |L.| caerulea |(L.) Gouan}}
{{../for1|1927|Anagallis arvensis        |L.| caerulea |(L.) Lüdi}}
{{../sp-1|2009|'''Lysimachia arvensis''' |(L.) U.Manns &amp; Anderb.}}
&lt;!-- -- --&gt;
| en1     = Scarlet pimpernel
| en2     = Poor-man's weatherglass
| en3     = Shepherd's clock
| en4     = Scarlet yellow-loosestrife
| en5     = Common pimpernel
| fr1     = Mouron rouge
| fr2     = Mouron des champs
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| from1   = northern Africa
| status1 = Potentially invasive
| status2 = 
| imap    = 
| ipa-us  = 57368
| gbif-e  = 8184903
| gbif-i  = 
| gbif-n  = 
| ny-tier =
| c-rank  = 
| nwi1    = FACU
| nwi2    = UPL
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6481|X|}}
| usda    = {{../usda|ANAR|XX|}}
| gbif    = 8184903
| vascan  = 28273
| gobot   = lysimachia/arvensis
| its-id  = 
| ars-id  = 316552
| fna-id  = 
| tro-id  = 100397700
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Lysimachia%20arvensis
| map     =  nymap.svg
| image1  = 20170525Lysimachia arvensis1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group E&lt;/small&gt;=====

{{:Flora of New York/txt
|img=Lysimachia terrestris.jpg
|cap=''Lysimachia terrestris''
|''Lysimachia'' subg. ''Lysimachia'' '''group E'''
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia terrestris
| author  = (L.) Britton, Sterns &amp; Poggenb.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Viscum terrestre      |L.}}
{{../sp-1|1788|Lysimachia vulgaris   |Walter (sensu auct)}}
{{../sp-1|1789|Lysimachia stricta    |Aiton}}
{{../sp-1|1789|Lysimachia bulbifera  |Curtis}}
{{../sp-1|1792|Lysimachia racemosa   |Lam.}}
{{../sp-1|1836|Lysimachia michauxii  |F.Dietr.}}
{{../sp-1|1888|'''Lysimachia terrestris''' |(L.) Britton, Sterns &amp; Poggenb.}}
{{../var1|1922|Lysimachia terrestris ||ovata |(E.L.Rand &amp; Redfield) Fernald}}
&lt;!-- -- --&gt;
| en1     = Swamp candles
| en2     = Swamp loosestrife
| en3     = Swamp yellow loosestrife
| en4     = Bog loosestrife
| en5     = Bulblet loosestrife 
| en6     = Earth loosestrife 
| en7     = Lake loosestrife
| fr1     = Lysimaque terrestre
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|2528|5|}}
| usda    = {{../usda|LYTE2|NN|}}
| gbif    =
| pow-id  = 148158-2
| wfo-id  = 
| vascan  = 6693
| gobot   = Lysimachia/terrestris
| inat    = 
| its-id  = 
| ars-id  = 400269
| fna-id  = 250092260
| tro-id  = 48062-Lysimachia-terrestris
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia terrestris 3-eheep (5097935642).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia quadrifolia
| au-abbr = L.
| au-full = 
| au-pub  = Sp. Pl.: 147 (1753)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|'''Lysimachia quadrifolia''' |L.}}
{{../sp-1|1777|Anagallis flava              |Houtt.}}
{{../sp-1|1788|Lysimachia punctata          |Walter}}
{{../sp-1|1803|Lysimachia hirsuta           |Michx.}}
{{../var1|1894|Lysimachia quadrifolia       |L. |variegata |Peck}}
{{../for1|1924|Lysimachia quadrifolia       |L. |variegata |(Peck) House}}
&lt;!-- -- --&gt;
| en1     = Whorled loosestrife
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|2532|5|Lysimachia quadrifolia - whorled loosestrife - Acidic dry-mesic to mesic hardwood forests and forest edges. Sometimes it grows in more open sites but generally it is a forest herb.}}
| usda    = {{../usda|LYQU2|NN|}}
| vascan  = 6692
| gobot   = Lysimachia/quadrifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia quadrifolia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia thyrsiflora
| author  = L. (1753)
| sn0     = {{../sn||Naumburgia thyrsiflora|(L.) Duby}}
| ---
| en1     = Tufted loosestrife
| en2     = Tufted yellow loosestrife
| en3     = Water loosestrife
| en4     = Swamp loosestrife
| en5     = Lysimaque thyrsiflore
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|2534|4|}}
| usda    = {{../usda|LYTH2|NN|}}
| vascan  = 
| gobot   = Lysimachia/thyrsiflora
| its-id  = 24000
| ars-id  = 457193
| fna-id  = 200017111
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia thyrsiflora 2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia × commixta
| author  = Fernald
| hp1     = Lysimachia terrestris 
| hp2     = Lysimachia thyrsiflora
| ---
| en1     = {{../hybrid-of|Swamp loosestrife|Tufted loosestrife}}
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|2538|2?|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia × producta
| author  = (A. Gray) Fernald (pro sp.)
| hp1     = Lysimachia quadrifolia 
| hp2     = Lysimachia terrestris
| ---
| en1     = {{../hybrid-of|Whorled loosestrife|Swamp loosestrife}}
| ---
| status  = Native
| status1 = Vulnerable
| nyfa    = {{../nyfa|2526|3?|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia vulgaris
| author  = L.
| sn0     = {{../sn|1753|Lysimachia vulgaris|L.}}
| ---
| en1     = Garden loosestrife
| en2     = Garden yellow loosestrife
| ---
| fr1     = Lysimaque commune
| status  = Introduced
| status1 = Highly invasive
| status2 = Prohibited&lt;ref name=prohibit&gt;{{fny-prohibit-ref}}&lt;/ref&gt;
| nyis    = 73%&lt;ref&gt;{{../nyis-ref|Lysimachia vulgaris|Moderate|73|first}}&lt;/ref&gt;
| nyfa    = {{../nyfa|6484|X|}}
| usda    = {{../usda|LYVU|XX|}}
| vascan  = 
| gobot   = lysimachia/vulgaris
| its-id  = 
| ars-id  = 
| fna-id  = 200017121
| tro-id  = 26400022
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = 20150609Lysimachia punctata1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Palladia + Lysimachiopsis''=====

{{:Flora of New York/txt
|img=Lysimachia clethroides2.jpg
|cap=''Lysimachia clethroides''
|The ''Lysimachia'' subgenera ''Palladia and Lysimachiopsis'' together are thought to form a clade but may be paraphyletic with each other.&lt;ref&gt;[http://www.bgbm.org/sites/default/files/documents/wi37-2Anderberg%2Bal.pdf A. A. Anderberg, U. Manns, and M. Källersjö (2007). &quot;Phylogeny and floral evolution of the Lysimachieae (Ericales, Myrsinaceae): evidence from ''ndhF'' sequence data.&quot; ''Willdenowia'' '''37:'''407-421.]&lt;/ref&gt; The only New York species of this group appears to be the Asian introducion ''Lysimachia clethroides'' ('''goose-neck loosestrife''').
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Palladia + Lysimachiopsis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Loosestrife|17|12|subg=Palladia + Lysimachiopsis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia clethroides
| au-abbr = Duby
| au-full = Jean Étienne Duby (1798-1885)
| au-pub  = Prodromus Systematis Naturalis Regni Vegetabilis 8:61. 1844.
&lt;!-- -- --&gt;
| syns    = 
{{../spa1|1844|Lysimachia clethroides|Duby}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Goose-neck loosestrife|NYFA}}
{{../vn1|Gooseneck loosestrife|ARS-GRIN}}
{{../vn1|Gooseneck yellow loosestrife|USDA NRCS National Plant Data Team / VASCAN}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Lysimaque faux-clèthre|VASCAN}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Potentially invasive
| nyis    = not assessed&lt;ref&gt;{{../nyis-ref|Lysimachia clethroides|Not Assessable|NA}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2533|Xm|}}
| usda    = {{../usda|LYCL2|XX|}}
| gbif    = 3169331
| vascan  = 6686
| gobot   = 
| inat    = 132224-Lysimachia-clethroides
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Livingston (2009), Nassau (1978), Queens (1988), Rensselaer (1999), Suffolk (1941), Westchester (1993)&quot;&gt;6 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Cattaraugus (2021), Essex (2021), Greene (2021), Kings (2021), Ontario (2021), Tompkins (2021), Saratoga (2021), Schenectady (2021), Westchester (2021)&quot;&gt;9+ counties&lt;/abbr&gt;
| image1  = Lysimachia clethroides 01.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Symplocaceae==

The {{../w|Symplocaceae}} (sweetleaf family).&lt;ref&gt;{{../nyfa-fam|Symplocaceae}}&lt;/ref&gt;

===''Symplocos''===

{{:Flora of New York/txt
|img=Symplocos paniculata 2015-05-29 OB 004.jpg
|cap=''Symplocos paniculata''
|'''Sweetleaf'''.
}}

{{../table|order=Ericales|Symplocaceae||||Symplocos|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Symplocos|Sweetleaf|113|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Symplocos tinctoria
| au-abbr = (L.) L'Hér.
| au-full = Charles Louis L'Héritier de Brutelle (1746-1800).
| au-pub  = Transactions of the Linnean Society of London 1: 176. 1791.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1767|Hopea tinctoria|L.}}
{{../sp-1|1791|Symplocos tinctoria|(L.) L'Hér.}}
{{../sp-1|1879|Protohopea tinctoria|(L.) Miers}}
{{../sp-1|1891|Eugeniodes tinctorium|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Sweetleaf|2022 iNaturalist, New York Flora Atlas}}
{{../vn1|Sweet-leaf|}}
{{../vn1|Common sweetleaf|}}
{{../vn1|Horsesugar|2022 New York Flora Atlas}}
{{../vn1|Horse-sugar|}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|3024|X|}}
| usda    = {{../usda|SYTI|N0|}}
| vascan  = 
| gobot   = 
| inat    = 141855-Symplocos-tinctoria
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 30900076
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Symplocos%20tinctoria
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Suffolk (1991-92)&quot;&gt;Suffolk&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Cultivated in Central Park (New York County)&quot;&gt;Cultivated NYC&lt;/abbr&gt;
| image1  = Symplocos tinctoria AR-01 (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Symplocos paniculata
| author  = (Thunb.) Miq.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1784|Prunus paniculata|Thunb.}}
{{../sp-1|1867|Symplocos paniculata|(Thunb.) Miq.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Sapphire berry|}}
{{../vn1|Sapphire-berry|2022 iNaturalist}}
{{../vn1|Asiatic sweetleaf|}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Highly invasive
| status2 = 
| imap    =
| ipa-us  = 
| gbif-e  = 7157705
| ny-tier = 2
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|3023|X|}}
| usda    = {{../usda|SYPA12|X0|}}
| gbif    = 7157705
| vascan  = 
| gobot   = 
| inat    = 169497-Symplocos-paniculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Symplocos%20paniculata
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Nassau (1977, '87, '92, '95)&quot;&gt;Nassau county&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Dutchess (2021), Nassau  (2019), Queens (2018), Westchester (2020)&quot;&gt;4 counties&lt;/abbr&gt;
| image1  = Symplocos paniculata (9958808546).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Styracaceae==

{{:Flora of New York/txt
|The {{../w|Styracaceae}} (snowbell family) &lt;ref&gt;{{../nyfa-fam|Styracaceae}}&lt;br&gt;{{../usda-fam|Styracaceae}}&lt;/ref&gt; contains snowbells and silverbell trees.
}}

===''Halesia''===

{{:Flora of New York/txt
|img=Halesia carolina 22zz.jpg
|cap=''Halesia carolina''
|
}}

{{../table|order=Ericales|Styracaceae||||Halesia|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Halesia|Silverbell|1030|n|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Halesia carolina
| author  = L.
| sn0     = {{../sn|1759|Halesia carolina|L.}}
| sn1     = {{../sn|1761|Halesia tetraptera|J.Ellis}}
| sn2     = {{../sn|1803|Halesia parviflora|Michx.}}
| sn3     = {{../sn|1914|Halesia carolina|var=monticola|Rehder}}
| sn4     = {{../sn|1921|Halesia monticola|(Rehder) Sarg.}}
| ---
| en1     = Silver bells 
| en2     = Possomwood
| en3     = Carolina silverbell
| ---
| status  = Introduced
| status1 = US South native
| habit0  = Perennial
| habit1  = Tree
| nyfa    = {{../nyfa|6480|X|State exotic or non-native}}
| usda    = {{../usda|HACA3|N0|}}
| vascan  = 
| gobot   = halesia/carolina
| its-id  =
| ars-id  = 18215
| fna-id  = 220005985
| tro-id  = 
| ipn-id  = 
| nse-id  = Halesia+carolina
| bna-id  = Halesia%20carolina
| map     =  nymap.svg
| image1  = Halesia carolina var. monticola Ośnieża karolińska 2017-05-01 02.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

===''Styrax''===

{{../table|order=Ericales|Styracaceae||||Styrax|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Styrax|Snowbell|162|1|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Styrax americanus
| author  = Lam.
| sn0     = {{../sn|1783|Styrax americanus|Lam.}}
| sn1     = {{../sn|1803|Styrax pulverulentus|Michx.}}
| sn2     = {{../sn|1917|Styrax americanus|var1=pulverulentus|Rehder}}
| ---
| en1     = American snowbell
| en2     = Mock orange
| ---
| fr1     = 
| status  = Introduced
| status1 = US South native
| status2 = Cultivated
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|STAM4|N0|}}
| gbif    =  
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 313878
| fna-id  = 
| tro-id  = 30800057
| ipn-id  = 
| nse-id  = Styrax+americanus
| bna-id  = Styrax%20americanus
| map     =  nymap.svg
| image1  = Styrax americanus.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Styrax japonicus
| author  = Sieb. &amp; Zucc.
| ---
| en1     = Japanese snowbell
| en2     = Japanese storax
| ---
| status  = Introduced
| status1 = 
| nyfa    = {{../nyfa|3021|X|State exotic or non-native}}
| usda    = {{../usda|STJA5|X0|}}
| gbif    = 5371678
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 505974
| fna-id  = 200017748
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Nassau (1977, 2010), Suffolk (1996, 2011)&quot;&gt;2 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Kings, Nassau, New York, Queens, Suffolk, Westchester&quot;&gt;6 counties&lt;/abbr&gt;
| image1  = Styrax japonicus Styrak japoński 2018-08-12 01.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

==Family Diapensiaceae==

The {{../w|Diapensiaceae}}  (diapensia family).&lt;ref&gt;{{../nyfa-fam|Diapensiaceae}}&lt;br&gt;{{../usda-fam|Diapensiaceae}}&lt;/ref&gt;

===''Diapensia''===

{{../table|order=Ericales|Diapensiaceae||||Diapensia|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Diapensia|Diapensia|676|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Diapensia lapponica
| author  = L.
| ssp     = lapponica
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Lapland diapensia
| en2     = Pincushion plant
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1233|2|}}
| usda    = {{../usda|DILAL||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Diapensia lapponica plant upernavik.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===''Galax''===
{{../txt
|img=Galax urceolata (cropped).jpg
|cap=''Galax urceolata''&lt;br&gt;beetleweed 
|''Galax'' ('''beetleweed''') is an eastern North America native, but is considered to be an introduction in the Northeast. It is cultivated as a &quot;native&quot; groundcover and may persist as an escapee on Long Island, but is not considered to be truly naturalized in New York State.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Diapensiaceae||||Galax|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Galax|Galax|586|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Galax urceolata
| author  = (Poir.) Brummitt 
&lt;!-- -- --&gt;
| sn0     = {{../sn|auct|Galax aphylla|non=L.}}
&lt;!-- -- --&gt;
| en1     = Beetleweed
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Not naturalized
| status3 = Nassau, Suffolk Counties
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1232|X|State exotic or non-native}}
| usda    = {{../usda|GAUR2|N0|Galax urceolata (Poir.) Brummitt - beetleweed}}
| vascan  = 
| gobot   = galax/urceolata
| its-id  = 502705
| fna-id  = 220005401
| tro-id  = 
| ipn-id  = 
| lbj-id  = GAUR2
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     =  nymap.svg
| image1  = Galax - Flickr - pellaea (1).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===''Pyxidanthera''===
{{../txt
|img=Pyxidanthera barbulata.jpg
|cap=''Pyxidanthera barbulata''&lt;br&gt;pyxie moss
|Both species of ''Pyxidanthera'' ('''pixie moss''') are rare natives of the east coast. The smaller ''P. brevifolia'' (little-leaf pixiemoss) is only found in the Carolinas, but the larger ''P. barbulata'' ('''flowering pixiemoss''') can be found as far north as Long Island, where it is considered endangered.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Diapensiaceae||||Pyxidanthera|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Pyxidanthera|Pyxie moss|243|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyxidanthera barbulata
| author  = {{../w|Michx.}}
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Common pyxiemoss
| en2     = Flowering moss
| en3     = Flowering pixiemoss
| en4     = Big pyxie
| en5     = Pyxies
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| status2 = Suffolk County only
| c-rank  = 6
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6479|1|especially vulnerable in New York State}}
| usda    = {{../usda|PYBA|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 23798
| fna-id  = 220011320
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Pyxidanthera barbulata BB-1913.png
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Sarraceniaceae==

{{../txt|The {{../w|Sarraceniaceae}} (pitcher-plant family) are carnivorous plants, consisting of three New-World genera. The largest genus, ''Heliamphora'', is endemic to South America, and the monospecific ''Darlingtonia'' is native to the West Coast of the U.S.  The third genus,  ''Sarracenia'', with about eight species, is native to eastern North America.&lt;ref&gt;{{../nyfa-fam|Sarraceniaceae}}&lt;/ref&gt;}}

===''Sarracenia''===

{{../txt
|img=Purple Pitcher Plant - Sarracenia purpurea (18239408828).jpg
|cap=''Sarracenia purpurea''&lt;br&gt;purple pitcher-plant
|The most widespread and only cold-tolerant species of ''Sarracenia'' is ''{{../w|Sarracenia purpurea}}'' (the '''purple pitcher plant''').  Of its two subspecies, ''S. purpurea'' ssp. ''venosa'' is only found south of New York, but ''S. purpurea'' ssp. ''purpurea'' is fairly widespread in New York peatlands as well as farther north.}}

{{../table|order=Ericales|Sarraceniaceae||||Sarracenia|
}} 
{{../genus|Sarracenia|Pitcher plant|282|1|
}}
{{../taxon
| species = Sarracenia purpurea
| author  = L.
| ssp     = purpurea
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Sarracenia purpurea|L.}}
| sn1     = {{../sn|1827|S. purpurea|var=terrae-novae|var-au=LaPylaie}}
| sn2     = {{../sn|1933|S. purpurea|ssp=gibbosa|ssp-au=(Raf.) Wherry}}
| sn3     = {{../sn|1933|S. purpurea|var=stolonifera|var-au=Macfarl. &amp; Steckbeck}}
| sn4     = {{../sn|1951|S. purpurea|var=ripicola|var-au=B.Boivin}}
&lt;!-- -- --&gt;
| en1     = Purple pitcher-plant
| en2     = Northern pitcher-plant
| en3     = Side-saddle plant
| en4     = Common pitcherplant
| en5     = Huntsman's-cup
| en6     = Huntsman's horn
| en7     = Decumbent pitcher plant
| en8     = Frog's-britches
| fr1     = Sarracénie pourpre
&lt;!-- -- --&gt;
| status  = Native
| status1 = Vulnerable
| c-rank  = 9
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Sun
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6370|3-4|Sarracenia purpurea L. - purple pitcherplant - Acidic to alkaline peatlands. Limited acreage or Apparently secure in New York State.}}
| usda    = {{../usda|SAPU4|NN|shows ssp. gibbosa in western &amp; northest US (incl. NY) but not Canada. Shows ssp. purpurea in Southeast US &amp; Canada.}}
| vascan  = 9222
| gobot   = Sarracenia/purpurea
| its-id  = 195652
| ars-id  = 409890
| fna-id  = 250092288
| tro-id  = 
| ipn-id  = 
| lbj-id  = SAPU4
| nse-id  = 
| bna-id  = Sarracenia%20purpurea
| cpc-id  = 
| cab-id  = 
| map     =  nymap.svg
| image1  = PurplepitcherplantMN.jpg
}}
{{../end table|Sarraceniaceae|
}}

==Family Actinidiaceae==
{{../txt|The {{../w|Actinidiaceae}} (Chinese gooseberry family) has only two species that have been found outside of cultivation in New York. Only one of these are considered to be naturalized.&lt;ref&gt;{{../nyfa-fam|Actinidiaceae}}&lt;/ref&gt;}}

===''Actinidia''===
{{../txt
|img=Actinidia arguta 0906.JPG
|cap=''Actinidia arguta''&lt;br&gt;taravine or hardy kiwi
|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Actinidiaceae|Actinidioideae|||Actinidia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Actinidia|Kiwifruit|1072|1|actin3||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Actinidia arguta
| au-abbr = (Siebold &amp; Zucc.) Planch. ex Miq.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1867|Actinidia arguta|(Siebold &amp; Zucc.) Planch. ex Miq.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Hardy kiwi|2018 New York Flora Atlas}}
{{../vn1|Baby kiwi|2018 ARS-GRIN: Porcher, M. H. et al. 2000}}
{{../vn1|Tara vine|2018 USDA NRCS National Plant Data Team}}
{{../vn1|Taravine|2019 iMapInvasive}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Highly invasive
| status2 = Naturalized
| imap    = 1383
| ipa-us  = 51426 
| ny-tier = 2
| c-rank  = X
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7242|Xn|}}
| usda    = {{../usda|ACAR10|X0|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 1389
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Westchester (2013)&quot;&gt;Westchester (2013)&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Queens (2010), Westchester (2018-19)&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Actinidia-arguta-habit.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Actinidia polygama
| au-abbr = (Siebold &amp; Zucc.) Planch. ex Maxim. 
| au-full = 
| au-pub  = Mém. Acad. Imp. Sci. St.-Pétersbourg Divers Savans 9:64. 1859 (Prim. fl. amur.)
&lt;!-- -- --&gt;
| sn0     = {{../sn|1843|Trochostigma polygamum|Siebold &amp; Zucc.}}
| sn1     = {{../sn|1859|Actinidia polygama|(Siebold &amp; Zucc.) Planch. ex Maxim.}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Silver vine
| en2     = Cat powder
| en3     = Matatabi
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Potentially invasive
| status2 = Not naturalized
| imap    = 1399
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|33|X|Orange (only)}}
| usda    = {{../usda|ACPO9|X0|NY (only)}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 1411
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Actinidia polygama.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Clethraceae==

{{../txt|The {{../w|Clethraceae}} (clethra family)&lt;ref&gt;{{../nyfa-fam|Clethraceae}}&lt;/ref&gt; contains just two species, only one of which (''Clethra'') is found in New York.}}

===''Clethra''===

{{../txt
|img=Zimterle (Clethra alnifolia)@Heilpflanzengarten Celle 20160728 01.jpg
|cap=''Clethra alnifolia''
|''Clethra'' contains two North American species: 
*''Clethra acuminata'', '''mountain pepperbush''', is limited to the Appalachians from Alabama to southern Pennsylvania, and has not been known to naturalize as far north as New York State.
*''Clethra alnifolia'', '''coastal sweet pepperbush''', is found primarily along the East Coast, including parts of New York, particularly in wet acidic environments.

''Clethra alnifolia'' cultivars are available and are usually either more compact than the wild form or have pink flowers.&lt;ref&gt;[http://www.clemson.edu/extension/hgic/plants/landscape/shrubs/hgic1090.html Clemson University Cooperative Extension Service, Home &amp; Garden Information Center, Clemson, South Carolina.]&lt;/ref&gt;}}

{{../table|order=Ericales|Clethraceae||||Clethra|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Clethra|Sweetpepperbush|112|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Clethra alnifolia
| author  = L.
&lt;!-- -- --&gt;
| syns    = {{../snl
|{{../spa1|1753|Clethra alnifolia  |L.}}
|{{../spa1|1786|Clethra tomentosa  |Lam.}}
|{{../spa1|1789|Clethra paniculata |Aiton}}
|{{../var1|1803|Clethra alnifolia  |L. |tomentosa|Michx.}}
|{{../var1|1803|Clethra alnifolia  |L. |glabella|Michx.}}
|{{../var1|1900|Clethra alnifolia  |L. |paniculata|Rehder}}
|{{../var1|1903|Clethra alnifolia  |L. |michauxii|Zabel}}
}}
&lt;!-- -- --&gt;
| en1     = Coastal sweet-pepperbush
| en2     = Coast pepperbush
| en3     = Sweet Pepperbush
| en4     = Summersweet
| en5     = Summersweet clethra
| en6     = Pink spires (cv.)
| en7     = Anne Bidwell (cv.)
| fr1     = Clèthre à feuilles d'aulne
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = Sun - shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|852|5|Clethra alnifolia L. - coastal sweet-pepperbush - Edges of acidic ponds, acidic sphagnum wetlands, and bog edges. Often with other shrubs including Rhododendron viscosum}}
| usda    = {{../usda|CLAL3|NN|}}
| vascan  = 4589
| gobot   = clethra/alnifolia
| its-id  = 23454
| ars-id  = 10901
| fna-id  = 220003027
| tro-id  = 7700002
| ipn-id  = 319146-2
| lbj-id  = CLAL3
| nse-id  = Clethra+alnifolia
| bna-id  = Clethra%20alnifolia
| map     = Clethra alnifolia nymap.svg
| image1  = Clethra alnifolia2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Ericaceae==

{{../txt|The {{../w|Ericaceae}} (heath family).&lt;ref&gt;{{../nyfa-fam|Ericaceae}}&lt;br&gt;{{../usda-fam|Ericaceae}}&lt;/ref&gt;&lt;ref&gt;The subfamily (and tribal) organization used here is from of [http://www.plantsystematics.org/reveal/pbio/fam/MagnoliidaeOnLine.html James L. Reveal (2012) An outline of a classification scheme for extant flowering plants. ''Phytoneuron'' 2012–37: 1–221.]&lt;/ref&gt;}}

===Subfamily Pyroloideae===

{{../txt|The subfamily '''Pyroloideae''' Kosteltsky. Perennial herbs, rhizomatous.
}}

====Tribe Pyroleae====

{{../txt|The native New York '''Pyroleae''' consist of about nine species of &quot;wintergreen&quot; or &quot;shineleaf&quot; plants. The organization of the Pyroleae used here is based primarily on Liu (2011).&lt;ref&gt;[http://link.springer.com/article/10.1007%2Fs10265-010-0376-8  Zhen-wen Liu, Ze-huan Wang, Jing Zhou, Hua Peng (2011). &quot;Phylogeny of Pyroleae (Ericaceae): implications for character evolution.&quot; ''Journal of Plant Research'' May 2011, Volume 124, Issue 3, pp 325-337.]&lt;/ref&gt;}}

=====''Orthilia''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Orthilia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Orthilia|Orthilia|330|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Orthilia secunda
| author  = (L.) House
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Pyrola secunda|L.}}
| sn1     = {{../sn|1858|Ramischia secunda|(L.) Garcke}}
| sn2     = {{../sn|1914|Ramischia elatior|Rydb.}}
| sn3     = {{../sn|1921|Orthilia secunda|(L.) House}}
&lt;!-- -- --&gt;
| en1     = One-sided wintergreen
| en2     = Sidebells wintergreen
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1338|5|}}
| usda    = {{../usda|ORSE|NN|}}
| vascan  = 
| gobot   = Orthilia/secunda
| its-id  =
| fna-id  = 242334848
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = OrthiliaSecunda.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Pyrola''=====
{{../txt
|img=Pyrola asarifolia, americana, elliptica WFNY-151.jpg
|cap=''Pyrola'' (wintergreen) photos from the 1918 ''Wild Flowers of New York''
|New York ''Pyrola'' species go by the various colloquial names: '''wintergreen''', '''shinleaf''', or sometimes '''shineleaf'''.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Pyrola|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Pyrola|Wintergreen|30|5|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola americana
| author  = Sweet
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1830|Pyrola americana|Sweet}}
{{../sp-1|1844|Pyrola obovata|Bertol.}}
{{../sp-1|auct|Pyrola rotundifolia|p.p.|non=L.}}
{{../var1|1920|Pyrola rotundifolia|L.|americana|Fernald}}
{{../ssp1|1966|Pyrola asarifolia|Michx.|americana|Křísa}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|American wintergreen|}}
{{../vn1|American shinleaf|}}
{{../vn1|Wild lily-of-the-valley|}}
{{../vn1|Round-leaved shineleaf|}}
{{../vn1|Round-leaved wintergreen|}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Pyrole d'Amérique|}}
{{../vn1|Pyrole à feuilles rondes|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1323|5|Demonstrably secure in New York State.}}
| usda    = {{../usda|PYAM|NN|}}
| gobot   = pyrola/americana
| its-id  = 504691
| ars-id  = 407190 &lt;!-- Pyrola rotundifolia L. var. americana (Sweet) Fernald --&gt;
| fna-id  = 250092297
| tro-id  = 26800163
| bna-id  = Pyrola%20americana
| image1  = Pyrola americana (31887890964) 3x4.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola chlorantha
| author  = Sw.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1810|Pyrola chlorantha|Sw.}}
{{../sp-1|1811|Pyrola virens|Schweigg.}}
{{../sp-1|1867|Pyrola oxypetala|Austin ex A. Gray}}
{{../var1|1920|Pyrola chlorantha|Sw.|convoluta|(W.P.C. Barton) Fernald}}
{{../var1|1920|Pyrola chlorantha|Sw.|paucifolia|Fernald}}
{{../var1|1920|Pyrola chlorantha|Sw.|revoluta|Jenn.}}
{{../var1|1941|Pyrola virens|Schweigg.|convoluta|(W.P.C. Barton) Fernald}}
{{../var1|1941|Pyrola virens|Schweigg.|saximontana|(Fernald) Fernald}}
&lt;!-- -- --&gt;
| en1     = Green-flowered wintergreen
| en2     = Green-flowered shineleaf
| en3     = Greenish-flowered wintergreen
| en4     = Pale-green wintergreen
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1325|5|}}
| usda    = {{../usda|PYCH|NN|}}
| its-id  =
| fna-id  =
| map     = nymap.svg
| image1  = Rohekas uibuleht.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola elliptica
| author  = Nutt.
| ---
| en1     = Shinleaf
| en2     = Waxflower shinleaf
| en3     = Large-leaved shineleaf
| en4     = Elliptic shineleaf
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1334|5|}}
| usda    = {{../usda|PYEL|NN|}}
| its-id  = 23759
| image1  = Pyrola elliptica 4 (5097357221).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola asarifolia
| author  = Michx.
| ssp     = asarifolia
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|yyyy|Pyrola californica|Krísa}}
{{../sp-1|yyyy|Pyrola elata|Nutt.}}
{{../sp-1|yyyy|Pyrola uliginosa|Torr. &amp; A. Gray ex Torr.}}
{{../var1|yyyy|Pyrola uliginosa|Torr. &amp; A. Gray ex Torr.|gracilis|Jennings}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|asarifolia}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|incarnata|(DC.) Fernald}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|ovata|Farw.}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|purpurea|(Bunge)Fernald}}
{{../ssp1|yyyy|Pyrola rotundifolia|L.|asarifolia|(Michx.) Á. Löve &amp; D. Löve}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Pink shinleaf|New York Flora Atlas}}
{{../vn1|Pink wintergreen|FNA Ed. Comm., 2009}}
{{../vn1|Bog wintergreen|FNA Ed. Comm., 2009}}
{{../vn1|Liverleaf wintergreen|USDA NRCS National Plant Data Team}}
{{../vn1|Pink pyrola|Hinds, 2000 via VASCAN}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Pyrole à feuilles d'asaret|Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| en1     = 
| en2     = 
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|1324|2|}}
| usda    = {{../usda|PYASA|NN|}}
| vascan  = 5546
| gobot   = 
| its-id  = 23760
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Pyrola asarifolia (6187038332).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola minor
| author  = L.
| ---
| en1     = Snowline wintergreen
| en2     = Lesser wintergreen
| ---
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1336|1|}}
| usda    = {{../usda|||}}
| its-id  =
| fna-id  =
| map     = nymap.svg
| image1  = Pyrola minor 090612a.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Moneses''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Moneses|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Moneses|Single-delight|1056|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Moneses uniflora
| author  = (L.) {{../w|A.Gray}}
| sn0     = {{../sn||Pyrola uniflora|L.}}
| ---
| en1     = One-flowered wintergreen
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1328|4|}}
| usda    = {{../usda|MOUN2|NN|}}
| vascan  = 
| gobot   = Moneses/uniflora
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Moneses uniflora kz1.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Chimaphila''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Chimaphila|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Chimaphila|Prince's-pine|253|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chimaphila umbellata
| author  = (L.) {{../w|Benjamin Smith Barton|W.P.C.Barton}}
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Pyrola umbellata|L.}}
{{../sp-1|1817|Chimaphila umbellata|W.P.C.Barton}}
{{../sp-1|1891|Pseva umbellata|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en1     = Common wintergreen
| en2     = Pipsissewa
| en3     = Noble prince's-pine
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1316|5|Native. Apparently secure in New York State.}}
| usda    = {{../usda|CHUM|NN|}}
| vascan  = 
| gobot   = chimaphila/umbellata
| its-id  = 
| ars-id  = 402538
| fna-id  = 
| tro-id  = 26800037
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Chimaphila umbellata (3817695201).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chimaphila maculata
| author  = (L.) {{../w|Frederick Traugott Pursh|Pursh}}
| ---
| en1     = Spotted wintergreen
| en2     = Striped prince's pine
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1347|4|Apparently secure in New York State.}}
| usda    = {{../usda|CHMA3|NN|Native, eastern US &amp; Canada}}
| vascan  = 
| gobot   = Chimaphila/maculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Striped Wintergreen - Chimaphila maculata, Leesylvania State Park, Woodbridge, Virginia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Monotropoideae===

{{../txt|The subfamily '''Monotropoideae''' consists exclusively of taxa which are both herbaceous and achlorophyllous. Because they are symbiotic with mycorrhizal fungi and a photosynthetic host, these taxa are known as mycoheterotrophs.&lt;ref&gt;[http://rave.ohiolink.edu/etdc/view?acc_num=osu1417442819 Broe, Michael Brian (2014) Phylogenetics of the Monotropoideae (Ericaceae) with Special Focus on the Genus Hypopitys Hill, together with a Novel Approach to Phylogenetic Inference Using Lattice Theory.]&lt;/ref&gt;
}}

====Tribe Monotropeae====

''Monotropeae'' are mycotrophs, which are parasitic plants that obtain their nutrients trough fungi instead photosynthesis.

=====''Monotropa''=====

{{../txt
|''Monotropa''
}}

{{../table|order=Ericales|Ericaceae|Monotropoideae|Monotropeae||Monotropa|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Monotropa|Indianpipe|480|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Monotropa uniflora
| author  = L.
| sn0     = {{../sn|1753|Monotropa uniflora|L.}}
| sn1     = {{../sn|1927|Monotropa brittonii|Small}}
| ---
| en1     = Indian-pipe
| en2     = One-flowered Indian pipe
| en3     = Ghost pipe
| en4     = Convulsion root
| ---
| fr1     = Monotrope uniflore
| fr2     = Monotrope à une fleur
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1327|5|}}
| usda    = {{../usda|MOUN3|NN|}}
| vascan  = 5537
| gobot   = monotropa/uniflora
| its-id  = 
| ars-id  = 431514
| fna-id  = 200016137
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Monotropa uniflora nymap.svg
| image1  = Monotropa uniflora in Penwood State Park 3, 2009-07-03.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Hypopitys''=====

{{../txt
|img=Monotropa hypopitys WFNY-153A.jpg
|cap=''Hypopitys monotropa '' - pinesap
|''Hypopitys'' is often included in ''Monotropa'' (above).&lt;ref&gt;[http://rave.ohiolink.edu/etdc/view?acc_num=osu1417442819 Broe, Michael Brian (2014) Phylogenetics of the Monotropoideae (Ericaceae) with Special Focus on the Genus Hypopitys Hill, together with a Novel Approach to Phylogenetic Inference Using Lattice Theory.]&lt;/ref&gt;
}}

{{../table|order=Ericales|Ericaceae|Monotropoideae|Monotropeae||Hypopitys|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Hypopitys|Pinesap|480|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Hypopitys monotropa
| author  = Crantz
| sn0     = {{../sn|1753|Monotropa hypopitys|L.}}
| sn1     = {{../sn|1766|Hypopitys monotropa|Crantz}}
| sn2     = {{../sn|1772|Hypopitys multiflora|Scop.}}
| sn3     = {{../sn|1803|Monotropa lanuginosa|Michx.}}
| sn4     = {{../sn|1810|Hypopitys lanuginosa|(Michx.) Raf.}}
| sn5     = {{../sn|1843|Hypopitys americana|(DC.) Small}}
| sn6     = {{../sn|1894|Hypopitys hypopitys|(L.) Small ''taut''.}}
| sn7     = {{../sn|1897|Hypopitys multiflora|(Scop.) Fritsch}}
| sn8     = {{../sn|1901|Hypopitys fimbriata|(A.Gray) Howell}}
| sn9     = {{../sn|1910|M. hypopitys|var=lanuginosa|Purah.}}
| sn10    = {{../sn|1956|M. hypopitys|ssp=lanuginosa|H.Hara}}
| ---
| en1     = Pinesap
| en2     = American pinesap
| en3     = Yellow pinesap
| en4     = Yellow bird's-nest
| en5     = False beechdrops
| ---
| fr1     = Monotrope du pin
| fr2     = Monotrope à grappe
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1318|4|}}
| usda    = {{../usda|MOHY3|NN|}}
| vascan  = 5527      | Hypopitys monotropa  Crantz
| gobot   = hypopitys/monotropa
| its-id  = 23775     | Hypopitys monotropa  Crantz
| ars-id  = 456116    | Monotropa hypopitys L.
| fna-id  = 200016135 | Monotropa hypopitys L.
| tro-id  = 26800126  | Monotropa hypopitys L.
| nse-id  = 
| ipn-id  = 
| map     =  nymap.svg
| image1  = Fichtenspargel Monotropa hypopitys.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Hypopitys lanuginosa
| au-abbr = (Michx.) Raf.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Monotropa lanuginosa|Michx.}}
{{../sp-1|1810|Hypopitys lanuginosa|(Michx.) Raf.}}
{{../sp-1|1818|H. lanuginosa|(Michx.) Nutt. ''isonym''}}
{{../ssp1|1910|Monotropa hypopitys|L.|lanuginosa|(Michx.) Purah.}}
{{../ssp1|1956|Monotropa hypopitys|L.|lanuginosa|(Michx.) H.Hara}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Red pinesap|}}
{{../vn1|hairy pine-sap|}}
{{../vn1||}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1||}}
{{../vn1||}}
{{../vn1||}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Unranked
| c-rank  = 8
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7404|Nu|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = Hypopitys/lanuginosa
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Fringed Pinesap (1056971122).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Pterosporeae====

=====''Pterospora''=====

{{../table|order=Ericales|Ericaceae|Monotropoideae|Pterosporeae||Pterospora|
}}  
{{../genus|Pterospora|Pinedrops|288|n|
}}
{{../taxon
| species = Pterospora andromedea
| author  = Nutt.
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Pinedrops
| en2     = Pine-drops
| en3     = Woodland pinedrops
| en4     = Giant pinedrops
| en5     = Giant birds's-nest
| en6     = Albany beechdrops
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1322|1|Native especially vulnerable in New York State.}}
| usda    = {{../usda|PTAN2|NN|}}
| vascan  = 
| gobot   = Pterospora/andromedea
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 26800015
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Pterospora andromedea Sprague Lake.jpg|width1=120
}}
{{../end table|Ericaceae|Monotropoideae|Pterosporeae|
}}

===Subfamily Arbutoideae===

====''Arctostaphylos''====

{{../txt
|img=Arctostaphylos uva-ursi 38450.JPG
|cap=''Arctostaphylos uva-ursi''&lt;br&gt;bearberry or kinnikinnick
|'''Bearberry''' is a trailing evergreen shrub, native to rocky and sandy upland parts of the Northern Hemisphere, including suitable parts of New York State. 
}}

{{../table|order=Ericales|Ericaceae|Arbutoideae|||Arctostaphylos|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Arctostaphylos|Manzanita|411|1|ARCTO3|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Arctostaphylos uva-ursi
| author = (L.) Spreng.
| sn0     = {{../sn|1753|Arbutus uva-ursi|L.}}
| sn1     = {{../sn|1812|Arbutus buxifolia|Stokes}}
| sn2     = {{../sn|1813|Mairania uva-ursi|(L.) Desv.}}
| sn3     = {{../sn|1821|Uva-ursi buxifolia|(Stokes) Gray}}
| sn4     = {{../sn|1825|Arctostaphylos uva-ursi|(L.) Spreng.}}
| sn5     = {{../sn|1913|Uva-ursi uva-ursi|(L.) Britton ''taut''.}}
| ---
| en1     = Red bearberry
| en2     = Kinnikinnick
| en3     = Pinemat manzanita
| en4     = Mealberry
| en5     = Hog cranberry
| ---
| fr1     = Raisin d’ours
| fr2     = Arctostaphyle raisin-d'ours
| fr3     = Busserole
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = UPL
| habit0  = Perennial
| habit1  = Shrub-subshrub
| light   = Sun - shade
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1348|5|}}
| usda    = {{../usda|ARUV|NN|}}
| vascan  = 5492
| gobot   = arctostaphylos/uva-ursi
| its-id  = 23530
| ars-id  = 3866
| fna-id  = 220001038
| tro-id  = 12300006
| ipn-id  = 
| lbj-id  = ARUV
| nse-id  = Arctostaphylos+uva-ursi
| bna-id  = Arctostaphylos%20uva-ursi 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =  nymap.svg
| cos     = Alba Bron Clin Colu Dutc Erie Esse Fran Fult Gene Gree Hami Jeff Lewi Nass Niag Onei Onta Oran Oswe Putn Quee Rens Rich Rock Stla Steu Suff Tomp Ulst Warr Wash
| image1  = Arctostaphylos uva-ursi 28273.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Harrimanelloideae===

======''Harrimanella''======

{{../table|order=Ericales|Ericaceae|Harrimanelloideae|||Harrimanella|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Harrimanella|Harrimanella|538|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Harrimanella hypnoides
| author  = (L.) Coville
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Moss bell-heather
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely extirpated
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1335|Z|}}
| usda    = {{../usda|HAHY2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 404379
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Harrimanella hypnoides Kilpisjärvi 2012-07.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Ericoideae===

====Tribe Phyllodoceae====

=====''Epigaea''=====

{{../txt
|img=Epigaea repens WFNY-153B.jpg
|cap=''Epigaea repens'' L.&lt;br&gt;trailing arbutus
|''Epigaea'' contains three species, only one of which, '''trailing arbutus''' (''Epigaea repens''), is native to the New World. It is a low shrub/subshrub found in acidic upland hemlock-hardwood forests and forest edges through much of eastern North America. It flowers in early spring and does well in disturbed sites such as logging roads. Trailing arbutus is the state plant of Massachusetts.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Phyllodoceae||Epigaea|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Epigaea|Trailing-arbutus||1|EPIGA|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Epigaea repens
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Epigaea repens|L.}}
{{../var1|1837|Epigaea repens|L.|rubicunda|D.Don}}
{{../var1|1939|Epigaea repens|L.|glabrifolia|Fernald}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Trailing arbutus|2021 New York Flora Atlas -- 2021 VASCAN-ACC: Hinds, 2000}}
{{../vn1|Trailing-arbutus|2021 Native Plant Trust(Go Botany)}}
{{../vn1|Mayflower|2021 New York Flora Atlas -- 2021 VASCAN-SYN:  FNA Ed. Comm., 2009}}
{{../vn1|Creeping mayflower|2021 VASCAN-SYN: Newmaster et al., 1998}}
{{../vn1|Gravelroot|2021 VASCAN-SYN: GRIN 2010}}
{{../vn1|Ground laurel|2021 VASCAN-SYN:  FNA Ed. Comm., 2009}}
| fr-vns  = 
{{../vn1|Épigée rampante|2021 VASCAN-ACC: Marie-Victorin, 1995}}
{{../vn1|Épigée fleur-de-mai|2021 VASCAN-SYN: Lamoureux, 2002}}
{{../vn1|Fleur de mai|2021 VASCAN-ACC: Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = S4, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1312|4|}}
| usda    = {{../usda|EPRE2|NN|}}
| vascan  = 5516
| gobot   = Epigaea/repens
| inat    = 83075-Epigaea-repens
| its-id  = 
| ars-id  = 
| fna-id  = 220004812
| tro-id  = 12300017
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Epigaea repens 5-sphillips (5097282717).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Kalmia''=====

{{../txt
|img=Kalmia angustifolia 4500.JPG
|cap=Sheep-laurel (''Kalmia angustifolia'')
|''Kalmia'' ('''laurel''') is a North American genus with about 10 species, four of which are considered to be native to New York.}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Phyllodoceae||Kalmia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Kalmia|Laurel|449|4|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia angustifolia
| author  = L.
| var     = angustifolia
&lt;!-- -- --&gt;
| en1     = Sheep-laurel
| en2     = Sheep-kill
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1304|5|Sub-alpine forests, wet acidic peatlands, and dry sandy forests and forest edges. Primarily a species of acidic soils it grows in dry to wet open or slightly shaded habitats.}}
| usda    = {{../usda|KAAN|NN|}}
| vascan  = 
| gobot   = Kalmia/angustifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = Kalmia angustifolia NY-dist-map.png
| image1  = Kalmia angustifolia 4502.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia latifolia
| author  = L.
| ---
| en1     = Mountain laurel
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6474|5|}}
| usda    = {{../usda|KALA|N0|}}
| vascan  = 
| gobot   = Kalmia/latifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Sugarcamp Mountain (7) (27749726806).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia polifolia
| author  = Wang.
&lt;!-- -- --&gt;
| en1     = Pale laurel
| en2     = Bog laurel
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1326|4|}}
| usda    = {{../usda|KAPO|NN|}}
| vascan  = 
| gobot   = Kalmia/polifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Bog laurel (Kalmia polifolia) (23945002063).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia procumbens
| author  = 
| sn0     = {{../sn||Loiseleuria procumbens|(L.) Desv.}} 
&lt;!-- -- --&gt;
| en1     = Alpine azalea
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1332|1|}}
| usda    = {{../usda|LOPR|NN|}}
| vascan  = 
| gobot   = Kalmia/procumbens
| its-id  = 23556
| ars-id  = 22486
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Loiseleuria procumbens Kilpisjärvi 2012-07.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Kalmia|Laurel|449|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia buxifolia
| author  = (Bergius) Gift &amp; Kron 2008 
&lt;!-- -- --&gt;
| sn0     = {{../sn|1777|Ledum buxifolium|Bergius}}
| sn1     = {{../sn|1813|Dendrium buxifolium|Desv.}}
| sn2     = {{../sn|1817|'''{{../w|Leiophyllum buxifolium}}'''|Elliott}}&lt;ref&gt;NYFA and NRCS both list ''Kalmia buxifolia'' as ''Leiophyllum buxifolium'' (Berg) Elliott.  ITIS, GRIN and Flora of North America, place it in ''Kalmia''.&lt;/ref&gt;
| sn3     = {{../sn|1839|Leiophyllum lyonii|Sweet}}
| sn4     = {{../sn|1901|Leiophyllum hugeri|(Small) K. Schum.}}
&lt;!-- -- --&gt;
| en1     = Sandmyrtle
| en2     = Sand-myrtle
&lt;!-- -- --&gt;
| status  = N. America native
| of      = southern U. S.
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|851|EXCLUDED}}
| usda    = {{../usda|LEBU|N0|}}
| vascan  = 
| gobot   = Kalmia/buxifolia
| its-id  = 894448
| ars-id  = 451598
| fna-id  = 250065674
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = Excluded nymap.svg
| image1  = Kalmia buxifolia NRCS-1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Ericeae====

=====''Calluna''=====

{{../txt
|img=CallunaVulgaris.jpg
|cap=''Calluna vulgaris'' (L.) Hull&lt;br&gt;heather
|''Calluna'' is a monotypic plant genus whose sole species is ''Calluna vulgaris'' or '''heather''', an Old-World plant.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Ericeae||Calluna|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Calluna|Heather|531|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Calluna vulgaris
| author  = (L.) Hull
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Erica vulgaris|L.}}
| sn1     = {{../sn|1808|Calluna vulgaris|(L.) Hull}}
| sn2     = {{../sn|1906|Ericoides vulgaris|(L.) Merino}}
&lt;!-- -- --&gt;
| en1     = Heather
| en2     = Scotch heather
| en3     = Scots heather
| en4     = Common heather
| en5     = Ling
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| from1   = northern Africa
| status1 = Naturalized
| c-rank  = X
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1337|X|}}
| usda    = {{../usda|CAVU|XX|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 8605
| fna-id  = 
| tro-id  = 12300012
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Calluna%20vulgaris
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Calluna vulgaris sl3.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Rhodoreae====

=====''Rhododendron''=====

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Pentanthera''======

{{../txt
|img=Rhododendron periclymenoides WFNY-154.jpg
|cap=''Rhododendron periclymenoides''&lt;br&gt;pinxter flower
|''Rhododendron'' sect. ''Pentanthera'' contains the '''azaleas''', three of which are native to New York.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pentanthera|Azalea|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron prinophyllum
| author  = (Small) Millais 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1914|Azalea prinophylla|Small}}
{{../sp-1|1917|R. prinophyllum|(Small) Millais}}
{{../sp-1|1921|R. roseum|(Loisel.) Rehder superfl.}}
{{../var1|1924|R. nudiflorum|Torr.|roseum|(Loisel.) Wiegand}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Early azalea|NYFA 2017 / FNA Ed. Comm., 2009}}
{{../vn1|Roseshell azalea|FNA Ed. Comm., 2009 / Castanea, 1994 per ARS-GRIN}}
{{../vn1|Woolly azalea|LBJ 2017}}
{{../vn1|Election-pink|Castanea, 1994 per ARS-GRIN / FNA Ed. Comm., 2009}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = Shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1339|5|Rhododendron prinophyllum (Small) Millais - early azalea - Dry to dry-mesic forests, forest edges, bluffs, hummocks and edges of swamps, and utility rights-of-way. Primarily a species of slightly open dry acidic oak-dominated forests but also somewhat frequent on hummocks in swamps.}}
| usda    = {{../usda|RHPR|N0|}}
| vascan  = 5562
| gobot   = rhododendron/prinophyllum
| its-id  = 
| ars-id  = 105058
| fna-id  = 242417131
| tro-id  = 12300652
| ipn-id  = 
| lbj-id  = RHPR
| nse-id  = 
| bna-id  = Rhododendron%20prinophyllum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron prinophyllum (Roseshell Azalea) (27254491900).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron periclymenoides
| author  = (Michx.) Shinners
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1762|Azalea nudiflora|L. illeg.}}
{{../sp-1|1803|A. periclymenoides|Michx.}}
{{../sp-1|1824|R. nudiflorum|(L.) Torr. illeg.}}
{{../for1|1941|R. nudiflorum|(L.) Torr.|glandiferum|(Porter) Fernald}}
{{../sp-1|1962|R. periclymenoides|(Michx.) Shinners}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Pinxter flower|NYFA}}
{{../vn1|Pinxter-flower|ARS-GRIN}}
{{../vn1|Pinkster|}}
{{../vn1|Pink azalea|}}
{{../vn1|Election-pink|Castanea 1994 per ARS-GRIN}}
{{../vn1|Pinxterbloom azalea|Castanea 1994 per ARS-GRIN}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1343|5|}}
| usda    = {{../usda|RHPE4|N0|}}
| vascan  = 
| gobot   = rhododendron/periclymenoides
| its-id  = 
| ars-id  = 316275
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron periclymenoides - Wild Pink Azalea 2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron viscosum
| author  = (L.) Torr. 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Azalea serrulata|Small|}}
{{../sp-1||Azalea viscosa|L.|}}
{{../sp-1|1824|R. viscosum|(L.) Torr.}}
{{../sp-1||R. serrulatum|(Small) Millais}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Swamp azalea|}}
{{../vn1|Clammy azalea |}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1352|5|}}
| usda    = {{../usda|RHVI2|N0|}}
| vascan  = 
| gobot   = rhododendron/viscosum
| its-id  = 
| ars-id  = 
| fna-id  = 250065656
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20viscosum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron viscosum (28332827000).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron calendulaceum
| author  = (Michx.) Torrey
| ---
| en1     = Flame azalea
| ---
| status  = Introduced
| from    = southeastern US
| status1 = N. America native
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1319|X|}}
| usda    = {{../usda|RHCA4|N0|}}
| vascan  = 
| gobot   = rhododendron/calendulaceum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20calendulaceum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron calendulaceum.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pentanthera|Azalea|228|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron arborescens
| author  = (Pursh) Torr.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1813|Azalea arborescens|Pursh}}
{{../sp-1|1824|R. arborescens|(Pursh) Torr.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Smooth azalea|}}
{{../vn1|Sweet Azalea|}}
&lt;!-- -- --&gt;
| status  = N. America native
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|398|EXCLUDED}}
| usda    = {{../usda|RHAR3|N0|NY: Mitchell, R.S. (ed.). 1986.}}
| note    = 1986&lt;ref&gt;1986: Mitchell, R.S. (ed.). ''A checklist of New York State plants. Contributions of a Flora of New York State, Checklist III. New York State Bulletin No. 458.'' New York State Museum, Albany.&lt;/ref&gt;
| vascan  = 
| gobot   = rhododendron/arborescens
| its-id  = 
| ars-id  = 105055
| fna-id  = 
| tro-id  = 12300627
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20arborescens
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| map     =  nymap.svg
| image1  = Rhododendron arborescens - Sweet Azalea.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron canescens
| author  = (Michx.) Sweet
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Azalea canescens|Michx.}}
{{../sp-1|1830|R. canescens|(Michx.) Sweet}}
{{../var1|1900|Azalea nudiflora|L.|canescens|Rehder in L.H.Bailey}}
{{../sp-1|1901|Azalea candida|Small}}
{{../sp-1|1916|R. candidum|(Small) Rehder}}
{{../var1|1921|R. canescens|(Michx.) Sweet|candidum|(Small) Rehder}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Mountain azalea|WFNY 1918}}
{{../vn1|Hoary azalea|WFNY 1918 / Tropicos 2017}}
{{../vn1|Wild azalea|}}
{{../vn1|Honeysuckle azalea|}}
{{../vn1|Piedmont azalea|Castanea 1994 per ARS-GRIN / Tropicos 2017}}
{{../vn1|Sweet azalea|}}
{{../vn1|Southern pinxterflower|Castanea 1994 per ARS-GRIN}}
{{../vn1|Sweet-Florida pinxter|Tropicos 2017}}
&lt;!-- -- --&gt;
| status  = N. America native
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| usda    = {{../usda|RHCA7|N0|}}
| note    = 1918&lt;ref&gt;1918: Homer D. House, New York State Botanist, ''Wild Flowers of New York'', Vol 2, page 201, &quot;Mountain or Hoary Azalea, ''Azalea canescens'' Michaux ... eastern and southern New York....&quot; This appears to be a misidentification of the similar species ''Rhododendron periclymenoides'', which is common in that range.&lt;/ref&gt;
| vascan  = 
| gobot   = 0
| its-id  = 23712
| ars-id  = 5020
| fna-id  = 
| tro-id  = 12300635
| ipn-id  = 27637-2
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20canescens
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| map     =  nymap.svg
| image1  = Rhododendron canescens (Ericaceae).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Rhodora''======

{{../txt
|img=Rhododendron canadense flowers.JPG
|cap=''Rhododendron canadense''&lt;br&gt;rhodora
|''Rhododendron'' sect. ''Rhodora'' contains the single species ''Rhododendron canadense'' ('''rhodora'''), which is native to northeastern North America. 
}}
{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Rhodora|Rhodora|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron canadense
| author  = (L.) Torr.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1762|Rhodora canadensis|L.}}
{{../sp-1|1841|Rhododendron canadense|(L.) Torr.}}
{{../sp-1|1891|Azalea canadensis|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Rhodora|2017 NYFA}}
{{../vn1|Canada rosebay|2017 New England Wild Flower Society Go Botany}}
{{../vn1|Canada rhododendron|2017 VASCAN: Anions, M., Proposition de nom anglais (pers. comm.)}}
{{../vn1|Canadian rhododendron|2017 VASCAN: Darbyshire S.J., M. Favreau &amp; M. Murray (revu et augmenté par). 2000. Noms populaires et scientifiques des plantes nuisibles du Canada. Agriculture et Agroalimentaire Canada. Publication 1397. 132 pp.}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Rhododendron du Canada|2017 VASCAN (accepted): Darbyshire S.J., M. Favreau &amp; M. Murray (revu et augmenté par). 2000. Noms populaires et scientifiques des plantes nuisibles du Canada. Agriculture et Agroalimentaire Canada. Publication 1397. 132 pp.}}
{{../vn1|Rhodora|2017 VASCAN: Marie-Victorin, Fr. 1995. Flore laurentienne. 3e éd. Mise à jour et annotée par L. Brouillet, S.G. Hay, I. Goulet, M. Blondeau, J. Cayouette et J. Labrecque. Gaétan Morin éditeur. 1093 pp.}}
{{../vn1|Rhodora du Canada|2017 VASCAN: Cunningham, G.C. 1958. Flore fiorestière du Canada. Ministère du Nord canadien et des Ressources nationales. 144 pp.}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| c-rank  = 9
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1311|2|	Rhododendron canadense (L.) Torr. - Rhodora - Bogs, edges of bogs, and wet thickets.}}
| usda    = {{../usda|RHCA6|NN|}}
| vascan  = 5556
| gobot   = rhododendron/canadense
| its-id  = 
| ars-id  = 5019
| fna-id  = 
| tro-id  = 50134272
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20canadense
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron canadense (27196782616).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Pontica''======

{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pontica|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron maximum
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Rhododendron maximum|L.}}
{{../sp-1|1935|Rhododendron ashleyi|Coker}}
{{../sp-1|1943|Hymenanthes maxima|(L.) H.F.Copel.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Great rosebay|NYFA-1}}
{{../vn1|Great laurel|NYFA-2}}
{{../vn1|Great rhododendron|Terrell, E. E. et al. AH 505 1994 per ARS-GRIN}}
{{../vn1|Rose bay|Castanea 1994 per ARS-GRIN}}
{{../vn1|Wild rhododendron|}}
{{../vn1|Rosebay rhododendron|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 9
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1341|4|}}
| usda    = {{../usda|RHMA4|NN|}}
| vascan  = 
| gobot   = rhododendron/maximum
| its-id  = 
| ars-id  = 31441
| fna-id  = 
| tro-id  = 12300644
| ipn-id  = 
| lbj-id  = RHMA4
| nse-id  = 
| bna-id  = Rhododendron%20maximum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Budding (9249172175).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Rhododendron'' sect. ''Rhododendron''======
{{../txt
|img=Lapland Rose-Bay (3815996423).jpg
|cap=''Rhododendron lapponicum''&lt;br&gt;Lapland rosebay
|''Rhododendron'' sect. ''Rhododendron'' contains the two New York natives '''Labrador tea''' (''R. groenlandicum'') and '''Lapland rosebay''' (''R. lapponicum'').
}}
{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Rhododendron|sect=Rhododendron|subsect=Ledum|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron groenlandicum
| author  = (Oeder) Kron &amp; Judd
&lt;!-- ========= --&gt;
| syns    = &lt;poem&gt;
1771. ''Ledum groenlandicum ''Oeder in G.C.Oeder &amp; al. (eds.), Fl. Dan. 4(10):5
1892. ''Ledum palustre      ''var. ''groenlandicum ''(Oeder) Rosenv. in Meddel. Grønland
1948. ''Ledum palustre      ''ssp. ''groenlandicum ''(Oeder) Hultén in Acta Univ. Lund.
1990. '' '''Rhododendron groenlandicum''' ''(Oeder) Kron &amp; Judd in  Syst. Bot. 15:67
&lt;/poem&gt;
&lt;!-- ========= --&gt;
| en1     = Labrador tea
| en2     = Bog Labrador tea
| en3     = Common Labrador tea
| ---
| status  = Native
| status1 = Likely secure
| c-rank  = 9
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1351|4-5|}}
| inatc   = {{../inat|180413-Rhododendron-groenlandicum||}}
| usda    = {{../usda|LEGR|NN|}}
| vascan  = 
| gobot   = rhododendron/groenlandicum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 50303842
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20groenlandicum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Ledem groenlandicum C.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Rhododendron|sect=Rhododendron|subsect=Lapponica|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron lapponicum
| author  = (L.) Wahlenb.
| sn0     = {{../sn||Azalea lapponica|L.}}
| ---
| en1     = Lapland rosebay
| en2     = Lapland azalea
| ---
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1353|1|}}
| usda    = {{../usda|RHLA2|NN|}}
| vascan  = 
| gobot   = rhododendron/lapponicum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20lapponicum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Lapland Rose-Bay (3816823222).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Empetroideae===

'''Empetroideae''' Sweet, ''Hort. Brit.'': 491. Sep–Oct 1826.

====Tribe Empetreae====

Empetreae Horan., ''Char. Ess. Fam.'': 109. 17 Jun 1847.

=====''Empetrum''=====

{{../table|order=Ericales|Ericaceae|Empetroideae|Empetreae||Empetrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Empetrum|Crowberry|701|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Empetrum nigrum
| author  = L.
| sn0     = {{../sn|1753|Empetrum nigrum|L.}}
| sn1     = {{../sn|1913|E. eamesii|ssp=hermaphroditum|}}
| sn2     = {{../sn|1927|E. hermaphroditum|Lange ex Hagerup}}
| sn3     = {{../sn|1933|E. nigrum|var=hermaphroditum|}}
| sn4     = {{../sn|1952|E. nigrum|ssp=hermaphroditum|}}
| ---
| en1     = Black crowberry
| en2     = Crakeberry   
| en3     = Curlew-berry
| ---
| fr1     = Camarine noire
| status  = Native
| status1 = Rare
| nyfa    = {{../nyfa|1313|3|}}
| usda    = {{../usda|EMNI|NN|}}
| vascan  = 5514
| gobot   = empetrum/nigrum
| its-id  = 23743
| ars-id  = 15127
| fna-id  = 200012671
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Empetrum nigrum nymap.svg
| image1  = Empetrum nigrum (1).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Empetrum atropurpureum
| author  = Fernald &amp; Wiegand
| sn0     = {{../sn|1913|E. atropurpureum|Fernald &amp; Wiegand}}
| sn1     = {{../sn|1927|E. rubrum'' var. ''atropurpureum|}}
| sn2     = {{../sn|1960|E. eamesii'' ssp. ''atropurpureum|}}
| sn3     = {{../sn|1966|E. nigrum'' var. ''atropurpureum|}}&lt;ref&gt;USDA-NRCS-PLANTS and ITIS treat ''E. atropurpureum'' as a synonym of ''E. nigrum'', but NYFA, FNA, and ''Flora Novae Angliae'' all treat ''E. atropurpureum'' as a separate species.&lt;/ref&gt;
| ---
| en1     = Purple crowberry
| en2     = Red crowberry
| ---
| fr1     = Camarine noire-pouprée
| fr2     = Camarine atropourpre
| fr3     = Camarine pourpre
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1314|1|}}
| usda    = {{../usda|EMEAA|NN|as Empetrum eamesii Fernald &amp; Wiegand ssp. atropurpureum (Fernald &amp; Wiegand) D. Löve }}
| vascan  = 5511
| gobot   = empetrum/atropurpureum
| its-id  = 23743
| ars-id  = 
| fna-id  = 250065677
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Empetrum atropurpureum nymap.svg
| image1  = Empetrum nigrum (Crowberry) - unripe berries - Flickr - S. Rae.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Corema''=====

{{../table|order=Ericales|Ericaceae|Empetroideae|Empetreae||Corema|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Corema|Crowberry|992|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Corema conradii
| author  = (Torr.) Torr.
| sn0     = {{../sn|1837|Empetrum conradii|Torr.}}
| sn1     = {{../sn|1842|Corema conradii|(Torr.) Torr.}}
| ---
| en1     = Broom crowberry 
| en2     = Broom-crowberry 
| en3     = Poverty-grass 
| ---
| fr1     = Corème de Conrad
| fr2     = Camarine de Conrad
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1315|1|}}
| usda    = {{../usda|COCO9|NN|}}
| vascan  = 5509
| gobot   = corema/conradii
| its-id  = 23750
| ars-id  = 400579
| fna-id  = 250065678
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Corema conradii nymap.svg
| image1  = Corema conradii buds.jpeg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Vaccinoideae===

====Tribe Oxydendreae====

=====''Oxydendrum''=====

{{../txt
|img=Oxydendrum arboreum 3zz.jpg
|cap=''Oxydendrum arboreum'' (L.) DC.&lt;br&gt;sourwood tree
|''Oxydendrum'' contains a single species, the '''sourwood''' tree (''O. arboreum''), which is endemic to the southeastern U.S., as far north as southwestern Pennsylvania. Its flowers grow in racemes and look similar to lily-of-the-valley flowers.  It is cultivated as an ornamental in southern New York, where it has escaped.}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Oxydendreae||Oxydendrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Oxydendrum|Sourwood|117|1|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Oxydendrum arboreum
| author  = (L.) DC.
| syns    = {{../snl
|{{../spa1|1753|Andromeda arborea|L.}}
|{{../spa1|1839|Oxydendrum arboreum|(L.) DC.}}
}}
&lt;!-- -- --&gt;
| en1     = Sourwood
| en2     = Sorreltree
| en3     = Lily-of-the-valley tree
| en4     = Swamp-cranberry
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| nyfa    = {{../nyfa|1333|X|State exotic or non-native}}
| usda    = {{../usda|OXAR|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 23690
| ars-id  = 311632
| fna-id  = 220009741
| tro-id  = 12300622
| nse-id  = Oxydendrum+arboreum
| ipn-id  = 1175876-2
| map     = Oxydendrum arboreum nymap.svg
| image1  = Sourwood leaves and flowers.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Oxydendreae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Lyonieae====

=====''Lyonia''=====

{{../txt
|img=Lyonia ligustrina 1120554.jpg
|cap=''Lyonia ligustrina'' - maleberry
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Lyonieae||Lyonia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lyonia|Staggerbush|136|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lyonia ligustrina
| author  = (L.) DC.
| var     = ligustrina
| sn0     = {{../sn|1753|Vaccinium ligustrinum|L.}}
| sn1     = {{../sn|1813|Andromeda ligustrina|(L.) Muhl.}}
| sn2     = {{../sn|1839|Lyonia ligustrina|(L.) DC.}}
| sn3     = {{../sn|1894|Xolisma ligustrina|(L.) Britton}}
| sn4     = {{../sn|1913|Arsenococcus ligustrinus|(L.) Small}}
&lt;!-- -- --&gt;
| en1     = Maleberry
| en2     = Seedy buckberry
| en3     = Privet andromeda
| fr1     = Lyonie faux-troène
| fr2     = Lyonie ligustrine
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1331|5|}}
| usda    = {{../usda|LYLI|N0|}}
| vascan  = 23768
| gobot   = lyonia/ligustrina
| its-id  = 23559
| ars-id  = 22993
| fna-id  = 250065687
| tro-id  = 12300077
| nse-id  = Lyonia+ligustrina
| ipn-id  = 
| map     =  nymap.svg
| image1  = Lyonia ligustrina 2zz.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lyonia mariana
| author  = (L.) D.Don
| sn0     = {{../sn|1753|Andromeda mariana|L.}}
| sn1     = {{../sn|1834|Lyonia mariana|(L.) D.Don}}
| sn2     = {{../sn|1876|Pieris mariana|(L.) Benth. &amp; Hook.f.}}
| sn3     = {{../sn|1913|Neopieris mariana|(L.) Britton}}
| sn4     = {{../sn|1924|Xolisma mariana|(L.) Rehder}}
&lt;!-- -- --&gt;
| en1     = Stagger-bush
| en2     = Piedmont staggerbush
| en3     = Maryland staggerbush
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1329|4|}}
| usda    = {{../usda|LYMA2|N0|}}
| vascan  = | south of Canada
| gobot   = lyonia/mariana
| its-id  = 23564
| ars-id  = 22995
| fna-id  = 242416810
| tro-id  = 12302715
| nse-id  = Lyonia+mariana
| ipn-id  = 331154-1
| map     =  nymap.svg
| image1  = Lyonia mariana WFNY-155B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Lyonieae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Andromedeae====

=====''Andromeda''=====

{{../txt
|img=Andromeda polifolia 2016-04-28 9221.jpg
|cap=''Andromeda polifolia''&lt;br&gt;bog rosemary
|''Andromeda polifolia'' ('''bog-rosemary''') is generally treated as the only member of its genus and is found in acidic bogs throughout the northern hemisphere.  Two varieties are considered to be present in North America, but only the '''glaucous-leaved bog rosemary''', ''A. polifolia'' var. ''latifolia'' (syn. ''A. glaucophylla'') has been reported in New York. 
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Andromedeae||Andromeda|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Andromeda|Bog-rosemary|824|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Andromeda polifolia
| author  = L.
| var     = latifolia
| var-au  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|A. polifolia|var=latifolia|var-au=Aiton}}
| sn1     = {{../sn|1821|A. glaucophylla|Link}}
| sn2     = {{../sn|1839|A. polifolia|var=glaucophylla|var-au=(Link) DC.}}
| sn3     = {{../sn|1914|A. canescens|Small}}
| sn4     = {{../sn|1916|A. glaucophylla|var=iodandra|var-au=Fernald}}
| sn5     = {{../sn|1924|A. glaucophylla|fo=latifolia|fo-au=(Aiton) Rehder}}
| sn6     = {{../sn|1927|A. glaucophylla|var=latifolia|var-au=(Aiton) Rehder}}
| sn7     = {{../sn|1948|A. polifolia|ssp=glaucophylla|ssp-au=(Link) Hultén}}
&lt;!-- -- --&gt;
| en1     = Bog rosemary
| en2     = Glaucous-leaved bog rosemary
| fr1     = Andromède glauque
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 10
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1330|5|Andromeda polifolia L. var. latifolia Aiton - bog rosemary - Acidic bogs. Often growing mixed with other shrubs and forming a dense low shrub thicket in bogs. Also growing on hummocks or rises in bogs in small shrub ''islands''}}
| usda    = {{../usda|ANPOG|NN|}}
| vascan  = 5486
| gobot   = Andromeda/polifolia
| its-id  = 894555 
| ars-id  = 467871
| fna-id  = 250065693
| tro-id  = 50294868
| ipn-id  = 
| lbj-id  = ANPOG
| nse-id  = 
| bna-id  = 
| map     = Andromeda polifolia var glaucophylla NY-dist-map.png
| image1  = Andromeda polifolia var. latifolia WFNY-159A.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Andromedeae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Gaultherieae====

=====''Chamaedaphne''=====

{{../txt
|img=Chamaedaphne calyculata 1 (5097220701).jpg
|cap=''Chamaedaphne calyculata''&lt;br&gt;leatherleaf
|''Chamaedaphne calyculata'' ('''leatherleaf''') is the sole species of the genus and is native to much of the Northern Hemisphere, including New York State.
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Chamaedaphne|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Chamaedaphne|Leatherleaf|446|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chamaedaphne calyculata
| author  = (L.) Moench
| ---
| en1     = Leatherleaf
| ---
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1349|5|Chamaedaphne calyculata (L.) Moench - leatherleaf - Bogs, edges of ponds, and acidic peaty open sites. Mostly confined to acidic peatlands where it can form dense extensive monospecific stands or become mixed with other low shrubs to from dense shrub thickets.}}
| usda    = {{../usda|CHCA2|NN|}}
| vascan  = 
| gobot   = Chamaedaphne/calyculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Chamaedaphne calyculata WFNY-157A.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Gaultheria''=====
{{../txt
|img=Gaultheria procumbens.JPG
|cap=''Gaultheria procumbens''&lt;br&gt;wintergreen
|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Gaultheria|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gaultheria|Snowberry|160|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaultheria procumbens
| author  = L.
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Wintergreen|2018 New York Flora Atlas 1 / Flora of North America}}
{{../vn1|Teaberry|2018 New York Flora Atlas 2}}
{{../vn1|Eastern teaberry|2018 USDA NRCS National Plant Data Team / Flora of North America}}
{{../vn1|Checkerberry|2018 Flora of North America}}
{{../vn1|Eastern spicy-wintergreen|2018 New England Wild Flower Society, Go Botany}}
{{../vn1|Creeping wintergreen|2018 VASCAN Canadensys}}
{{../vn1|Spring wintergreen|2018 VASCAN Canadensys}}
{{../vn1|Mountain-tea|2018 ARS GRIN Huxley, A., ed. Dict. Gard 1994}}

| en3     = American wintergreen

| ---
| fr1     = Thé des bois
| fr2     = Thé rouge
| fr3     = Gaulthérie couchée
| status  = Native
| status1 = Secure
| habit0  = Perennial
| habit1  = Shrub, subshrub
| nyfa    = {{../nyfa|1310|5|}}
| usda    = {{../usda|GAPR2|NN|}}
| vascan  = 5520
| gobot   = Gaultheria/procumbens
| its-id  = 
| ars-id  = 360
| fna-id  = 220005469
| tro-id  = 12300028
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Golteria1-pl.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaultheria hispidula
| author  = (L.) Muhl. ex Bigel.
| sn0     = Chiogenes hispidula
| ---
| en1     = Creeping snowberry
| ---
| status  = Native
| status1 = Secure
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1317|5|}}
| usda    = {{../usda|GAHI2|NN|}}
| vascan  = 
| gobot   = Gaultheria/hispidula
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Gaultheria hispidula 7847 (cropped).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Eubotrys''=====

{{:Flora of New York/txt
|img=Eubotrys racemosa - Swamp doghobble 2.jpg
|cap=''Eubotrys racemosa''
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Eubotrys|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Eubotrys|Fetterbush|1133|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Eubotrys racemosa
| author  = (L.) Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1842|Eubotrys racemosa|(L.) Nutt.}}
{{../sp-1|1856|Leucothoe racemosa|(L.) A. Gray}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Swamp fetterbush|2019 New York Flora Atlas}}
{{../vn1|Swamp doghobble|2019 Native Plant Trust, USDA NRCS National Plant Data Team}}
{{../vn1|Swamp deciduous dog-laurel|2019 Native Plant Trust}}
{{../vn1|Deciduous swamp fetterbush|Flora of North America}}
{{../vn1|Sweetbells|2019 Native Plant Trust}}
{{../vn1|Swamp sweetbells|2019 Native Plant Trust}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1346|4|}}
| usda    = {{../usda|EURA5|N0|}}
| vascan  = 
| gobot   = Eubotrys/racemosa
| its-id  = 
| ars-id  = 
| fna-id  = 250065697
| tro-id  = 12302763
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Leucothe racemosa (5625200643).gif
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Eubotrys recurva
| author  = (Buckley) Britton 
| ---
| en1     = Mountain fetterbush 
| en2     = Deciduous mountain fetterbush 
| en3     = Redtwig doghobble
| en4     = Red-twigged doghobble
| ---
| status  = Introduced
| status1 = US South native
| habit0  = Perennial
| habit1  = Shrub
| nyfa    = {{../nyfa|1342|X|}}
| usda    = {{../usda|EURE10|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Eubotrys%recurva
| map     =  nymap.svg
| image1  = Eubotrys recurva BB-1913.png
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Leucothoe''=====

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Leucothoe|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Leucothoe|Doghobble|991|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Leucothoe fontanesiana
| author  = (Steud.) Sleumer
| ---
| en1     = Highland dog-hobble
| ---
| status  = Introduced
| status1 = US South native
| status2 = Impersistent
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1344|Xm|}}
| usda    = {{../usda|LEFO|N0|}}
| vascan  = 
| gobot   = leucothoe/fontanesiana &lt;!--exotic in MA--&gt;
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Leucothoe%20fontanesiana
| map     =  nymap.svg
| image1  = Leucothe fontanesia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Leucothoe axillaris
| author  = (Lam.) D.Don 
&lt;!-- -- --&gt;
| sn0     = {{../sn|1783|Andromeda axillaris|Lam.}}
| sn1     = {{../sn|1788|Andromeda catesbaei|Walter}}
| sn2     = {{../sn|1809|Andromeda walteri|Willd.}}
| sn3     = {{../sn|1834|Leucothoe axillaris|D.Don}}
| sn4     = {{../sn|1856|Leucothoe catesbaei|A.Gray|}}
&lt;!-- -- --&gt;
| en1     = Coastal doghobble
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Cultivated
| status3 = No NY reports
&lt;!-- -- --&gt;
| habit0  = Perennial
| habit1  = Shrub
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|LEAX|N0|}}
| vascan  = 
| gobot   = &lt;!--Not listed in New Enland--&gt;
| its-id  = 
| ars-id  = 22006
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = Leucothoe+axillaris
| bna-id  = Leucothoe%20axillaris
| map     =  nymap.svg
| image1  = Leucothoe axillaris3.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====Tribe Vaccinieae====

{{../txt
|The Vaccinieae contain the morphologically similar '''huckleberries''' (''Gaylussacia'') and '''blueberries''' (''Vaccinium'').
}}

=====''Gaylussacia''=====

{{../txt
|img=Gaylussacia dumosa (6009920604).jpg
|cap=''Gaylussacia bigeloviana''&lt;br&gt;bog huckleberry
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Gaylussacia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gaylussacia|Huckleberry|491|3|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia baccata
| author  = ({{../w|Wangenh.}}) {{../w|K.Koch}}
| sn0     = {{../sn|1787|Andromeda baccata|Wangenh.}}
| sn1     = {{../sn|1872|Gaylussacia baccata|K.Koch}}
| sn2     = {{../sn|1900|G. resinosa|var=glaucocarpa|B.L.Rob.}}
| sn3     = {{../sn|1907|G. baccata|var=glaucocarpa|Mack.}}
| sn4     = {{../sn|1933|Decachaena baccata|Small}}
| ---
| en1     = Black huckleberry
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1307|5|}}
| usda    = {{../usda|GABA|NN|}}
| vascan  = 
| gobot   = gaylussacia/baccata
| its-id  = 
| ars-id  = 
| fna-id  = 242416580
| tro-id  = 12300598
| nse-id  = 
| ipn-id  = 
| map     =  
| image1  = Gaylussacia baccata 5497208.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia frondosa
| author  = (L.) Torr. &amp; A.Gray
| sn0     = {{../sn|1753|Vaccinium frondosum|L.}}
| sn1     = {{../sn|1843|Gaylussacia frondosa|Torr. &amp; A.Gray}}
| sn2     = {{../sn|1933|Decachaena frondosa|ex Small}}
| ---
| en1     = Blue huckleberry
| en2     = Dangleberry
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1308|4|Apparently secure in New York State.}}
| usda    = {{../usda|GAFR2|N0|}}
| vascan  = 
| gobot   = gaylussacia/frondosa
| its-id  = 
| ars-id  = 376
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     =  
| image1  = Gaylussacia frondosa NRCS-01.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia bigeloviana
| author  = (Fernald) Sorrie &amp; Weakley
| sn0     = {{../sn|1911|'''{{../w|Gaylussacia dumosa}}'''|var1=bigeloviana|Fernald}}
| sn1     = {{../sn|1933|Lasiococcus dumosus|var=bigelovianus||Fernald}}
| sn2     = {{../sn|2007|Gaylussacia bigeloviana|Sorrie &amp; Weakley}}
| sn3     = {{../sn|auct|Gaylussacia dumosa|'''non''' (Andrews) A.Gray}}
| ---
| en1     = Dwarf huckleberry
| en2     = Bog huckleberry
| ---
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1306|1-2|}}
| usda    = {{../usda|GADU|NN|as Gaylussacia dumosa (Andrews) Torr. &amp; A.Gray 1846}}
| vascan  = 
| gobot   = gaylussacia/bigeloviana
| its-id  = 894464
| ars-id  = 319642
| fna-id  = 250065725
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Gaylussacia bigeloviana nymap.svg
| image1  = Gaylussacia bigeloviana WFNY-157B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Vaccinieae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Vaccinium''=====

{{../txt|The genus '''''Vaccinium''''' comprises the blueberries, cranberries, huckleberries and similar shrubs.  It is here broken down into sections as described in Flora of North America.&lt;ref&gt;[http://efloras.org/florataxon.aspx?flora_id=1&amp;taxon_id=134285 ''Vaccinium.'' Flora of North America Editorial Committee, eds. 1993+. Flora of North America North of Mexico.  20+ vols.  New York and Oxford.]&lt;/ref&gt;}}

======''Vaccinium'' sect. ''Cyanococcus''======

{{../txt|Section ''Cyanococcus'' was described by Asa Gray in ''Memoirs of the American Academy of Arts and Science'' in 1846.&lt;ref&gt;[http://www.tropicos.org/Name/50140387 ''Vaccinium'' sect. ''Cyanococcus'' A. Gray. Tropicos.org. Missouri Botanical Garden. 13 Oct 2016]&lt;/ref&gt; This section contains all of the New York '''blueberry''' species except the alpine blueberry.}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Cyanococcus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Blueberry|832|11|VACCI|14|sect=Cyanococcus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium angustifolium
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium angustifolium|Aiton}}
| sn1     = {{../sn|1861|V. pensylvanicum|var=nigrum|var-au=Alph. Wood}}
| sn2     = {{../sn|1894|V. nigrum|(Alph.Wood) Britton}}
| sn3     = {{../sn|1914|V. brittonii|Porter ex E. P. Bicknell}}
| sn4     = {{../sn|1931|Cyanococcus angustifolium|(Aiton) Rydb.}}
| sn5     = {{../sn|1943|V. lamarckii|Camp}}
&lt;!-- -- --&gt;
| en1     = Lowbush blueberry
| en2     = Late lowbush blueberry
| en3     = Low sweet blueberry
| en4     = Late sweet blueberry
| en5     = Sweet-hurts
&lt;!-- -- --&gt;
| fr1     = Bleuet à feuilles étroites
| fr2     = Airelle de Pennsylvanie
| status  = Native
| status1 = Secure
| c-rank  = 4
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = Sun - shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6470|5|Onon-Tomp-Cort-Oswe-Onei}}
| usda    = {{../usda|VAAN|NN|}}
| vascan  = 5566
| gobot   = Vaccinium/angustifolium
| its-id  = 23579
| ars-id  = 40981
| fna-id  = 250065717
| tro-id  = 12300044
| ipn-id  = 
| lbj-id  = VAAN
| nse-id  = 
| bna-id  = Vaccinium%20angustifolium
| map     =  nymap.svg
| image1  = 2016-09 Monts Valin 23.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium corymbosum
| author  = L.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Vaccinium corymbosum|L.}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Highbush blueberry
| en2     = Swamp blueberry
| en3     = Whortleberry
| en4     = New Jersey blueberry
| en5     = Southern blueberry
| ---
| fr1     = Bleuet en corymbe 
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6472|5|}}
| usda    = {{../usda|VACO|NN|}}
| vascan  = 5569
| gobot   = Vaccinium/corymbosum
| its-id  = 23573
| ars-id  = 41002
| fna-id  = 242417401
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20corymbosum
| map     =  nymap.svg
| image1  = 2013-08-25 14 02 03 Closeup of blueberries along the shore of Spring Lake at 88 Lake Road in Berlin, New York.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium × atlanticum
| author  = E.P.Bicknell (pro sp.)
| hp1     = Vaccinium angustifolium
| hp2     = Vaccinium corymbosum
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Atlantic huckleberry
| en2     = {{../hybrid-of|Lowbush blueberry|Highbush blueberry}}
| ---
| status  = Native
| status1 = Unranked
| c-rank  = 
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1345|U|}}
| usda    = {{../usda|VAAT3|N0|×atlanticum NY &amp; CT only}}
| vascan  = 
| gobot   = 
| its-id  = 505629
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20X%20atlanticum
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium myrtilloides
| author  = Michx.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1803|Vaccinium myrtilloides|Michx.}}
| sn1     = {{../sn|1823|V. canadense|Kalm ex Richardson}}
| sn2     = {{../sn|1917|Cyanococcus canadensis|(Kalm ex Richardson) Rydb.}}
| sn3     = {{../sn|1921|V. angustifolium|var=myrtilloides|var-au=(Michx.) House}}
| sn4     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Velvet-leaved blueberry
| en2     = Velvet-leaved huckleberry
| en3     = Velvetleaf huckleberry
| en4     = Sour-top blueberry
| en5     = Sourtop
| en6     = Canada blueberry
| fr1     = Bleuet fausse-myrtille
| fr2     = Bleuet rameau-velouté
| fr3     = Bleuet du Canada
| fr4     = Bleuets
| fr5     = Airelle fausse-myrtille
| fr6     = Airelle faux-myrtille
| fr7     = Airelle du Canada
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6471|5|Hummocks in swamps, edges of swamps, cool northern forests, edges of forests, forest openings, barrens, and bluffs. More common in the northern and cooler parts of New York.}}
| usda    = {{../usda|VAMY|NN|}}
| vascan  = 5573
| gobot   = vaccinium/myrtilloides
| its-id  = 
| ars-id  = 41039
| fna-id  = 250065719
| tro-id  = 12300054
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20myrtilloides
| map     =  nymap.svg
| image1  = Vaccinium myrtilloides berries.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium pallidum
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium pallidum|Aiton}}
| sn1     = {{../sn|1856|V. corymbosum|var=pallidum|var-au=(Aiton) A. Gray}}
| sn2     = {{../sn|1933|Cyanococcus pallidus|(Aiton) Small}}
&lt;!-- -- --&gt;
| en1     = Early lowbush blueberry
| en2     = Late lowbush blueberry
| en3     = Blue Ridge blueberry
| en4     = Hillside blueberry
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1320|5|}}
| usda    = {{../usda|VAPA4|NN|}}
| vascan  = 
| gobot   = Vaccinium/pallidum
| its-id  = 
| ars-id  = 41049
| fna-id  = 242417402
| tro-id  = 12300678
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20pallidum
| map     =  nymap.svg
| image1  = Vaccinium pallidum - Blue Ridge Blueberry.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium boreale
| author  = I.V.Hall &amp; Aalders
| sn0     = {{../sn|1848|V. pensylvanicum|var=angustifolium|var-au=(Aiton) A. Gray}}
| sn1     = {{../sn|1861|V. pensylvanicum|var=alpinum|var-au=Wood}}
| sn2     = {{../sn|1961|V. boreale|I.V.Hall &amp; Aalders}}
| ---
| en1     = Northern blueberry
| en2     = High-mountain blueberry
| en3     = Sweet hurts
| ---
| fr1     = Bleuet boréal
| status  = Native
| status1 = Threatened
| c-rank  = 10
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nynhp   = 2&lt;ref&gt;{{../nynhp-ref|9021|Vaccinium boreale|Threatened|S2|G4}}&lt;/ref&gt;
| nyfa    = {{../nyfa|1309|2|}}
| usda    = {{../usda|VABO|NN|}}
| vascan  = 5567
| gobot   = vaccinium/boreale
| its-id  = 23583 
| ars-id  = 315259
| fna-id  = 250065716
| tro-id  = 50077834
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20boreale
| map     =  nymap.svg
| image1  = 
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Vaccinium''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Vaccinium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|sect=Vaccinium|Blueberry|832|11|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium uliginosum
| author  = L. (1753)
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Alpine blueberry
| en2     = Bog blueberry
| en3     = Bog bilberry
| en4     = Bog whortleberry
| en5     = Northern bilberry
| en6     = Airelle de marécages
| status  = Native
| status1 = Rare
| nyfa    = {{../nyfa|6478|3|}}
| usda    = {{../usda|VAUL|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23574
| ars-id  = 41063
| fna-id  = 200016726
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Ripe bilberries, Dipton Wood - geograph.org.uk - 1472503.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Myrtillus''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Myrtillus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Dwarf blueberry|832|11|VACCI|14|sect=Myrtillus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium cespitosum
| author  = Michx.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1803|V. caespitosum|Michx.}}
| sn1     = {{../sn|1818|V. geminiflorum|Kunth}}
| sn2     = {{../sn|1899|V. arbuscula|(A.Gray) Merriam|}}
| sn3     = {{../sn|1942|V. nivictum|Camp|}}
| sn4     = {{../sn|1942|V. paludicola|Camp|}}
&lt;!-- -- --&gt;
| en1     = Dwarf bilberry
| en2     = Dwarf blueberry
| fr1     = Airelle gazonnante
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1340|1|}}
| usda    = {{../usda|VACE|NN|}}
| vascan  = 
| gobot   = Vaccinium/cespitosum
| its-id  = 23587
| ars-id  = 40996
| fna-id  = 250065710
| tro-id  = 12300045
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium caespitosum 3-eheep (5097507665).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Oxycoccus''======

'''Cranberries''' (''Vaccinium'' sect. ''Oxycoccus'') can be found in acidic bogs throughout the state.

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Oxycoccus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Cranberry|832|11|VACCI|14|sect=Oxycoccus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium macrocarpon
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium macrocarpon|Aiton}}
| sn1     = {{../sn|1803|V. oxycoccos|var=oblongifolium|var-au=Michx.}}
| sn2     = {{../sn|1805|Oxycoccus macrocarpos|(Aiton) Pers.}}
| sn3     = {{../sn|1805|O. palustris|var=macrocarpos|var-au=(Aiton) Pers.}}
| sn4     = {{../sn|1894|Schollera macrocarpos|(Aiton) Britton}}
&lt;!-- -- --&gt;
| en1     = Cranberry
| en2     = Large cranberry
| en3     = American cranberry
| fr1     = Canneberge à gros fruits
| fr2     = Ronce d'Amérique
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub, subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1305|5|}}
| usda    = {{../usda|VAMA|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23599
| ars-id  = 41030
| fna-id  = 250065705
| tro-id  = 12300051
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium macrocarpon01.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium oxycoccos
| author  = L.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Vaccinium oxycoccos|L.}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Cranberry
| en2     = Small cranberry
| en3     = European cranberry
| fr1     = Canneberge commune
| status  = Native
| status1 = Secure
| status2 = (circumboreal)
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub, subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1321|5|}}
| usda    = {{../usda|VAOX|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 505635
| ars-id  = 41047
| fna-id  = 200016697
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium oxycoccos 09072.JPG|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Polycodium''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Polycodium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Deerberry|832|11|VACCI|14|sect=Polycodium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium stamineum
| author  = L. (1753)
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Deerberry
| en2     = Southern-gooseberry
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|6477|5|}}
| usda    = {{../usda|VAST|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23615
| ars-id  = 41061
| fna-id  = 242417403
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccineum stamineum 1120600.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

{{BookCat}}

==References==
{{Reflist}}</text>
      <sha1>lpfb9w9gce89n5xit8kbsh7f4bmsfvj</sha1>
    </revision>
    <revision>
      <id>4631231</id>
      <parentid>4631230</parentid>
      <timestamp>2026-04-18T13:57:24Z</timestamp>
      <contributor>
        <username>Nonenmac</username>
        <id>6290</id>
      </contributor>
      <comment>/* Primula subg. Auriculastrum */</comment>
      <origin>4631231</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="169666" sha1="r5hzsi12reirq24walqvhas6xq2f23r" xml:space="preserve">{{../header
| this      = Ericales
| prev-link = Cornales
| next-link = Gentianales
}}

==Ericales introduction==

{{:Flora of New York/Clade1 Ericales}}
{{:Flora of New York/TOC-Ericales}}

==Family Balsaminaceae==

{{../txt|The {{../w|Balsaminaceae}} ('''balsam family''') contains the two genera: ''Impatiens'' with about 1000 species and ''Hydrocera'' with a single species. Of these, four ''Impatiens'' species have been reported growing wild in New York.&lt;ref&gt;{{../nyfa-fam|Balsaminaceae}}&lt;/ref&gt;&lt;ref&gt;{{../usda-fam|Balsaminaceae}}&lt;/ref&gt;&lt;ref&gt;[http://www.mobot.org/mobot/research/apweb/orders/ericalesweb.htm#Balsaminaceae P.F. Stevens, P. F. (2001-2015). ''Angiosperm Phylogeny Website''. Version 14, University of Missouri, St Louis, and Missouri Botanical Garden, April 2015.]&lt;/ref&gt;}}

===''Impatiens''===

{{../txt|[[File:Impatiens capensis and-or pallida SCA-4330.jpg|thumb|right|Jewelweed with bright white water droplets at the edges of the leaves.]]
Of the four ''Impatiens'' species found outside of cultivation in New York, all are annual herbaceous plants The two native species, '''spotted jewelweed''' (''Impatiens capensis'') and '''pale jewelweed''' (''Impatiens pallida''), are common in wet wooded areas throughout much of the state.

Two non-native ''Impatiens'' species may also be found, but are less common. '''Himalayan balsam''' (''Impatiens glandulifera'') is considered to be moderately invasive in New York.  '''Garden balsam''' (''Impatiens balsamina'' has been reported, but whether it has truly naturalized in the state is in question.}}

{{../table|order=Ericales|Balsaminaceae||||Impatiens|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Impatiens|Touch-me-not|90|4|IMPAT|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens capensis
| au-abbr = Meerb.
| au-full = Nicolaas Meerburgh
| au-pub  = Afb. Zeldz. Gew.: 8 (1775)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1775|Impatiens capensis    |Meerb.}}
{{../sp-1|1788|Impatiens biflora     |Walter}}
{{../sp-1|1813|Impatiens maculata    |Muhl. (not validly publ.)}}
{{../sp-1|1818|Impatiens fulva       |Nutt.}}
{{../sp-1|1849|Balsamina fulva       |(Nutt.) Ser.}}
{{../sp-1|1910|Impatiens nortonii    |Rydb.}}
{{../sp-1|1916|Chrysaea biflora      |(Walter) Nieuwl. &amp; Lunell}}
{{../ssp1|1967|Impatiens noli-tangere|L. |biflora |(Walter) Hultén}}
&lt;!-- -- --&gt;
| en1     = Spotted jewelweed
| en2     = Orange touch-me-not
| en3     = Spotted touch-me-not
| en4     = Spotted snapweed
| ---
| fr1     = Impatiente du Cap
| fr2     = Chou sauvage
| fr3     = Impatiente biflore
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 3
| wetland = FACW
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = S5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6618|5|Impatiens capensis Meerb. - spotted jewelweed, touch-me-not, snapweed - Floodplain forests, wet forests, seepage areas, swamps, marshes, fens, stream banks, thickets, disturbed areas, shaded roadsides and trail edges, and ditches. A very common annual of wet to mesic soils, preferring wetter sites. It can form dense large stands.}}
| usda    = {{../usda|IMCA|NN|}}
| vascan  = 3655
| gobot   = impatiens/capensis
| its-id  = 29182
| ars-id  = 19813
| fna-id  = 
| tro-id  = 3100006
| map     = Impatiens capensis NY-dist-map.png
| image1  = Jewelweed - Impatiens capensis, Julie Metz Wetlands, Woodbridge, Virginia - 27275915403.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens pallida
| author  = Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Impatiens noli-tangere |Michx. (sensu auct.)}}
{{../sp-1|1813|Impatiens aurea        |Muhl.}}
{{../sp-1|1818|Impatiens pallida      |Nutt.}}
{{../var1|1904|Impatiens pallida      |Nutt. |alba |Clute}}
{{../sp-1|1916|Chrysaea aurea         |(Muhl.) Nieuwl. &amp; Lunell}}
&lt;!-- -- --&gt;
| en1     = Pale jewel-weed
| en2     = Pale snapweed
| en3     = Pale touch-me-not
| en4     = Yellow touch-me-not
| ---
| fr1     = Impatiente pâle
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| wetland = FACW
| c-rank  = 3
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = S4, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|522|4|Impatiens pallida Nutt. - pale jewel-weed - Floodplain forests, wet forests, seepage areas, swamps, marshes, fens, stream banks, thickets, disturbed areas, shaded roadsides and trail edges, and ditches. Similar habitat to I. capensis and sometimes occurring together but not as common in at least parts of NY. It perhaps prefers more calcareous soils. Like I. capensis it can form dense large patches.}}
| usda    = {{../usda|IMPA|NN|}}
| vascan  = 3659
| gobot   = impatiens/pallida
| its-id  = 29189
| ars-id  = 417739
| fna-id  = 
| tro-id  = 3100027
| map     = 
| image1  = Impatiens pallida SCA-5938.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens glandulifera
| author  = Royle
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1835|Impatiens glandulifera |Royle}}
{{../sp-1|1840|Impatiens glanduligera |Lindl.}}
{{../sp-1|1842|Impatiens roylei       |Walp. (nom. superfl.)}}
{{../sp-1|1849|Balsamina roylei       |(Walp.) Ser.}}
{{../var1|2021|Impatiens sulcata      |Wall. |glandulifera|(Royle) R.Kr.Singh &amp; D.Borah}}
&lt;!-- -- --&gt;
| en1     = Ornamental jewelweed
| en2     = Policeman's helmet
| en3     = Himalayan balsam
| en4     = Himalayan touch-me-not
| ---
| fr1     = Balsamie de l'Himmalaya
| fr2     = Impatiente glanduleuse
| fr3     = Balsamine géante
&lt;!-- -- --&gt;
| status  = Introduced
| from    = India, Nepal
| from1   = Pakistan
| status1 = Moderately invasive
| c-rank  = X
| habit0  = Annual
| habit1  = Herb-forb
| nyis    = 67%&lt;ref&gt;{{../nyis-ref|Impatiens glandulifera|Moderate|67|first}}&lt;/ref&gt;
| ny-rank = SNA, GNR
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|521|X|Impatiens glandulifera Royle - policeman's helmet}}
| usda    = {{../usda|IMGL|XX|Impatiens glandulifera Royle - ornamental jewelweed }}
| vascan  = 3657
| gobot   = impatiens/glandulifera
| its-id  = 29187
| ars-id  = 19826
| fna-id  = 
| tro-id  = 3100049
| map     = 
| cos     = Columbia, Essex, Jefferson, Lewis, Madison, Schuyler, Sullivan
| image1  = Impatiens glandulifera on way from Gangria to Valley of Flowers National Park - during LGFC - VOF 2019 (12).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens balsamina
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Impatiens balsamina |L.}}
{{../sp-1|1790|Balsamina foemina   |Gaertn.}}
{{../sp-1|1922|Impatiens giorgii   |De Wild.}}
{{../sp-1|1962|Impatiens eriocarpa |Launert}}
&lt;!-- -- --&gt;
| en1     = Garden balsam
| en2     = Garden touch-me-not
| en3     = Spotted snapweed
| en4     = Rose balsam
&lt;!-- -- --&gt;
| status  = Introduced
| from    = India, Myanmar
| c-rank  = 
| wetland = UPL
| habit0  = Annual
| habit1  = Herb-forb
| ny-rank = SNA, GNR
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|520|X|Impatiens balsamina L. - garden touch-me-not, garden balsam, rose balsam}}
| usda    = {{../usda|IMBA|XX|}}
| vascan  = 
| gobot   = 
| its-id  = 29185
| ars-id  = 19810
| fna-id  = 
| tro-id  = 3100102
| map     = 
| cos     = Erie, Kings, Nassau, Suffolk
| image1  = Balsam Malaysia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Impatiens balfourii
| au-abbr = Hook. f.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1903|Impatiens balfourii |Hook. f.}}
{{../sp-1|1928|Impatiens mathildae |Chiov.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Balfour’s touch-me-not |2022 New York Flora Atlas}}
{{../vn1|Kashmir Balsam         |2023 iNaturalist}}
{{../vn1|Poor Man's Orchid      |2023 iNaturalist}}
| fr-vns  = 
{{../vn1|Impatience de Balfour  |2023 iNaturalist}}
{{../vn1|Impatiente des jardins |2023 iNaturalist}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Potentially invasive
| status2 = 
| imap    =
| ipa-us  = 
| griisus = 
| ny-tier =
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7807|Xu|}}
| usda    = {{../usda|||}}
| gbif    =
| pow-id  = 373976-1
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| fna-i1  = 
| fna-i2  = 
| tro-id  =
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;New York (2015)&quot;&gt;New York (2015)&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Nassau (2020), New York (2017-21)&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Impatiens balfourii Niecierpek Balfoura 2007-08-11 01.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

==Family Polemoniaceae==

The {{../w|Polemoniaceae}} (phlox family).&lt;ref&gt;{{../nyfa-fam|Polemoniaceae}}&lt;br&gt;{{../usda-fam|Polemoniaceae}}&lt;/ref&gt;

===Subamily Polemonioideae===

====Tribe Polemonieae====

=====''Polemonium''=====

{{../txt
|img=Jacob's ladder (24735231220).jpg
|cap=''Polemonium reptans''
|Although the New York Flora Atlas does not include ''Polemonium caeruleum'', Go Botany reports that it has escaped cultivation in New England.}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Polemonieae|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Polemonium|Jacob's ladder|474|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium reptans
| author  = L.
| var0    = reptans
&lt;!-- -- --&gt;
| en1     = Common Jacob's-ladder
| en2     = Creeping Jacob's-ladder
| en3     = Spreading Jacob's-ladder
| en4     = Greek valerian
| en5     = Charity
| fr1     = Polémoine rampante
| fr2     = Valériane grecque
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = FAC
| nwi2    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| habit2  = Subshrub
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6894|4|}}
| usda    = {{../usda|PORER|NN|}}
| vascan  = 8087               &lt;!-- Creeping Jacob's-ladder --&gt;
| gobot   = polemonium/reptans &lt;!-- Spreading Jacob's-ladder --&gt;
| its-id  = 529739             &lt;!-- Greek valerian --&gt;
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20reptans
| map     = Polemonium reptans var reptans nymap.svg
| image1  = Jacob's ladder (25910631796).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium vanbruntiae
| author  = Britton 
&lt;!-- -- --&gt;
| en1     = Vanbrunt's Jacob's ladder
| en2     = Vanbrunt's polemonium
| en3     = Bog Jacob's ladder
| fr1     = Polémoine de Van-Brunt
&lt;!-- -- --&gt;
| status  = Native
| status1 = Rare
| c-rank  = 7
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2418|3|}}
| usda    = {{../usda|POVA5|NN|}}
| vascan  = 8088
| gobot   = polemonium/vanbruntiae
| its-id  = 504486
| ars-id  = 310258
| fna-id  = 
| tro-id  = 25800477
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20vanbruntiae
| map     = Polemonium vanbruntiae nymap.svg
| image1  = Polemonium vanbruntiae FWS-2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Polemonium|Jacob's ladder|474|X|txt=(unlisted&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Polemonium caeruleum
| author  = L.
&lt;!-- -- --&gt;
| en1     = Blue Jacob's-ladder
| en2     = Charity
| en3     = Greek-valerian
| fr1     = Polémoine bleue
| fr2     = Valériane grecque
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| status1 = Cultivated
| c-rank  = 
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|POCA2||}}
| vascan  = 8080
| gobot   = polemonium/caeruleum
| its-id  = 31005 
| ars-id  = 29176
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Polemonium%20caeruleum
| map     = Excluded nymap.svg
| nyfa-co = &lt;abbr title=&quot; &quot;&gt;Not listed&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot; &quot;&gt;Jefferson County 2017&lt;/abbr&gt;
| image1  = Polemonium caeruleum BotGardBln 20170610 I.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Phlocideae====

=====''Phlox''=====

{{:Flora of New York/txt
|The sectional classification of the genus ''Phlox'' used here (Wherry 1955, Grant 1959)&lt;ref&gt;[http://w3.biosci.utexas.edu/prc/pdfs/Grant02_Lundellia04.pdf Verne Grant (2001). &quot;Nomenclature of the Main Subdivisions of ''Phlox'' (Polemoniaceae).&quot; ''Lundellia'' '''4:25''', 2001.]&lt;/ref&gt; is:
* Sect. '''''Phlox''''' (''alpha-Phlox''). Perennial herbs with soft deciduous leaves and long styles (''P. glaberrima, P. paniculata, P. subulata''...)
* Sect. '''''Divaricatae''''' (''Protophlox''). Perennial or annual herbs with soft deciduous leaves and short styles (''P. divaricata, P. drummondii, P. nivalis, P. pilosa''...)
* Sect. '''''Occidentales''''' (''Microphlox''). Cespitose or cushion-like subshrubs with stiff evergreen leaves and short styles. (''P. caespitosa, P. hoodii, P. sibirica''...)
Of these sections, only the herbaceous North American sections ''Phlox'' and ''Divaricatae'' contain native or naturalized species reported in New York.
}}

======''Phlox'' sect. ''Divaricatae''======

{{../txt
|img=Phlox divaricata 3.jpg
|cap=''Phlox divaricata''
|Sect. ''Divaricatae'' contains perennial or annual herbs with soft deciduous leaves and short styles.
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Phlocideae||Phlox||Divaricatae|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Phlox|Phlox|297|6|sect=Divaricatae|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox divaricata
| author  = L.
| ssp0    = divaricata
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Timber phlox|2021 New York Flora Atlas}}
{{../vn1|Wild blue phlox|2021 New York Flora Atlas}}
{{../vn1|Woodland phlox|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 8
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6891|4|Phlox divaricata L. ssp. divaricata - wild blue phlox - Native. Apparently secure in New York State. - Rich mesic to dry-mesic forests, floodplain forests, and forest edges.}}
| usda    = {{../usda|PHDID3|NN|Endangered in New Jersey}}
| vascan  = 
| gobot   = Phlox/divaricata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Phlox divaricata - Wild Blue Phlox 2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox pilosa
| author  = L.
| ssp0    = pilosa
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox pilosa|L.}}
{{../sp-1|1911|Phlox argillacea|Clute &amp; Ferriss}}
{{../var1|1931|Phlox pilosa|L.|amplexicaulis|(Raf.) Wherry}}
{{../var1|1931|Phlox pilosa|L.|virens|(Michx.) Wherry}}
&lt;!-- -- --&gt;
| en1     = Downy phlox
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| status2 = Impersistent
| c-rank  = 10
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = SH
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6892|1z|Phlox pilosa L. ssp. pilosa - downy phlox - Historically known from New York State, but not seen in the past 15 years.}}
| usda    = {{../usda|PHPIP2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 30975
| ars-id  = 432155
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = Phlox+pilosa
| bna-id  = Phlox%20pilosa
| map     =  nymap.svg
| cos     = Niagara (1838)
| image1  = Phlox pilosa2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

======''Phlox'' sect. ''Phlox''======

{{../txt
|img=2021-04-17 12 22 06 Creeping Phlox along Old Dairy Road in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg
|cap=''Phlox subulata''
|Section '''''Phlox''''' contains perennial herbs with soft deciduous leaves and long styles, including native '''moss phlox''' (''Phlox subulata''), endangered '''wild sweet William''' (''Phlox maculata''), as well as common introduced '''garden phlox''' (''Phlox paniculata'').
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Phlocideae||Phlox||Phlox|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Phlox|Phlox|297|6|sect=Phlox|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox subulata
| author  = L.
| ssp0    = subulata
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox subulata   |L.}}
{{../sp-1|1823|Phlox aristata   |Lodd.}}
{{../sp-1|1891|Armeria subulata |(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Moss phlox|2021 New York Flora Atlas - 2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening }}
{{../vn1|Creeping phlox|2021 ARS-GRIN: Locklear, J. H. 2011. Phlox: a natural history and gardener's guide Timber Press, Portland, Oregon. 248-253. Note: recognizes subsp. setacea (Linnaeus) Locklear and brittonii (Small) Wherry as well }}
{{../vn1|Moss pink|2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Ground pink|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Mountain phlox|2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening }}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S4
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6893|4|}}
| usda    = {{../usda|PHSUS2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 314438
| fna-id  = 
| tro-id  = 25800441
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Mauve Flowers 2 (4540691499).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox maculata
| author  = L.
| ssp0    = maculata
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox maculata|L.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Wild sweet-william|2021 New York Flora Atlas - 2021 ARS-GRIN: Huxley, A., ed. 1992. The new Royal Horticultural Society dictionary of gardening. - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Wild sweetwilliam|2021 USDA NRCS National Plant Data Team}}
{{../vn1|Meadow phlox|2021 ARS-GRIN: Gleason, H. A. &amp; A. Cronquist. 1991. Manual of vascular plants of northeastern United States and adjacent Canada, ed. 2. - 2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Spotted phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967-. The vascular flora of Ohio.}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 7
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S2, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2416|1|}}
| usda    = {{../usda|PHMAM|NX|}}
| gbif    =
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 524492
| ars-id  = 405814
| fna-id  =
| tro-id  = 25800411
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = Phlox+maculata
| bna-id  = Phlox%20maculata
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = Phlox maculata ssp maculata NY-dist-map.png
| image1  = Phlox maculata NRCS-1 (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox paniculata
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Phlox paniculata|L.}}
{{../sp-1|1813|Phlox acuminata|Pursh}}
{{../sp-1|1813|Phlox decussata|Lyon ex Pursh }}
&lt;!-- -- --&gt;
| en1     = Fall phlox
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = 
| c-rank  = 
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Sun
| chro-no = 
| ny-rank = SNA, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2414|X|}}
| usda    = {{../usda|PHPA9|NX|}}
| gobot   = Phlox/paniculata
| its-id  =
| ars-id  = 401812
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = PHPA9
| nse-id  = Phlox+paniculata
| bna-id  = Phlox%20paniculata
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  =  
| map     =
| image1  = Phlox paniculata Tatjana20140704 022.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Phlox stolonifera
| author  = Sims
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1892|Phlox stolonifera|Sims}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Creeping phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967. The vascular flora of Ohio. - 2021 New York Flora Atlas.}}
{{../vn1|Cherokee phlox|2021 ARS-GRIN: Ohio Flora Committee (E. L. Braun, T. S. Cooperrider, T. R. Fisher, J. J. Furlow). 1967. The vascular flora of Ohio.}}
| en1     = Creeping phlox
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Not naturalized
| c-rank  = X
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = SNA, G4-5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2417|X|}}
| usda    = {{../usda|PHST3||}}
| vascan  = 
| gobot   = Phlox/stolonifera
| its-id  = 
| ars-id  = 405816
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = Phlox+stolonifera
| bna-id  = Phlox%20stolonifera
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| cos     = Fulton, Rensselaer (1947)
| image1  = Creeping Phlox Phlox stolonifera Flowers 3008px (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====Tribe Gilieae====

=====''Collomia''=====

{{:Flora of New York/txt
|img=Collomia linearis - Flickr - aspidoscelis (1).jpg
|cap=''Collomia linearis''
|''Collomia'' species, commonly known as '''trumpets''' or '''mountain trumpets''', are native to western North America. Of these the only species reported in the northeast is the '''tiny trumpet''' (''Collomia linearis'').
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Gilieae||Collomia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Collomia|Trumpet|86|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Collomia linearis
| author  = (Cav.) Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1800|Phlox linearis|Cav.}}
{{../sp-1|1818|Collomia linearis|(Cav.) Nutt.}}
{{../sp-1|1882|Gilia linearis (Nutt.)|A. Gray}}
{{../sp-1|1891|Navarretia linearis|(Nutt.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Tiny trumpet|2021 New York Flora Atlas - 2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
{{../vn1|Narrow-leaved collomia|2021 Tropicos: Canada Weed Committee. 1969. Comm. bot. nam. weeds Canada 1–67.}}
{{../vn1|Narrow leaved collomia|2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
{{../vn1|Narrow-leaved mountain-trumpet|2021 Native Plant Trust, Go Botany}}
{{../vn1|Slenderleaf collomia|2021 Calflora, https://www.calflora.org/app/taxon?crn=2303}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = western N. America
| status1 = N. America native
| status2 = Impersistent
| status3 = Not naturalized
| c-rank  = 
| nwi1    = FACU
| nwi2    = UPL
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| ny-rank = SNA, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2413|Xm|}}
| usda    = {{../usda|COLI2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 316631
| fna-id  = 
| tro-id  = 25800001
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Collomia%20linearis
| map     = 
| cos     = Cattaraugus (1899)
| image1  = Collomia linearis (1).jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Gilia''=====

{{:Flora of New York/txt
|img=Gilia achilleifolia-2.jpg
|cap=''Gilia achilleifolia''
|About 35 ''Gilia'' species are native to western North America.  Of these, only a few have been reported in the northeast. The New York Flora Atlas shows only ''Gilia achilleifolia'' ('''California gilia'''), collected in Brooklyn in 1880. 

Although excluded by the New York Flora Atlas, &quot;reaserch-grade&quot; iNaturalist observations of ''Gilia capitata'' ('''bluehead gilia''') were made in 2020 in Nassau County&lt;ref&gt;[https://www.inaturalist.org/observations/49079716 ''Gilia capitata'' (Bluehead Gilia) iNaturalist observation, West Hempstead, NY, Jun 9, 2020]&lt;/ref&gt; and in 2021 in Ulster County.&lt;ref&gt;[https://www.inaturalist.org/observations/82928314 ''Gilia capitata'' (Bluehead Gilia) iNaturalist observation, Kingson, NY, Jun 13, 2021]&lt;/ref&gt;
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Gilieae||Gilia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gilia|Gilia|1025|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gilia achilleifolia
| author  = Benth.
| ssp     = achilleifolia
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1833|Gilia achilleifolia|Benth.}}
&lt;!-- -- --&gt;
| en1     = California gilia
| en2     = Blue gilia
&lt;!-- -- --&gt;
| status  = Introduced
| from    = California
| status1 = N. America native
| status2 = Impersistent
| c-rank  = 
| nwi1    = 
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| ny-rank = SNA
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7195|Xm|}}
| usda    = {{../usda|GIACA|X0|}}
| vascan  = 
| gobot   = Gilia/achilleifolia
| its-id  = 31096
| ars-id  = 
| fna-id  = 
| tro-id  = 25800683
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Gilia%20achilleifolia
| map     =  nymap.svg
| cos     = Kings (1880)
| image1  = Gilia achilleifolia 042.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gilia|Gilia|1025|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gilia capitata
| au-abbr = Sims
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1826|Gilia capitata|Sims}}
{{../sp-1|1891|Navarretia capitata|(Sims) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Bluehead gilia|2021 iNaturalist}}
{{../vn1|Gillyflower|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
{{../vn1|Field gilia|2021 Tropicos: Brako, L., A.Y. Rossman &amp; D.F. Farr. 1995. Sci. Comm. Names 1–294.}}
| fr-vns  = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = N. America native
| status2 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|799|EXCLUDED}}
| usda    = {{../usda|GICA5|NX|}}
| gbif    =
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 25800126
| nwg-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Gilia%20capitata
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     = Excluded nymap.svg
| cos     = Nassau (2020),&lt;br&gt;Ulster (2021)
| image1  = Gilia capitata (468859352).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Loeselieae====

======''Ipomopsis''======

{{:Flora of New York/txt
|img=Ipomopsis rubra - Standing Cypress Gilia rubra (8470988667).jpg
|cap=''Ipomopsis rubra''
|
}}

{{../table|order=Ericales|Polemoniaceae|Polemonioideae|Loeselieae||Ipomopsis|
}}
{{../genus|Ipomopsis|Ipomopsis|479|X|txt=(excluded&amp;nbsp;taxa)|
}}
{{../taxon
| species = Ipomopsis rubra
| author  = (L.) Wherry
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Polemonium rubrum|L.}}
{{../sp-1|1891|Navarretia rubra|(L.) Kuntze}}
{{../sp-1|1895|Gilia rubra|(L.) Heller}}
{{../sp-1|1936|Ipomopsis rubra|(L.) Wherry}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Red standing-cypress|}}
{{../vn1|Standing-cypress|}}
&lt;!-- -- --&gt;
| status  = N. America native
| of      = southern U.S.
| status1 = Excluded
| nyfa    = {{../nyfa|X|846|}}
| usda    = {{../usda|IPRU2|NX|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 316899
| fna-id  = 
| tro-id  = 25800231
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = Excluded nymap.svg
| cos     = excluded
| image1  = Ipomopsis rubra.jpg
}}
{{../end table|Polemoniaceae|Polemonioideae|Loeselieae|
}}

==Family Ebenaceae==

{{../txt|The '''{{../w|Ebenaceae}}''' (ebony family) is primarily a tropical and warmer-temperate family of trees and shrubs. Its only species reported outside of cultivation in New York is the American persimmon tree, which consists mainly of escapes from cultivation. It is believed that the only native persimmon trees in the state are on Staten Island (Richmond County) and Long Island (Nassau and Suffolk Counties).&lt;ref&gt;{{../nyfa-fam|Ebenaceae}}&lt;/ref&gt;&lt;ref&gt;{{../usda-fam|Ebenaceae}}&lt;/ref&gt;}}

===Subfamily Ebenoideae===

====''Diospyros''====

{{../txt
|img=Persimmon American3 Asit.jpg
|cap=American Persimmons (''Diospyros virginiana'') in Tampa, Florida
|''Diospyros'' is primarily a tropical genus of trees, known generally as '''ebony''', but ''Diospyros virginiana'' ('''persimmon''') is native to the central and eastern United States, including New York.

In New York State, the only known native persimmon populations are in New York City and Long Island. Most of the naturalized trees in the state are assumed to be garden escapes. &lt;ref name=nynhp&gt;{{../nynhp-ref|9001|Diospyros virginiana|Threatened|S2|G5}}&lt;/ref&gt;

The genus name ''Diospyros'' was derived from the Greek ''dios'' (divine) and ''pyros'' (wheat or grain) in reference to the tree's &quot;divine fruit.&quot;

Persimmons are dioecious, requiring the presence of both male and female trees to produce fruit.
}}

{{../table|order=Ericales|Ebenaceae|Ebenoideae|||Diospyros}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Diospyros|Persimmon|673|1}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Diospyros virginiana
| au-abbr = L.
| au-full = Carl von Linnaeus (1707-1778).
| au-pub  = Species Plantarum 2: 1057–1058. 1 May 1753.
&lt;!-- -- --&gt;
| syns    = &lt;poem&gt;
1753.'' '''Diospyros virginiana'''   ''L.              in Sp. Pl.:1057
1817.''    Diospyros caroliniana  ''Muhl. ex Raf.   in Fl. Ludov.:139
1834.''    Diospyros persimon     ''Wikstr.         in Jahresber. Königl. Schwed. Akad. Wiss. Fortschr. Bot.
1838.''    Persimon virginiana    ''(L.) Raf.       in Sylva Tellur.:164 
1840.''    Diospyros angustifolia ''Audib. ex Spach in Hist. Nat. Vég. 9:405
1921.''    Diospyros mosieri      ''Small           in J. New York Bot. Gard. 22:33
1921.''    Diospyros virginiana   ''var. ''mosieri ''(Small) Sarg. in J. Arnold Arbor. 2(3): 170.
&lt;/poem&gt;
&lt;!-- -- --&gt;
| en1     = Persimmon
| en2     = Common persimmon
| en3     = American persimmon
| fr1     = Plaqueminier d'Amérique
&lt;!-- ========= --&gt;
| status  = Native
| status1 = Threatened
| nynhp   = '''S2'''&lt;ref&gt;[https://guides.nynhp.org/persimmon/ New York Natural Heritage Program. 2024. Online Conservation Guide for Diospyros virginiana.]&lt;/ref&gt;
| c-rank  = 8
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Tree
| light   = Heliophily: 6 &lt;br&gt; Part shade 
| chro-no = 2n = 60, 90
| ny-rank = S2, G5
| ns-rank = 
&lt;!-- ========= --&gt;
| nyfa    = {{../nyfa|1287|8 counties|Bronx, Nassau, New York, Richmond, Rockland, Suffolk, Westchester}}
| inatc   = {{../inat|83435-Diospyros-virginiana|13 counties|13 counties: Bronx, Dutchess, Kings, Monroe, Nassau, New York, Putnam, Queens, Richmond, Rockland, Suffolk, Ulster, Westchester}}
| gbifc   = {{../gbif||present in NY|present in New York State}}
| usda    = {{../usda|DIVI5|N0|NY is at the northern edge of this tree's US range}}
&lt;!-- ========= --&gt;
| col-id  = 
| pow-id  = 
| wfo-id  = 
| fsus    = 671
| vascan  = 
| gobot   = Diospyros/virginiana
| its-id  = 
| ars-id  = 14329
| fna-i1  = 242416451 | fna-i2 = Diospyros_virginiana
| tro-id  = 11500206
| lbj-id  = DIVI5
| ipn-id  = 
| nse-id  = Diospyros+virginiana
| bna-id  = diospyros%20virginiana
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| red-id  =
| mbg-id  = h740
| map     = nymap.svg
| image1  = Diospyros virginia (Ebenaceae) (leaves).JPG|width1=96
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Primulaceae==

{{../txt|The '''{{../w|Primulaceae}}''' {{../au|Batsch ex Borkh.}} (primrose family).&lt;ref&gt;{{../nyfa-fam|Primulaceae}}&lt;br&gt;{{../usda-fam|Primulaceae}}&lt;/ref&gt;  Note that this family does not include evening primroses (''Oenothera''), which are in the Onagraceae (evening primrose or willowherb family) in the order Myrtales.

The classification used here for Primulaceae is based primariliy on [http://www.mobot.org/MOBOT/research/APweb/orders/ericalesweb.htm#Ericales P. F. Stevens (2001-2015). ''Angiosperm Phylogeny Website''. Version 14, April 2015.]
}}

===Subfamily Theophrastoideae===

The Theophrastoideae (brookweeds &amp;&amp;) ...

====Tribe Samoleae====

The {{../w|Samoleae}} (brookweeds) ...

=====''Samolus''=====

{{../txt
|img=SamolusValerandiFlowers.jpg
|cap=''Samolus valerandi'' flowers
|The native ''Samolus valerandi'' ('''Water pimpernel''') is the only species of the genus ''Samolus'' reported in the wild in New York.
}}

{{../table|order=Ericales|Primulaceae|Theophrastoideae|Samoleae||Samolus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Samolus|Brookweed|816|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Samolus valerandi
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Samolus valerandi|L.}}
{{../sp-1|1818-1|S. parviflorus|Raf.}}
{{../sp-1|1818-2|S. floribundus|Kunth}}
{{../sp-1|1825|S. americanus|Spreng.}}
{{../ssp1|1971|S. valerandi|L.|parviflorus|Hultén}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Water pimpernel|}}
{{../vn1|Brookweed|}}
{{../vn1|Seaside brookweed|Go Botany}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|2539|4|as Samolus valerandi}}
| usda    = {{../usda|SAVAP|NN|Samolus valerandi ssp parviflorus}}
| vascan  = 21425       &lt;!-- Samolus parviflorus --&gt;
| gobot   = samolus/valerandi
| its-id  = 504999     &lt;!-- Samolus valerandi --&gt;
| ars-id  = 459207
| fna-id  = 242417215  &lt;!-- Samolus parviflorus --&gt;
| tro-id  = 26400042
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Samolus%20parviflorus &lt;!-- Samolus parviflorus --&gt;
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =  nymap.svg
| image1  = Samolus valerandi carex pre-a-pion-longpre-les-corps-saints 80 03062008 2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Primuloideae===

====''Primula''====

=====''Primula'' subg. ''Aleuritia''=====

{{../txt|[[File:Primula mistassinica.jpg|thumb|right|Lake Mistassini primrose, photographed in Wisconsin]] The only native species of ''Primula'' subg. ''Aleuritia'' is '''Lake Mistassini primrose''' or '''bird's eye primrose'''. {{../w|Lake Mistassini}} is the largest natural lake in Quebec.  Mistassini primrose is rare this far south. It is considered threatened in New York, where it is found only on cool wet cliff faces.&lt;ref&gt;{{../nynhp-ref|9257|Primula mistassinica|Threatened|S2|G5}}&lt;/ref&gt; 

The only non-native member of subg. ''Aleuritia'' reported to have naturalized in New York is '''Japanese primrose''', which has been found in Onondaga and Otsego counties.}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Aleuritia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|subg=Aleuritia|sect=Aleuritia|Primrose|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula mistassinica
| author  = Michx.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Primula mistassinica|Michx.}}
{{../var1|1888|Primula farinosa|L.|mistassinica|(Michx.) Pax}}
{{../var1|1894|Primula sibirica|Wulfen|mistassinica|(Michx.) Kurtz}}
{{../ssp1|1905|Primula farinosa|L.|mistassinica|(Michx.) Pax}}
{{../sp-1|1928|Primula intercedens|Fernald}}
{{../var1|1928|Primula mistassinica|Michx.|intercedens|Fernald}}
{{../var1|1966|Primula mistassinica|Michx.|intercedens|B.Boivin}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Lake Mistassini primrose|NYFA-1, Flora Novae Angliae, VASCAN-S}}
{{../vn1|Bird's-eye primrose|NYFA-2, NYNHP, VASCAN-S}}
{{../vn1|Mistassini primrose|VASCAN-A}}
{{../vn1|Dwarf Canadian primrose|VASCAN-S}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Primevère du lac Mistassini|VASCAN-A}}
{{../vn1|Primevère de Mistassini|VASCAN-S}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| nynhp   = 2&lt;ref&gt;{{../nynhp-ref|9257|Primula mistassinica|Threatened|S2|G5}}&lt;/ref&gt;
| c-rank  = 10
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2540|2|Cool calcareous perennially seepy north facing cliffs in relatively small isolated disjunct populations. Often found with Pinguicula vulgaris and Saxifraga aizoides.}}
| usda    = {{../usda|PRMI|NN|}}
| vascan  = 8375
| gobot   = primula/mistassinica
| its-id  = 24028
| ars-id  = 
| fna-id  = 250092230
| tro-id  = 26400185
| ipn-id  = 
| lbj-id  = PRMI
| nse-id  = 
| bna-id  = Primula%20mistassinica
| map     = 
| image1  = Primula mistassinica WFNY-159B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|subg=Aleuritia|sect=Proliferae|Primrose|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula japonica
| author  = A.Gray
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1858|Primula japonica|A.Gray}}
{{../sp-1|1980|Aleuritia japonica|(A.Gray) Soják}}
&lt;!-- -- --&gt;
| en1     = Japanese primrose
| fr1     = Primevère du Japon
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Japan
| status1 = Highly invasive
| status2 = Naturalized
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2527|X|Onondaga only}}
| usda    = {{../usda|PRJA2|X0|NY only}}
| gbif    = 5414324
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 29640
| fna-id  = 
| tro-id  = 26400450
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Onondaga, Otsego (2010)&quot;&gt;2 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Madison (2021), Nassau (2019), Oneida (2020), Tompkins (2017), Ulster (2019), Westchester (2020)&quot;&gt;6 counties&lt;/abbr&gt;
| image1  = Primula japonica 1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Primula'' subg. ''Primula''=====

{{../txt
|img=Cowslip (Primula veris) in Great Ashby District Park, Stevenage.jpg
|cap=''Primula veris'' - cowslip
|
}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Primula|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Primrose|subg=Primula|sect=Primula|255|4|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula veris
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Primula veris|L.}}
{{../sp-1|1765|Primula officinalis|(L.) Hill}}
&lt;!-- -- --&gt;
| en1     = Cowslip
| en2     = Cowslip primrose
| ---
| fr1     = Primevère officinale
| status  = Introduced
| from    = Europe
| from1   = temperate w. Asia
| status1 = Naturalized
| nyfa    = {{../nyfa|2543|X|}}
| usda    = {{../usda|PRVE2|XW|}}
| vascan  = 8378
| gobot   = 
| its-id  = 
| ars-id  = 400303
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Primula veris ENBLA03.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Primrose|subg=Primula|sect=Primula|255|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula × polyantha
| author  = Mill.
&lt;!-- -- --&gt;
| hp1     = Primula veris
| hp2     = Primula vulgaris
&lt;!-- -- --&gt;
| en1     = Elatior hybrid primrose
| en2     = False oxlip
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Excluded
| status2 = Only cultivated
| nyfa    = {{../nyfa|X|804|EXCLUDED}}
| usda    = {{../usda|PRPO2|XX|}}
| vascan  = 8379
| gobot   = 
| its-id  = 
| ars-id  = 29671 &lt;!-- as Primula polyantha Mill.  --&gt;
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     = Excluded nymap.svg
| image1  = Primula denticulata.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Primula'' subg. ''Auriculastrum''=====

{{../txt
|img=Dodecatheon meadia 01.JPG
|cap=''Primula meadia'' &lt;small&gt;(L.) A.R.Mast &amp; Reveal&lt;/small&gt;&lt;br&gt;sooting star
|Based on molecular evidence, A. R. Mast &amp; Reveal, in 2006, transferred the 17 species of genus ''Dodecatheon'' (the '''sooting stars''') to a new sect. ''Dodecatheon'' in ''Primula'' subg. ''Auriculastrum''.&lt;ref&gt;[http://www.botany.wisc.edu/courses/botany_940/11JClub/MastReveal2007.pdf Austin R. Mast and James L. Reveal (2007). &quot;Transfer of ''Dodecatheon'' to ''Primula'' (Primulaceae).&quot; ''Brittonia'', The New York Botanical Garden. 59(1):79-82. 2007.]&lt;/ref&gt; The transfer included the only (and impersistent) New York native, ''Primula meadia'' ('''Eastern shooting star'''). The shooting stars are still listed in genus ''Dodecatheon'' by many sources.

The New York Natural Heritage Program Rare Plant Status List - Active Inventory List listed
*in 2004: ''Dodecatheon meadia'' ssp. ''meadia'' (Shooting-star) STEU-X G5T5 SX U&lt;ref&gt;[https://www.dot.ny.gov/divisions/engineering/environmental-analysis/manuals-and-guidance/epm/repository/4-1-h.pdf New York Natural Heritage Program 2004 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2022: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/5/rare-plant-status-lists-2022.pdf New York Natural Heritage Program 2022 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2023: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/5/rare-plant-status-lists-2023.pdf New York Natural Heritage Program 2023 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
*in 2025: ''Primula meadia'' (Eastern Shooting Star) STEU-X G5 SX U&lt;ref&gt;[https://www.nynhp.org/documents/508/rare-plant-status-list-2025.pdf New York Natural Heritage Program 2025 Rare Plant Status List - Active Inventory List.]&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Primuloideae|||Primula|Auriculastrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Primula|Shooting-star|subg=Auriculastrum|sect=Dodecatheon|255|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Primula meadia
| au-abbr = (L.) A.R.Mast &amp; Reveal
| au-full = 
| au-pub  = Brittonia 59: 81 (2007)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Dodecatheon meadia        |L.}}
{{../sp-1|1768|Meadia dodecatheon        |Mill.}}
{{../sp-1|1818|Dodecatheon angustifolium |Raf. }}
{{../sp-1|1903|Dodecatheon hugeri        |Small}}
{{../sp-1|1903|Dodecatheon brachycarpum  |Small}}
{{../sp-1|2007|'''Primula meadia'''      |A.R.Mast &amp; Reveal  In: Brittonia 59(1):81}}
&lt;!-- -- --&gt;
| en1     = Eastern shooting star
| en2     = Shooting star
| en3     = Pride of Ohio
| en4     = American cowslip
| en5     = Mead's shootingstar
| fr1     = Gyroselle de Virginie
&lt;!-- -- --&gt;
| status  = Native
| status1 = Extirpated
| c-rank  = 10
| nwi1    = FACU
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Heliophily: 5
| chro-no = 
| ny-rank = SX, G5
&lt;!-- ========= --&gt;
| nyfa    = {{../nyfa|2544|Steuben |Steuben (no date)}}
| inatc   = {{../inat|549981-Primula-meadia|7 counties|7 counties}}
| gbifc   = {{../gbif|5640540|present in NY|present in New York State}}
| usda    = {{../usda|DOMEM3|NN|Steuben county only}} 
&lt;!-- ========= --&gt;
| gbif    =
| pow-id  = 60447606-2
| wfo-id  = 
| fsus    = 676
| vascan  = 8359      | Primula meadia (L.) A.R.Mast &amp; Reveal (2007)
| gobot   = primula/meadia
| its-id  = 836161    | Primula meadia (L.) A.R.Mast &amp; Reveal
| ars-id  = 459448    | Primula meadia (L.) A.R.Mast &amp; Reveal
| fna-id  = 220004300 | Dodecatheon meadia L.
| tro-id  = 50308078  | Primula meadia (L.) A.R.Mast &amp; Reveal
| ipn-id  = 
| nse-id  = Dodecatheon+meadia
| bna-id  = Dodecatheon%20meadia
| map     = nymap.svg
| image1  = Dodecatheon meadia habit.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====''Hottonia''====

{{../table|order=Ericales|Primulaceae|Primuloideae|||Hottonia|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Hottonia|Hottonia|494|1|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Hottonia inflata
| author  = Elliott 1817
| ---
| en1     = American featherfoil
| en2     = Featherfoil
| en3     = Hottonie enflée
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|2541|2|}}
| usda    = {{../usda|HOIN|NN|}}
| vascan  = 
| gobot   = hottonia/inflata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Hottonia inflata NRCS-2.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

====''Androsace''====

{{../table|order=Ericales|Primulaceae|Primuloideae|||Androsace|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Androsace|Rock jasmine|218|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Androsace maxima
| author  = L.
| ---
| en1     = Androsace
| en2     = Greater rock-jasmine
| en3     = Pussy-toes
| ---
| status  = Introduced
| status1 = 
| nyfa    = {{../nyfa|6482|X|}}
| usda    = {{../usda|ANMA14|X0|NY is only location shown}}
| image1  = Androsace maxima sl1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Myrsinoideae===

{{:Flora of New York/txt
|img=
|cap=
|The {{../w|Myrsinoideae}} subfamily has previously been grouped into its own family (Myrsinaceae). It contains about 35 genera worldwide, but of those, only '''''Lysimachia''''' is known to be native or naturlized in New York. The genera '''''Anagallis''''' and '''''Trientalis''''' are now treated as part of ''Lysimachia''.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51.&lt;/ref&gt;
}}

====''Lysimachia''====

{{../txt
|The grouping used here for genus ''Lysimachia'' is based on Anderberg (2007).&lt;ref&gt;[http://www.bgbm.org/sites/default/files/documents/wi37-2Anderberg%2Bal.pdf A. A. Anderberg, U. Manns, and M. Källersjö (2007). &quot;Phylogeny and floral evolution of the Lysimachieae (Ericales, Myrsinaceae): evidence from ''ndhF'' sequence data.&quot; ''Willdenowia'' '''37:'''407-421.]&lt;/ref&gt;}}

=====''Lysimachia'' subg. ''Trientalis''=====

{{:Flora of New York/txt
|img=Trientalis borealis 10137.JPG
|cap=''Lysimachia borealis''
|Based on phylogenetic research, members of the genus ''Trientalis'' ('''starflowers''') were transferred to this subgenus in 2009 by U. Manns &amp; A. A. Anderberg.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51. [''Trientalis borealis'' Raf. = ''Lysimachia borealis'' (Raf.) U.Manns &amp; Anderb., comb. nov.].&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Trientalis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Starflower|133|1|subg=Trientalis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia borealis
| au-abbr = (Raf.) U.Manns &amp; Anderb.
| au-full = 
| au-pub  = Willdenowia 39: 51 (2009)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Trientalis europaea   |Michx non L. 1753}}
{{../var1|1805|Trientalis europaea   |L.    |americana |Pers.}}
{{../sp-1|1808|Trientalis borealis   |Raf.}}
{{../sp-1|1813|Trientalis americana  |Pursh}}
{{../var1|1872|Lysimachia trientalis |Klatt |americana  |(Pursh) Klatt}}
{{../var1|1924|Trientalis borealis   |Raf.  |tenuifolia |House}}
{{../sp-1|2009|Lysimachia borealis   |(Raf.) U.Manns &amp; Anderb.}}
&lt;!-- -- --&gt;
| en1     = Northern starflower
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = FAC
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = S5 G5
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2536|49 counties|49 counties}}
| usda    = {{../usda|TRBOB|NN|}}
| gbif    =
| pow-id  = 77100389-1
| wfo-id  = 
| vascan  = 
| gobot   = lysimachia/borealis
| inat    = 204174-Lysimachia-borealis
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Trientalis borealis 1177.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group A&lt;/small&gt;=====

{{../txt
|img=Lysimachia hybrida iNat 140267462 (3x4).jpg
|cap=''Lysimachia hybrida''&lt;br&gt;at Alley Park in Queens&lt;br&gt;photographed by [https://www.inaturalist.org/observations/85269469 Zihao Wang]
|''Lysimachia'' subg. ''Lysimachia'' '''group A''' contains the two New-World sections: &lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51.&lt;/ref&gt;
*Sect. '''''Seleucia''''' (''Lysimachia ciliata'', ''L. quadriflora'' and ''L. hybrida'') 
*Sect. '''''Theopyxis''''' (''Lysimachia andina'', a South American species). 
The New York species of group A are in sect. ''Seleucia''.
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Seleucia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia ciliata
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|'''Lysimachia ciliata'''|L.}}
{{../sp-1|1843|Steironema ciliatum     |(L.) Baudo}}
{{../sp-1|1891|Nummularia ciliata      |(L.) Kuntze}}
&lt;!-- -- --&gt;
| en1     = Fringed loosestrife
| en2     = Fringed yellow loosestrife
| en3     = Ciliate loosestrife
| fr1     = Lysimaque ciliée
| fr2     = Lysimaque fimbriée
| fr3     = Stéironéma cilié
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 4
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6483|5|Fens, swamps, marshes, ditches, and wet thickets.}}
| usda    = {{../usda|LYCI|N|}}
| gbif    =
| pow-id  = 
| wfo-id  = 
| vascan  = 6685
| gobot   = lysimachia/ciliata
| inat    = 
| its-id  = 
| ars-id  = 468325
| tro-id  = 50297982
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Lysimachia ciliata WFNY-162.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia hybrida
| au-abbr = Michx.
| au-full = 
| au-pub  = Fl. Bor.-Amer. 1: 126 (1803)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1788|Lysimachia ciliata      |Walter non L. (nom. illeg)}}
{{../sp-1|1803|'''Lysimachia hybrida'''|Michx.}}
{{../var1|1860|Lysimachia ciliata      |L.              |hybrida  |(Michx.) Chapm.}}
{{../var1|1876|Lysimachia lanceolata   |Walter          |hybrida  |(Michx.) A.Gray}}
{{../sp-1|1877|Steironema lanceolatum  |(Walter) A.Gray (misapplied)}}
{{../var1|1877|Steironema lanceolatum  |(Walter) A.Gray |hybridum |(Michx.) A.Gray}}
{{../sp-1|1901|Steironema laevigatum   |Howell}}
{{../sp-1|1903|Steironema hybridum     |(Michx.) Raf. ex Small}}
{{../sp-1|1913|Steironema validulum    |Greene ex Wooton &amp; Standl.}}
{{../var1|1939|Lysimachia ciliata      |L.              |validula |Kearney &amp; Peebles}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Lowland loosestrife|2023 iNaturalist, 2023 New York Flora Atlas}}
{{../vn1|Lance-leaved loosestrife|}}
{{../vn1|Lowland yellow loosestrife|}}
{{../vn1|Mississippi loosestrife|}}
{{../vn1|Swamp candles|}}
| fr-vns  = 
{{../vn1|Lysimaque hybride |}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S1, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2530|1|}}
| usda    = {{../usda|LYHY|NN|}}
| gbif    =
| pow-id  = 148125-2
| wfo-id  = 
| vascan  = 
| gobot   = Lysimachia/hybrida
| inat    = 165002-Lysimachia-hybrida
| its-id  = 23990
| ars-id  = 
| fna-id  = 242416811 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =
| nyfa-co = &lt;abbr title=&quot;Allegany, Clinton, Kings, Nassau, Oneida, Orange, Queens, Richmond, Rockland, Suffolk, Washington&quot;&gt;11 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Queens and Nassau Counties only&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Lysimachia hybrida iNat 140267385.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia quadriflora
| au-abbr = Sims
| au-full = 
| au-pub  = Bot. Mag. 18: t. 660 (1803)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|'''Lysimachia quadriflora'''|Sims}}
{{../sp-1|1813|Lysimachia longifolia       |Pursh}}
{{../sp-1|1892|Steironema quadriflorum     |(Sims) C.L.Hitchc.}}
{{../sp-1|1913|Nummularia quadriflora      |(Sims) Farw.}}
&lt;!-- -- --&gt;
| en1     = Linear-leaved loosestrife
| en2     = Linear-leaf loosestrife
| en3     = Four-flowered loosestrife
| en4     = Fourflower yellow loosestrife
| en5     = Linear-leaf loosestrife
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = OBL
| nwi2    = FACW
| habit0  = Perennial
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = S1, G5?
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|2529|1|}}
| usda    = {{../usda|LYQU|NN|}}
| gbif    =
| pow-id  = 148140-2
| wfo-id  = 
| vascan  = 
| gobot   = 
| inat    = 165013-Lysimachia-quadriflora
| its-id  = 23996
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =
| nyfa-co = &lt;abbr title=&quot;Erie and Niagara Counties only as of Dec. 2023&quot;&gt;Erie, Niagara&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;No New York State observations as of Dec. 2023&quot;&gt;No observations&lt;/abbr&gt;
| image1  = Lysimachia quadriflora.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group B&lt;/small&gt;=====

{{../txt
|img=Lysimachia nummularia 5026.jpg
|cap=''Lysimachia nummularia''  L. - moneywort or creeping Jenny
|''Lysimachia'' subg. ''Lysimachia'' '''group B''' contains two members of sect. ''Nummularia'', which are introduced species that are listed as moderately invasive in New York: ''Lysimachia nummularia'' ('''moneywort''' or '''creeping Jenny'''), and ''Lysimachia punctata'' ('''spotted loosestrife'''). 
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Nummularia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia nummularia
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Lysimachia nummularia|L.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Moneywort|}}
{{../vn1|Creeping Jenny|}}
{{../vn1|Creeping loosestrife|}}
{{../vn1|Creeping yellow loosestrife|}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Lysimaque nummulaire|Darbyshire et al., 2000}}
{{../vn1|Herbe-aux-écus|Centre du réseau suisse de floristique, 2010}}
{{../vn1|Monnayère|Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Eurasia
| status1 = Moderately invasive
| habit0  = Perennial
| habit1  = Herb-forb
| nyis    = 64%&lt;ref&gt;{{../nyis-ref|Lysimachia nummularia|Moderate|64}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2535|X|Lysimachia nummularia L. - Thickets, successional forests, low forests, and swamps generally in wet to wet-mesic soils in shaded to partly shaded habitats. It does particularly well in compacted and high pH soils.}}
| usda    = {{../usda|LYNU|XX|}}
| vascan  = 6689
| gobot   = Lysimachia/nummularia
| its-id  = 
| ars-id  = 316741
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Lysimachia nummularia LJM040629-5447866.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia punctata
| author  = L.
| sn0     = {{../sn|1753|Lysimachia punctata|L.}}
| sn1     = {{../sn||L. punctata|var=verticillata|Klatt}}
| ---
| en1     = Spotted loosestrife
| en2     = Large yellow loosestrife
| ---
| status  = Introduced
| from    = temperate Eurasia
| status1 = Moderately invasive
| nyis    = 57%&lt;ref&gt;{{../nyis-ref|Lysimachia punctata|Moderate|57}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2537|X|}}
| usda    = {{../usda|LYPU2|XX|}}
| vascan  = 
| gobot   = Lysimachia/punctata
| its-id  = 23995 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| image1  = Lysimachia punctata-2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group C&lt;/small&gt;=====

{{../txt
|img=Lysimachia arvensis - EMPALOUS - IB-512 (Morró).jpg
|cap=Blue form of ''Lysimachia arvensis''
|Based on phylogenetic research, ''Anagallis arvensis'' ('''scarlet pimpernel''') was transferred to this subgenus in 2009 by U. Manns &amp; A. A. Anderberg.&lt;ref&gt;[http://www.jstor.org/stable/20699148 U. Manns &amp; A. A. Anderberg (2009). &quot;New combinations and names in ''Lysimachia'' (Myrsinaceae) for species of ''Anagallis'', ''Pelletiera'' and ''Trientalis''.&quot;] ''Willdenowia'' '''39:'''51. [''Anagallis arvensis'' L. = ''Lysimachia arvensis'' (Raf.) U.Manns &amp; Anderb., comb. nov.].&lt;/ref&gt;
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia |Pimpernel|292|2|subg=Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia arvensis
| author  = (L.) U.Manns &amp; Anderb.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Anagallis arvensis        |L.}}
{{../sp-1|1759|Anagallis caerulea        |L.}}
{{../var1|1764|Anagallis arvensis        |L.| caerulea |(L.) Gouan}}
{{../for1|1927|Anagallis arvensis        |L.| caerulea |(L.) Lüdi}}
{{../sp-1|2009|'''Lysimachia arvensis''' |(L.) U.Manns &amp; Anderb.}}
&lt;!-- -- --&gt;
| en1     = Scarlet pimpernel
| en2     = Poor-man's weatherglass
| en3     = Shepherd's clock
| en4     = Scarlet yellow-loosestrife
| en5     = Common pimpernel
| fr1     = Mouron rouge
| fr2     = Mouron des champs
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| from1   = northern Africa
| status1 = Potentially invasive
| status2 = 
| imap    = 
| ipa-us  = 57368
| gbif-e  = 8184903
| gbif-i  = 
| gbif-n  = 
| ny-tier =
| c-rank  = 
| nwi1    = FACU
| nwi2    = UPL
| habit0  = Annual
| habit1  = Herb-forb
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6481|X|}}
| usda    = {{../usda|ANAR|XX|}}
| gbif    = 8184903
| vascan  = 28273
| gobot   = lysimachia/arvensis
| its-id  = 
| ars-id  = 316552
| fna-id  = 
| tro-id  = 100397700
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Lysimachia%20arvensis
| map     =  nymap.svg
| image1  = 20170525Lysimachia arvensis1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Lysimachia'' &lt;small&gt;group E&lt;/small&gt;=====

{{:Flora of New York/txt
|img=Lysimachia terrestris.jpg
|cap=''Lysimachia terrestris''
|''Lysimachia'' subg. ''Lysimachia'' '''group E'''
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Yellow-loosestrife|17|12|subg=Lysimachia|sect=Lysimachia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia terrestris
| author  = (L.) Britton, Sterns &amp; Poggenb.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Viscum terrestre      |L.}}
{{../sp-1|1788|Lysimachia vulgaris   |Walter (sensu auct)}}
{{../sp-1|1789|Lysimachia stricta    |Aiton}}
{{../sp-1|1789|Lysimachia bulbifera  |Curtis}}
{{../sp-1|1792|Lysimachia racemosa   |Lam.}}
{{../sp-1|1836|Lysimachia michauxii  |F.Dietr.}}
{{../sp-1|1888|'''Lysimachia terrestris''' |(L.) Britton, Sterns &amp; Poggenb.}}
{{../var1|1922|Lysimachia terrestris ||ovata |(E.L.Rand &amp; Redfield) Fernald}}
&lt;!-- -- --&gt;
| en1     = Swamp candles
| en2     = Swamp loosestrife
| en3     = Swamp yellow loosestrife
| en4     = Bog loosestrife
| en5     = Bulblet loosestrife 
| en6     = Earth loosestrife 
| en7     = Lake loosestrife
| fr1     = Lysimaque terrestre
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|2528|5|}}
| usda    = {{../usda|LYTE2|NN|}}
| gbif    =
| pow-id  = 148158-2
| wfo-id  = 
| vascan  = 6693
| gobot   = Lysimachia/terrestris
| inat    = 
| its-id  = 
| ars-id  = 400269
| fna-id  = 250092260
| tro-id  = 48062-Lysimachia-terrestris
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia terrestris 3-eheep (5097935642).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia quadrifolia
| au-abbr = L.
| au-full = 
| au-pub  = Sp. Pl.: 147 (1753)
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|'''Lysimachia quadrifolia''' |L.}}
{{../sp-1|1777|Anagallis flava              |Houtt.}}
{{../sp-1|1788|Lysimachia punctata          |Walter}}
{{../sp-1|1803|Lysimachia hirsuta           |Michx.}}
{{../var1|1894|Lysimachia quadrifolia       |L. |variegata |Peck}}
{{../for1|1924|Lysimachia quadrifolia       |L. |variegata |(Peck) House}}
&lt;!-- -- --&gt;
| en1     = Whorled loosestrife
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|2532|5|Lysimachia quadrifolia - whorled loosestrife - Acidic dry-mesic to mesic hardwood forests and forest edges. Sometimes it grows in more open sites but generally it is a forest herb.}}
| usda    = {{../usda|LYQU2|NN|}}
| vascan  = 6692
| gobot   = Lysimachia/quadrifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia quadrifolia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia thyrsiflora
| author  = L. (1753)
| sn0     = {{../sn||Naumburgia thyrsiflora|(L.) Duby}}
| ---
| en1     = Tufted loosestrife
| en2     = Tufted yellow loosestrife
| en3     = Water loosestrife
| en4     = Swamp loosestrife
| en5     = Lysimaque thyrsiflore
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|2534|4|}}
| usda    = {{../usda|LYTH2|NN|}}
| vascan  = 
| gobot   = Lysimachia/thyrsiflora
| its-id  = 24000
| ars-id  = 457193
| fna-id  = 200017111
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Lysimachia thyrsiflora 2.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia × commixta
| author  = Fernald
| hp1     = Lysimachia terrestris 
| hp2     = Lysimachia thyrsiflora
| ---
| en1     = {{../hybrid-of|Swamp loosestrife|Tufted loosestrife}}
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|2538|2?|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia × producta
| author  = (A. Gray) Fernald (pro sp.)
| hp1     = Lysimachia quadrifolia 
| hp2     = Lysimachia terrestris
| ---
| en1     = {{../hybrid-of|Whorled loosestrife|Swamp loosestrife}}
| ---
| status  = Native
| status1 = Vulnerable
| nyfa    = {{../nyfa|2526|3?|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia vulgaris
| author  = L.
| sn0     = {{../sn|1753|Lysimachia vulgaris|L.}}
| ---
| en1     = Garden loosestrife
| en2     = Garden yellow loosestrife
| ---
| fr1     = Lysimaque commune
| status  = Introduced
| status1 = Highly invasive
| status2 = Prohibited&lt;ref name=prohibit&gt;{{fny-prohibit-ref}}&lt;/ref&gt;
| nyis    = 73%&lt;ref&gt;{{../nyis-ref|Lysimachia vulgaris|Moderate|73|first}}&lt;/ref&gt;
| nyfa    = {{../nyfa|6484|X|}}
| usda    = {{../usda|LYVU|XX|}}
| vascan  = 
| gobot   = lysimachia/vulgaris
| its-id  = 
| ars-id  = 
| fna-id  = 200017121
| tro-id  = 26400022
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = 20150609Lysimachia punctata1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Lysimachia'' subg. ''Palladia + Lysimachiopsis''=====

{{:Flora of New York/txt
|img=Lysimachia clethroides2.jpg
|cap=''Lysimachia clethroides''
|The ''Lysimachia'' subgenera ''Palladia and Lysimachiopsis'' together are thought to form a clade but may be paraphyletic with each other.&lt;ref&gt;[http://www.bgbm.org/sites/default/files/documents/wi37-2Anderberg%2Bal.pdf A. A. Anderberg, U. Manns, and M. Källersjö (2007). &quot;Phylogeny and floral evolution of the Lysimachieae (Ericales, Myrsinaceae): evidence from ''ndhF'' sequence data.&quot; ''Willdenowia'' '''37:'''407-421.]&lt;/ref&gt; The only New York species of this group appears to be the Asian introducion ''Lysimachia clethroides'' ('''goose-neck loosestrife''').
}}

{{../table|order=Ericales|Primulaceae|Myrsinoideae|||Lysimachia|Palladia + Lysimachiopsis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lysimachia|Loosestrife|17|12|subg=Palladia + Lysimachiopsis|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lysimachia clethroides
| au-abbr = Duby
| au-full = Jean Étienne Duby (1798-1885)
| au-pub  = Prodromus Systematis Naturalis Regni Vegetabilis 8:61. 1844.
&lt;!-- -- --&gt;
| syns    = 
{{../spa1|1844|Lysimachia clethroides|Duby}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Goose-neck loosestrife|NYFA}}
{{../vn1|Gooseneck loosestrife|ARS-GRIN}}
{{../vn1|Gooseneck yellow loosestrife|USDA NRCS National Plant Data Team / VASCAN}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Lysimaque faux-clèthre|VASCAN}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Potentially invasive
| nyis    = not assessed&lt;ref&gt;{{../nyis-ref|Lysimachia clethroides|Not Assessable|NA}}&lt;/ref&gt;
| nyfa    = {{../nyfa|2533|Xm|}}
| usda    = {{../usda|LYCL2|XX|}}
| gbif    = 3169331
| vascan  = 6686
| gobot   = 
| inat    = 132224-Lysimachia-clethroides
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Livingston (2009), Nassau (1978), Queens (1988), Rensselaer (1999), Suffolk (1941), Westchester (1993)&quot;&gt;6 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Cattaraugus (2021), Essex (2021), Greene (2021), Kings (2021), Ontario (2021), Tompkins (2021), Saratoga (2021), Schenectady (2021), Westchester (2021)&quot;&gt;9+ counties&lt;/abbr&gt;
| image1  = Lysimachia clethroides 01.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Symplocaceae==

The {{../w|Symplocaceae}} (sweetleaf family).&lt;ref&gt;{{../nyfa-fam|Symplocaceae}}&lt;/ref&gt;

===''Symplocos''===

{{:Flora of New York/txt
|img=Symplocos paniculata 2015-05-29 OB 004.jpg
|cap=''Symplocos paniculata''
|'''Sweetleaf'''.
}}

{{../table|order=Ericales|Symplocaceae||||Symplocos|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Symplocos|Sweetleaf|113|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Symplocos tinctoria
| au-abbr = (L.) L'Hér.
| au-full = Charles Louis L'Héritier de Brutelle (1746-1800).
| au-pub  = Transactions of the Linnean Society of London 1: 176. 1791.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1767|Hopea tinctoria|L.}}
{{../sp-1|1791|Symplocos tinctoria|(L.) L'Hér.}}
{{../sp-1|1879|Protohopea tinctoria|(L.) Miers}}
{{../sp-1|1891|Eugeniodes tinctorium|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Sweetleaf|2022 iNaturalist, New York Flora Atlas}}
{{../vn1|Sweet-leaf|}}
{{../vn1|Common sweetleaf|}}
{{../vn1|Horsesugar|2022 New York Flora Atlas}}
{{../vn1|Horse-sugar|}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|3024|X|}}
| usda    = {{../usda|SYTI|N0|}}
| vascan  = 
| gobot   = 
| inat    = 141855-Symplocos-tinctoria
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 30900076
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Symplocos%20tinctoria
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Suffolk (1991-92)&quot;&gt;Suffolk&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Cultivated in Central Park (New York County)&quot;&gt;Cultivated NYC&lt;/abbr&gt;
| image1  = Symplocos tinctoria AR-01 (3x4).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Symplocos paniculata
| author  = (Thunb.) Miq.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1784|Prunus paniculata|Thunb.}}
{{../sp-1|1867|Symplocos paniculata|(Thunb.) Miq.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Sapphire berry|}}
{{../vn1|Sapphire-berry|2022 iNaturalist}}
{{../vn1|Asiatic sweetleaf|}}
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = Highly invasive
| status2 = 
| imap    =
| ipa-us  = 
| gbif-e  = 7157705
| ny-tier = 2
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = 
| ns-rank = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|3023|X|}}
| usda    = {{../usda|SYPA12|X0|}}
| gbif    = 7157705
| vascan  = 
| gobot   = 
| inat    = 169497-Symplocos-paniculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Symplocos%20paniculata
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Nassau (1977, '87, '92, '95)&quot;&gt;Nassau county&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Dutchess (2021), Nassau  (2019), Queens (2018), Westchester (2020)&quot;&gt;4 counties&lt;/abbr&gt;
| image1  = Symplocos paniculata (9958808546).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Styracaceae==

{{:Flora of New York/txt
|The {{../w|Styracaceae}} (snowbell family) &lt;ref&gt;{{../nyfa-fam|Styracaceae}}&lt;br&gt;{{../usda-fam|Styracaceae}}&lt;/ref&gt; contains snowbells and silverbell trees.
}}

===''Halesia''===

{{:Flora of New York/txt
|img=Halesia carolina 22zz.jpg
|cap=''Halesia carolina''
|
}}

{{../table|order=Ericales|Styracaceae||||Halesia|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Halesia|Silverbell|1030|n|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Halesia carolina
| author  = L.
| sn0     = {{../sn|1759|Halesia carolina|L.}}
| sn1     = {{../sn|1761|Halesia tetraptera|J.Ellis}}
| sn2     = {{../sn|1803|Halesia parviflora|Michx.}}
| sn3     = {{../sn|1914|Halesia carolina|var=monticola|Rehder}}
| sn4     = {{../sn|1921|Halesia monticola|(Rehder) Sarg.}}
| ---
| en1     = Silver bells 
| en2     = Possomwood
| en3     = Carolina silverbell
| ---
| status  = Introduced
| status1 = US South native
| habit0  = Perennial
| habit1  = Tree
| nyfa    = {{../nyfa|6480|X|State exotic or non-native}}
| usda    = {{../usda|HACA3|N0|}}
| vascan  = 
| gobot   = halesia/carolina
| its-id  =
| ars-id  = 18215
| fna-id  = 220005985
| tro-id  = 
| ipn-id  = 
| nse-id  = Halesia+carolina
| bna-id  = Halesia%20carolina
| map     =  nymap.svg
| image1  = Halesia carolina var. monticola Ośnieża karolińska 2017-05-01 02.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

===''Styrax''===

{{../table|order=Ericales|Styracaceae||||Styrax|
}}&lt;!-- ---------------------------------------- --&gt;
{{../genus|Styrax|Snowbell|162|1|
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Styrax americanus
| author  = Lam.
| sn0     = {{../sn|1783|Styrax americanus|Lam.}}
| sn1     = {{../sn|1803|Styrax pulverulentus|Michx.}}
| sn2     = {{../sn|1917|Styrax americanus|var1=pulverulentus|Rehder}}
| ---
| en1     = American snowbell
| en2     = Mock orange
| ---
| fr1     = 
| status  = Introduced
| status1 = US South native
| status2 = Cultivated
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|STAM4|N0|}}
| gbif    =  
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 313878
| fna-id  = 
| tro-id  = 30800057
| ipn-id  = 
| nse-id  = Styrax+americanus
| bna-id  = Styrax%20americanus
| map     =  nymap.svg
| image1  = Styrax americanus.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../taxon
| species = Styrax japonicus
| author  = Sieb. &amp; Zucc.
| ---
| en1     = Japanese snowbell
| en2     = Japanese storax
| ---
| status  = Introduced
| status1 = 
| nyfa    = {{../nyfa|3021|X|State exotic or non-native}}
| usda    = {{../usda|STJA5|X0|}}
| gbif    = 5371678
| vascan  = 
| gobot   = 
| inat    = 
| its-id  = 505974
| fna-id  = 200017748
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| nyfa-co = &lt;abbr title=&quot;Nassau (1977, 2010), Suffolk (1996, 2011)&quot;&gt;2 counties&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Kings, Nassau, New York, Queens, Suffolk, Westchester&quot;&gt;6 counties&lt;/abbr&gt;
| image1  = Styrax japonicus Styrak japoński 2018-08-12 01.jpg
}}&lt;!-- ---------------------------------------- --&gt;
{{../end table}}

==Family Diapensiaceae==

The {{../w|Diapensiaceae}}  (diapensia family).&lt;ref&gt;{{../nyfa-fam|Diapensiaceae}}&lt;br&gt;{{../usda-fam|Diapensiaceae}}&lt;/ref&gt;

===''Diapensia''===

{{../table|order=Ericales|Diapensiaceae||||Diapensia|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Diapensia|Diapensia|676|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Diapensia lapponica
| author  = L.
| ssp     = lapponica
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Lapland diapensia
| en2     = Pincushion plant
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1233|2|}}
| usda    = {{../usda|DILAL||}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Diapensia lapponica plant upernavik.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===''Galax''===
{{../txt
|img=Galax urceolata (cropped).jpg
|cap=''Galax urceolata''&lt;br&gt;beetleweed 
|''Galax'' ('''beetleweed''') is an eastern North America native, but is considered to be an introduction in the Northeast. It is cultivated as a &quot;native&quot; groundcover and may persist as an escapee on Long Island, but is not considered to be truly naturalized in New York State.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Diapensiaceae||||Galax|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Galax|Galax|586|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Galax urceolata
| author  = (Poir.) Brummitt 
&lt;!-- -- --&gt;
| sn0     = {{../sn|auct|Galax aphylla|non=L.}}
&lt;!-- -- --&gt;
| en1     = Beetleweed
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Not naturalized
| status3 = Nassau, Suffolk Counties
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1232|X|State exotic or non-native}}
| usda    = {{../usda|GAUR2|N0|Galax urceolata (Poir.) Brummitt - beetleweed}}
| vascan  = 
| gobot   = galax/urceolata
| its-id  = 502705
| fna-id  = 220005401
| tro-id  = 
| ipn-id  = 
| lbj-id  = GAUR2
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| fws-id  = 
| bug-id  = 
| map     =  nymap.svg
| image1  = Galax - Flickr - pellaea (1).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===''Pyxidanthera''===
{{../txt
|img=Pyxidanthera barbulata.jpg
|cap=''Pyxidanthera barbulata''&lt;br&gt;pyxie moss
|Both species of ''Pyxidanthera'' ('''pixie moss''') are rare natives of the east coast. The smaller ''P. brevifolia'' (little-leaf pixiemoss) is only found in the Carolinas, but the larger ''P. barbulata'' ('''flowering pixiemoss''') can be found as far north as Long Island, where it is considered endangered.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Diapensiaceae||||Pyxidanthera|
}} &lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Pyxidanthera|Pyxie moss|243|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyxidanthera barbulata
| author  = {{../w|Michx.}}
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Common pyxiemoss
| en2     = Flowering moss
| en3     = Flowering pixiemoss
| en4     = Big pyxie
| en5     = Pyxies
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| status2 = Suffolk County only
| c-rank  = 6
| nwi1    = 
| habit0  = Perennial
| habit1  = Herb-subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6479|1|especially vulnerable in New York State}}
| usda    = {{../usda|PYBA|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 23798
| fna-id  = 220011320
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Pyxidanthera barbulata BB-1913.png
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Sarraceniaceae==

{{../txt|The {{../w|Sarraceniaceae}} (pitcher-plant family) are carnivorous plants, consisting of three New-World genera. The largest genus, ''Heliamphora'', is endemic to South America, and the monospecific ''Darlingtonia'' is native to the West Coast of the U.S.  The third genus,  ''Sarracenia'', with about eight species, is native to eastern North America.&lt;ref&gt;{{../nyfa-fam|Sarraceniaceae}}&lt;/ref&gt;}}

===''Sarracenia''===

{{../txt
|img=Purple Pitcher Plant - Sarracenia purpurea (18239408828).jpg
|cap=''Sarracenia purpurea''&lt;br&gt;purple pitcher-plant
|The most widespread and only cold-tolerant species of ''Sarracenia'' is ''{{../w|Sarracenia purpurea}}'' (the '''purple pitcher plant''').  Of its two subspecies, ''S. purpurea'' ssp. ''venosa'' is only found south of New York, but ''S. purpurea'' ssp. ''purpurea'' is fairly widespread in New York peatlands as well as farther north.}}

{{../table|order=Ericales|Sarraceniaceae||||Sarracenia|
}} 
{{../genus|Sarracenia|Pitcher plant|282|1|
}}
{{../taxon
| species = Sarracenia purpurea
| author  = L.
| ssp     = purpurea
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Sarracenia purpurea|L.}}
| sn1     = {{../sn|1827|S. purpurea|var=terrae-novae|var-au=LaPylaie}}
| sn2     = {{../sn|1933|S. purpurea|ssp=gibbosa|ssp-au=(Raf.) Wherry}}
| sn3     = {{../sn|1933|S. purpurea|var=stolonifera|var-au=Macfarl. &amp; Steckbeck}}
| sn4     = {{../sn|1951|S. purpurea|var=ripicola|var-au=B.Boivin}}
&lt;!-- -- --&gt;
| en1     = Purple pitcher-plant
| en2     = Northern pitcher-plant
| en3     = Side-saddle plant
| en4     = Common pitcherplant
| en5     = Huntsman's-cup
| en6     = Huntsman's horn
| en7     = Decumbent pitcher plant
| en8     = Frog's-britches
| fr1     = Sarracénie pourpre
&lt;!-- -- --&gt;
| status  = Native
| status1 = Vulnerable
| c-rank  = 9
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Herb-forb
| light   = Sun
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6370|3-4|Sarracenia purpurea L. - purple pitcherplant - Acidic to alkaline peatlands. Limited acreage or Apparently secure in New York State.}}
| usda    = {{../usda|SAPU4|NN|shows ssp. gibbosa in western &amp; northest US (incl. NY) but not Canada. Shows ssp. purpurea in Southeast US &amp; Canada.}}
| vascan  = 9222
| gobot   = Sarracenia/purpurea
| its-id  = 195652
| ars-id  = 409890
| fna-id  = 250092288
| tro-id  = 
| ipn-id  = 
| lbj-id  = SAPU4
| nse-id  = 
| bna-id  = Sarracenia%20purpurea
| cpc-id  = 
| cab-id  = 
| map     =  nymap.svg
| image1  = PurplepitcherplantMN.jpg
}}
{{../end table|Sarraceniaceae|
}}

==Family Actinidiaceae==
{{../txt|The {{../w|Actinidiaceae}} (Chinese gooseberry family) has only two species that have been found outside of cultivation in New York. Only one of these are considered to be naturalized.&lt;ref&gt;{{../nyfa-fam|Actinidiaceae}}&lt;/ref&gt;}}

===''Actinidia''===
{{../txt
|img=Actinidia arguta 0906.JPG
|cap=''Actinidia arguta''&lt;br&gt;taravine or hardy kiwi
|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Actinidiaceae|Actinidioideae|||Actinidia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Actinidia|Kiwifruit|1072|1|actin3||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Actinidia arguta
| au-abbr = (Siebold &amp; Zucc.) Planch. ex Miq.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1867|Actinidia arguta|(Siebold &amp; Zucc.) Planch. ex Miq.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Hardy kiwi|2018 New York Flora Atlas}}
{{../vn1|Baby kiwi|2018 ARS-GRIN: Porcher, M. H. et al. 2000}}
{{../vn1|Tara vine|2018 USDA NRCS National Plant Data Team}}
{{../vn1|Taravine|2019 iMapInvasive}}
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Highly invasive
| status2 = Naturalized
| imap    = 1383
| ipa-us  = 51426 
| ny-tier = 2
| c-rank  = X
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7242|Xn|}}
| usda    = {{../usda|ACAR10|X0|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 1389
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| nyfa-co = &lt;abbr title=&quot;Westchester (2013)&quot;&gt;Westchester (2013)&lt;/abbr&gt;
| inat-co = &lt;abbr title=&quot;Queens (2010), Westchester (2018-19)&quot;&gt;2 counties&lt;/abbr&gt;
| image1  = Actinidia-arguta-habit.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Actinidia polygama
| au-abbr = (Siebold &amp; Zucc.) Planch. ex Maxim. 
| au-full = 
| au-pub  = Mém. Acad. Imp. Sci. St.-Pétersbourg Divers Savans 9:64. 1859 (Prim. fl. amur.)
&lt;!-- -- --&gt;
| sn0     = {{../sn|1843|Trochostigma polygamum|Siebold &amp; Zucc.}}
| sn1     = {{../sn|1859|Actinidia polygama|(Siebold &amp; Zucc.) Planch. ex Maxim.}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Silver vine
| en2     = Cat powder
| en3     = Matatabi
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| from    = temperate Asia
| status1 = Potentially invasive
| status2 = Not naturalized
| imap    = 1399
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|33|X|Orange (only)}}
| usda    = {{../usda|ACPO9|X0|NY (only)}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 1411
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Actinidia polygama.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Clethraceae==

{{../txt|The {{../w|Clethraceae}} (clethra family)&lt;ref&gt;{{../nyfa-fam|Clethraceae}}&lt;/ref&gt; contains just two species, only one of which (''Clethra'') is found in New York.}}

===''Clethra''===

{{../txt
|img=Zimterle (Clethra alnifolia)@Heilpflanzengarten Celle 20160728 01.jpg
|cap=''Clethra alnifolia''
|''Clethra'' contains two North American species: 
*''Clethra acuminata'', '''mountain pepperbush''', is limited to the Appalachians from Alabama to southern Pennsylvania, and has not been known to naturalize as far north as New York State.
*''Clethra alnifolia'', '''coastal sweet pepperbush''', is found primarily along the East Coast, including parts of New York, particularly in wet acidic environments.

''Clethra alnifolia'' cultivars are available and are usually either more compact than the wild form or have pink flowers.&lt;ref&gt;[http://www.clemson.edu/extension/hgic/plants/landscape/shrubs/hgic1090.html Clemson University Cooperative Extension Service, Home &amp; Garden Information Center, Clemson, South Carolina.]&lt;/ref&gt;}}

{{../table|order=Ericales|Clethraceae||||Clethra|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Clethra|Sweetpepperbush|112|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Clethra alnifolia
| author  = L.
&lt;!-- -- --&gt;
| syns    = {{../snl
|{{../spa1|1753|Clethra alnifolia  |L.}}
|{{../spa1|1786|Clethra tomentosa  |Lam.}}
|{{../spa1|1789|Clethra paniculata |Aiton}}
|{{../var1|1803|Clethra alnifolia  |L. |tomentosa|Michx.}}
|{{../var1|1803|Clethra alnifolia  |L. |glabella|Michx.}}
|{{../var1|1900|Clethra alnifolia  |L. |paniculata|Rehder}}
|{{../var1|1903|Clethra alnifolia  |L. |michauxii|Zabel}}
}}
&lt;!-- -- --&gt;
| en1     = Coastal sweet-pepperbush
| en2     = Coast pepperbush
| en3     = Sweet Pepperbush
| en4     = Summersweet
| en5     = Summersweet clethra
| en6     = Pink spires (cv.)
| en7     = Anne Bidwell (cv.)
| fr1     = Clèthre à feuilles d'aulne
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = Sun - shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|852|5|Clethra alnifolia L. - coastal sweet-pepperbush - Edges of acidic ponds, acidic sphagnum wetlands, and bog edges. Often with other shrubs including Rhododendron viscosum}}
| usda    = {{../usda|CLAL3|NN|}}
| vascan  = 4589
| gobot   = clethra/alnifolia
| its-id  = 23454
| ars-id  = 10901
| fna-id  = 220003027
| tro-id  = 7700002
| ipn-id  = 319146-2
| lbj-id  = CLAL3
| nse-id  = Clethra+alnifolia
| bna-id  = Clethra%20alnifolia
| map     = Clethra alnifolia nymap.svg
| image1  = Clethra alnifolia2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

==Family Ericaceae==

{{../txt|The {{../w|Ericaceae}} (heath family).&lt;ref&gt;{{../nyfa-fam|Ericaceae}}&lt;br&gt;{{../usda-fam|Ericaceae}}&lt;/ref&gt;&lt;ref&gt;The subfamily (and tribal) organization used here is from of [http://www.plantsystematics.org/reveal/pbio/fam/MagnoliidaeOnLine.html James L. Reveal (2012) An outline of a classification scheme for extant flowering plants. ''Phytoneuron'' 2012–37: 1–221.]&lt;/ref&gt;}}

===Subfamily Pyroloideae===

{{../txt|The subfamily '''Pyroloideae''' Kosteltsky. Perennial herbs, rhizomatous.
}}

====Tribe Pyroleae====

{{../txt|The native New York '''Pyroleae''' consist of about nine species of &quot;wintergreen&quot; or &quot;shineleaf&quot; plants. The organization of the Pyroleae used here is based primarily on Liu (2011).&lt;ref&gt;[http://link.springer.com/article/10.1007%2Fs10265-010-0376-8  Zhen-wen Liu, Ze-huan Wang, Jing Zhou, Hua Peng (2011). &quot;Phylogeny of Pyroleae (Ericaceae): implications for character evolution.&quot; ''Journal of Plant Research'' May 2011, Volume 124, Issue 3, pp 325-337.]&lt;/ref&gt;}}

=====''Orthilia''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Orthilia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Orthilia|Orthilia|330|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Orthilia secunda
| author  = (L.) House
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Pyrola secunda|L.}}
| sn1     = {{../sn|1858|Ramischia secunda|(L.) Garcke}}
| sn2     = {{../sn|1914|Ramischia elatior|Rydb.}}
| sn3     = {{../sn|1921|Orthilia secunda|(L.) House}}
&lt;!-- -- --&gt;
| en1     = One-sided wintergreen
| en2     = Sidebells wintergreen
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1338|5|}}
| usda    = {{../usda|ORSE|NN|}}
| vascan  = 
| gobot   = Orthilia/secunda
| its-id  =
| fna-id  = 242334848
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = OrthiliaSecunda.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Pyrola''=====
{{../txt
|img=Pyrola asarifolia, americana, elliptica WFNY-151.jpg
|cap=''Pyrola'' (wintergreen) photos from the 1918 ''Wild Flowers of New York''
|New York ''Pyrola'' species go by the various colloquial names: '''wintergreen''', '''shinleaf''', or sometimes '''shineleaf'''.
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Pyrola|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Pyrola|Wintergreen|30|5|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola americana
| author  = Sweet
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1830|Pyrola americana|Sweet}}
{{../sp-1|1844|Pyrola obovata|Bertol.}}
{{../sp-1|auct|Pyrola rotundifolia|p.p.|non=L.}}
{{../var1|1920|Pyrola rotundifolia|L.|americana|Fernald}}
{{../ssp1|1966|Pyrola asarifolia|Michx.|americana|Křísa}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|American wintergreen|}}
{{../vn1|American shinleaf|}}
{{../vn1|Wild lily-of-the-valley|}}
{{../vn1|Round-leaved shineleaf|}}
{{../vn1|Round-leaved wintergreen|}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Pyrole d'Amérique|}}
{{../vn1|Pyrole à feuilles rondes|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1323|5|Demonstrably secure in New York State.}}
| usda    = {{../usda|PYAM|NN|}}
| gobot   = pyrola/americana
| its-id  = 504691
| ars-id  = 407190 &lt;!-- Pyrola rotundifolia L. var. americana (Sweet) Fernald --&gt;
| fna-id  = 250092297
| tro-id  = 26800163
| bna-id  = Pyrola%20americana
| image1  = Pyrola americana (31887890964) 3x4.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola chlorantha
| author  = Sw.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1810|Pyrola chlorantha|Sw.}}
{{../sp-1|1811|Pyrola virens|Schweigg.}}
{{../sp-1|1867|Pyrola oxypetala|Austin ex A. Gray}}
{{../var1|1920|Pyrola chlorantha|Sw.|convoluta|(W.P.C. Barton) Fernald}}
{{../var1|1920|Pyrola chlorantha|Sw.|paucifolia|Fernald}}
{{../var1|1920|Pyrola chlorantha|Sw.|revoluta|Jenn.}}
{{../var1|1941|Pyrola virens|Schweigg.|convoluta|(W.P.C. Barton) Fernald}}
{{../var1|1941|Pyrola virens|Schweigg.|saximontana|(Fernald) Fernald}}
&lt;!-- -- --&gt;
| en1     = Green-flowered wintergreen
| en2     = Green-flowered shineleaf
| en3     = Greenish-flowered wintergreen
| en4     = Pale-green wintergreen
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1325|5|}}
| usda    = {{../usda|PYCH|NN|}}
| its-id  =
| fna-id  =
| map     = nymap.svg
| image1  = Rohekas uibuleht.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola elliptica
| author  = Nutt.
| ---
| en1     = Shinleaf
| en2     = Waxflower shinleaf
| en3     = Large-leaved shineleaf
| en4     = Elliptic shineleaf
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1334|5|}}
| usda    = {{../usda|PYEL|NN|}}
| its-id  = 23759
| image1  = Pyrola elliptica 4 (5097357221).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola asarifolia
| author  = Michx.
| ssp     = asarifolia
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|yyyy|Pyrola californica|Krísa}}
{{../sp-1|yyyy|Pyrola elata|Nutt.}}
{{../sp-1|yyyy|Pyrola uliginosa|Torr. &amp; A. Gray ex Torr.}}
{{../var1|yyyy|Pyrola uliginosa|Torr. &amp; A. Gray ex Torr.|gracilis|Jennings}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|asarifolia}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|incarnata|(DC.) Fernald}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|ovata|Farw.}}
{{../var1|yyyy|Pyrola asarifolia|Michx.|purpurea|(Bunge)Fernald}}
{{../ssp1|yyyy|Pyrola rotundifolia|L.|asarifolia|(Michx.) Á. Löve &amp; D. Löve}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Pink shinleaf|New York Flora Atlas}}
{{../vn1|Pink wintergreen|FNA Ed. Comm., 2009}}
{{../vn1|Bog wintergreen|FNA Ed. Comm., 2009}}
{{../vn1|Liverleaf wintergreen|USDA NRCS National Plant Data Team}}
{{../vn1|Pink pyrola|Hinds, 2000 via VASCAN}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Pyrole à feuilles d'asaret|Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| en1     = 
| en2     = 
| ---
| status  = Native
| status1 = Threatened
| nyfa    = {{../nyfa|1324|2|}}
| usda    = {{../usda|PYASA|NN|}}
| vascan  = 5546
| gobot   = 
| its-id  = 23760
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Pyrola asarifolia (6187038332).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Pyrola minor
| author  = L.
| ---
| en1     = Snowline wintergreen
| en2     = Lesser wintergreen
| ---
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1336|1|}}
| usda    = {{../usda|||}}
| its-id  =
| fna-id  =
| map     = nymap.svg
| image1  = Pyrola minor 090612a.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Moneses''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Moneses|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Moneses|Single-delight|1056|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Moneses uniflora
| author  = (L.) {{../w|A.Gray}}
| sn0     = {{../sn||Pyrola uniflora|L.}}
| ---
| en1     = One-flowered wintergreen
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1328|4|}}
| usda    = {{../usda|MOUN2|NN|}}
| vascan  = 
| gobot   = Moneses/uniflora
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Moneses uniflora kz1.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Chimaphila''=====

{{../table|order=Ericales|Ericaceae|Pyroloideae|Pyroleae||Chimaphila|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Chimaphila|Prince's-pine|253|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chimaphila umbellata
| author  = (L.) {{../w|Benjamin Smith Barton|W.P.C.Barton}}
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Pyrola umbellata|L.}}
{{../sp-1|1817|Chimaphila umbellata|W.P.C.Barton}}
{{../sp-1|1891|Pseva umbellata|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en1     = Common wintergreen
| en2     = Pipsissewa
| en3     = Noble prince's-pine
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1316|5|Native. Apparently secure in New York State.}}
| usda    = {{../usda|CHUM|NN|}}
| vascan  = 
| gobot   = chimaphila/umbellata
| its-id  = 
| ars-id  = 402538
| fna-id  = 
| tro-id  = 26800037
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Chimaphila umbellata (3817695201).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chimaphila maculata
| author  = (L.) {{../w|Frederick Traugott Pursh|Pursh}}
| ---
| en1     = Spotted wintergreen
| en2     = Striped prince's pine
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1347|4|Apparently secure in New York State.}}
| usda    = {{../usda|CHMA3|NN|Native, eastern US &amp; Canada}}
| vascan  = 
| gobot   = Chimaphila/maculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Striped Wintergreen - Chimaphila maculata, Leesylvania State Park, Woodbridge, Virginia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Monotropoideae===

{{../txt|The subfamily '''Monotropoideae''' consists exclusively of taxa which are both herbaceous and achlorophyllous. Because they are symbiotic with mycorrhizal fungi and a photosynthetic host, these taxa are known as mycoheterotrophs.&lt;ref&gt;[http://rave.ohiolink.edu/etdc/view?acc_num=osu1417442819 Broe, Michael Brian (2014) Phylogenetics of the Monotropoideae (Ericaceae) with Special Focus on the Genus Hypopitys Hill, together with a Novel Approach to Phylogenetic Inference Using Lattice Theory.]&lt;/ref&gt;
}}

====Tribe Monotropeae====

''Monotropeae'' are mycotrophs, which are parasitic plants that obtain their nutrients trough fungi instead photosynthesis.

=====''Monotropa''=====

{{../txt
|''Monotropa''
}}

{{../table|order=Ericales|Ericaceae|Monotropoideae|Monotropeae||Monotropa|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Monotropa|Indianpipe|480|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Monotropa uniflora
| author  = L.
| sn0     = {{../sn|1753|Monotropa uniflora|L.}}
| sn1     = {{../sn|1927|Monotropa brittonii|Small}}
| ---
| en1     = Indian-pipe
| en2     = One-flowered Indian pipe
| en3     = Ghost pipe
| en4     = Convulsion root
| ---
| fr1     = Monotrope uniflore
| fr2     = Monotrope à une fleur
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1327|5|}}
| usda    = {{../usda|MOUN3|NN|}}
| vascan  = 5537
| gobot   = monotropa/uniflora
| its-id  = 
| ars-id  = 431514
| fna-id  = 200016137
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Monotropa uniflora nymap.svg
| image1  = Monotropa uniflora in Penwood State Park 3, 2009-07-03.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Hypopitys''=====

{{../txt
|img=Monotropa hypopitys WFNY-153A.jpg
|cap=''Hypopitys monotropa '' - pinesap
|''Hypopitys'' is often included in ''Monotropa'' (above).&lt;ref&gt;[http://rave.ohiolink.edu/etdc/view?acc_num=osu1417442819 Broe, Michael Brian (2014) Phylogenetics of the Monotropoideae (Ericaceae) with Special Focus on the Genus Hypopitys Hill, together with a Novel Approach to Phylogenetic Inference Using Lattice Theory.]&lt;/ref&gt;
}}

{{../table|order=Ericales|Ericaceae|Monotropoideae|Monotropeae||Hypopitys|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Hypopitys|Pinesap|480|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Hypopitys monotropa
| author  = Crantz
| sn0     = {{../sn|1753|Monotropa hypopitys|L.}}
| sn1     = {{../sn|1766|Hypopitys monotropa|Crantz}}
| sn2     = {{../sn|1772|Hypopitys multiflora|Scop.}}
| sn3     = {{../sn|1803|Monotropa lanuginosa|Michx.}}
| sn4     = {{../sn|1810|Hypopitys lanuginosa|(Michx.) Raf.}}
| sn5     = {{../sn|1843|Hypopitys americana|(DC.) Small}}
| sn6     = {{../sn|1894|Hypopitys hypopitys|(L.) Small ''taut''.}}
| sn7     = {{../sn|1897|Hypopitys multiflora|(Scop.) Fritsch}}
| sn8     = {{../sn|1901|Hypopitys fimbriata|(A.Gray) Howell}}
| sn9     = {{../sn|1910|M. hypopitys|var=lanuginosa|Purah.}}
| sn10    = {{../sn|1956|M. hypopitys|ssp=lanuginosa|H.Hara}}
| ---
| en1     = Pinesap
| en2     = American pinesap
| en3     = Yellow pinesap
| en4     = Yellow bird's-nest
| en5     = False beechdrops
| ---
| fr1     = Monotrope du pin
| fr2     = Monotrope à grappe
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1318|4|}}
| usda    = {{../usda|MOHY3|NN|}}
| vascan  = 5527      | Hypopitys monotropa  Crantz
| gobot   = hypopitys/monotropa
| its-id  = 23775     | Hypopitys monotropa  Crantz
| ars-id  = 456116    | Monotropa hypopitys L.
| fna-id  = 200016135 | Monotropa hypopitys L.
| tro-id  = 26800126  | Monotropa hypopitys L.
| nse-id  = 
| ipn-id  = 
| map     =  nymap.svg
| image1  = Fichtenspargel Monotropa hypopitys.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Hypopitys lanuginosa
| au-abbr = (Michx.) Raf.
| au-full = 
| au-pub  = 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Monotropa lanuginosa|Michx.}}
{{../sp-1|1810|Hypopitys lanuginosa|(Michx.) Raf.}}
{{../sp-1|1818|H. lanuginosa|(Michx.) Nutt. ''isonym''}}
{{../ssp1|1910|Monotropa hypopitys|L.|lanuginosa|(Michx.) Purah.}}
{{../ssp1|1956|Monotropa hypopitys|L.|lanuginosa|(Michx.) H.Hara}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Red pinesap|}}
{{../vn1|hairy pine-sap|}}
{{../vn1||}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1||}}
{{../vn1||}}
{{../vn1||}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Unranked
| c-rank  = 8
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|7404|Nu|}}
| usda    = {{../usda|||}}
| vascan  = 
| gobot   = Hypopitys/lanuginosa
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = nymap.svg
| image1  = Fringed Pinesap (1056971122).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Pterosporeae====

=====''Pterospora''=====

{{../table|order=Ericales|Ericaceae|Monotropoideae|Pterosporeae||Pterospora|
}}  
{{../genus|Pterospora|Pinedrops|288|n|
}}
{{../taxon
| species = Pterospora andromedea
| author  = Nutt.
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Pinedrops
| en2     = Pine-drops
| en3     = Woodland pinedrops
| en4     = Giant pinedrops
| en5     = Giant birds's-nest
| en6     = Albany beechdrops
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1322|1|Native especially vulnerable in New York State.}}
| usda    = {{../usda|PTAN2|NN|}}
| vascan  = 
| gobot   = Pterospora/andromedea
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 26800015
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Pterospora andromedea Sprague Lake.jpg|width1=120
}}
{{../end table|Ericaceae|Monotropoideae|Pterosporeae|
}}

===Subfamily Arbutoideae===

====''Arctostaphylos''====

{{../txt
|img=Arctostaphylos uva-ursi 38450.JPG
|cap=''Arctostaphylos uva-ursi''&lt;br&gt;bearberry or kinnikinnick
|'''Bearberry''' is a trailing evergreen shrub, native to rocky and sandy upland parts of the Northern Hemisphere, including suitable parts of New York State. 
}}

{{../table|order=Ericales|Ericaceae|Arbutoideae|||Arctostaphylos|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Arctostaphylos|Manzanita|411|1|ARCTO3|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Arctostaphylos uva-ursi
| author = (L.) Spreng.
| sn0     = {{../sn|1753|Arbutus uva-ursi|L.}}
| sn1     = {{../sn|1812|Arbutus buxifolia|Stokes}}
| sn2     = {{../sn|1813|Mairania uva-ursi|(L.) Desv.}}
| sn3     = {{../sn|1821|Uva-ursi buxifolia|(Stokes) Gray}}
| sn4     = {{../sn|1825|Arctostaphylos uva-ursi|(L.) Spreng.}}
| sn5     = {{../sn|1913|Uva-ursi uva-ursi|(L.) Britton ''taut''.}}
| ---
| en1     = Red bearberry
| en2     = Kinnikinnick
| en3     = Pinemat manzanita
| en4     = Mealberry
| en5     = Hog cranberry
| ---
| fr1     = Raisin d’ours
| fr2     = Arctostaphyle raisin-d'ours
| fr3     = Busserole
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = UPL
| habit0  = Perennial
| habit1  = Shrub-subshrub
| light   = Sun - shade
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1348|5|}}
| usda    = {{../usda|ARUV|NN|}}
| vascan  = 5492
| gobot   = arctostaphylos/uva-ursi
| its-id  = 23530
| ars-id  = 3866
| fna-id  = 220001038
| tro-id  = 12300006
| ipn-id  = 
| lbj-id  = ARUV
| nse-id  = Arctostaphylos+uva-ursi
| bna-id  = Arctostaphylos%20uva-ursi 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     =  nymap.svg
| cos     = Alba Bron Clin Colu Dutc Erie Esse Fran Fult Gene Gree Hami Jeff Lewi Nass Niag Onei Onta Oran Oswe Putn Quee Rens Rich Rock Stla Steu Suff Tomp Ulst Warr Wash
| image1  = Arctostaphylos uva-ursi 28273.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Harrimanelloideae===

======''Harrimanella''======

{{../table|order=Ericales|Ericaceae|Harrimanelloideae|||Harrimanella|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Harrimanella|Harrimanella|538|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Harrimanella hypnoides
| author  = (L.) Coville
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Moss bell-heather
| fr1     = 
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely extirpated
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1335|Z|}}
| usda    = {{../usda|HAHY2|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 404379
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     = 
| image1  = Harrimanella hypnoides Kilpisjärvi 2012-07.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Ericoideae===

====Tribe Phyllodoceae====

=====''Epigaea''=====

{{../txt
|img=Epigaea repens WFNY-153B.jpg
|cap=''Epigaea repens'' L.&lt;br&gt;trailing arbutus
|''Epigaea'' contains three species, only one of which, '''trailing arbutus''' (''Epigaea repens''), is native to the New World. It is a low shrub/subshrub found in acidic upland hemlock-hardwood forests and forest edges through much of eastern North America. It flowers in early spring and does well in disturbed sites such as logging roads. Trailing arbutus is the state plant of Massachusetts.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Phyllodoceae||Epigaea|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Epigaea|Trailing-arbutus||1|EPIGA|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Epigaea repens
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Epigaea repens|L.}}
{{../var1|1837|Epigaea repens|L.|rubicunda|D.Don}}
{{../var1|1939|Epigaea repens|L.|glabrifolia|Fernald}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Trailing arbutus|2021 New York Flora Atlas -- 2021 VASCAN-ACC: Hinds, 2000}}
{{../vn1|Trailing-arbutus|2021 Native Plant Trust(Go Botany)}}
{{../vn1|Mayflower|2021 New York Flora Atlas -- 2021 VASCAN-SYN:  FNA Ed. Comm., 2009}}
{{../vn1|Creeping mayflower|2021 VASCAN-SYN: Newmaster et al., 1998}}
{{../vn1|Gravelroot|2021 VASCAN-SYN: GRIN 2010}}
{{../vn1|Ground laurel|2021 VASCAN-SYN:  FNA Ed. Comm., 2009}}
| fr-vns  = 
{{../vn1|Épigée rampante|2021 VASCAN-ACC: Marie-Victorin, 1995}}
{{../vn1|Épigée fleur-de-mai|2021 VASCAN-SYN: Lamoureux, 2002}}
{{../vn1|Fleur de mai|2021 VASCAN-ACC: Marie-Victorin, 1995}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
| ny-rank = S4, G5
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1312|4|}}
| usda    = {{../usda|EPRE2|NN|}}
| vascan  = 5516
| gobot   = Epigaea/repens
| inat    = 83075-Epigaea-repens
| its-id  = 
| ars-id  = 
| fna-id  = 220004812
| tro-id  = 12300017
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Epigaea repens 5-sphillips (5097282717).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Kalmia''=====

{{../txt
|img=Kalmia angustifolia 4500.JPG
|cap=Sheep-laurel (''Kalmia angustifolia'')
|''Kalmia'' ('''laurel''') is a North American genus with about 10 species, four of which are considered to be native to New York.}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Phyllodoceae||Kalmia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Kalmia|Laurel|449|4|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia angustifolia
| author  = L.
| var     = angustifolia
&lt;!-- -- --&gt;
| en1     = Sheep-laurel
| en2     = Sheep-kill
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1304|5|Sub-alpine forests, wet acidic peatlands, and dry sandy forests and forest edges. Primarily a species of acidic soils it grows in dry to wet open or slightly shaded habitats.}}
| usda    = {{../usda|KAAN|NN|}}
| vascan  = 
| gobot   = Kalmia/angustifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = Kalmia angustifolia NY-dist-map.png
| image1  = Kalmia angustifolia 4502.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia latifolia
| author  = L.
| ---
| en1     = Mountain laurel
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6474|5|}}
| usda    = {{../usda|KALA|N0|}}
| vascan  = 
| gobot   = Kalmia/latifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Sugarcamp Mountain (7) (27749726806).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia polifolia
| author  = Wang.
&lt;!-- -- --&gt;
| en1     = Pale laurel
| en2     = Bog laurel
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1326|4|}}
| usda    = {{../usda|KAPO|NN|}}
| vascan  = 
| gobot   = Kalmia/polifolia
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Bog laurel (Kalmia polifolia) (23945002063).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia procumbens
| author  = 
| sn0     = {{../sn||Loiseleuria procumbens|(L.) Desv.}} 
&lt;!-- -- --&gt;
| en1     = Alpine azalea
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1332|1|}}
| usda    = {{../usda|LOPR|NN|}}
| vascan  = 
| gobot   = Kalmia/procumbens
| its-id  = 23556
| ars-id  = 22486
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| image1  = Loiseleuria procumbens Kilpisjärvi 2012-07.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Kalmia|Laurel|449|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Kalmia buxifolia
| author  = (Bergius) Gift &amp; Kron 2008 
&lt;!-- -- --&gt;
| sn0     = {{../sn|1777|Ledum buxifolium|Bergius}}
| sn1     = {{../sn|1813|Dendrium buxifolium|Desv.}}
| sn2     = {{../sn|1817|'''{{../w|Leiophyllum buxifolium}}'''|Elliott}}&lt;ref&gt;NYFA and NRCS both list ''Kalmia buxifolia'' as ''Leiophyllum buxifolium'' (Berg) Elliott.  ITIS, GRIN and Flora of North America, place it in ''Kalmia''.&lt;/ref&gt;
| sn3     = {{../sn|1839|Leiophyllum lyonii|Sweet}}
| sn4     = {{../sn|1901|Leiophyllum hugeri|(Small) K. Schum.}}
&lt;!-- -- --&gt;
| en1     = Sandmyrtle
| en2     = Sand-myrtle
&lt;!-- -- --&gt;
| status  = N. America native
| of      = southern U. S.
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|851|EXCLUDED}}
| usda    = {{../usda|LEBU|N0|}}
| vascan  = 
| gobot   = Kalmia/buxifolia
| its-id  = 894448
| ars-id  = 451598
| fna-id  = 250065674
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = Excluded nymap.svg
| image1  = Kalmia buxifolia NRCS-1.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Ericeae====

=====''Calluna''=====

{{../txt
|img=CallunaVulgaris.jpg
|cap=''Calluna vulgaris'' (L.) Hull&lt;br&gt;heather
|''Calluna'' is a monotypic plant genus whose sole species is ''Calluna vulgaris'' or '''heather''', an Old-World plant.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Ericeae||Calluna|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Calluna|Heather|531|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Calluna vulgaris
| author  = (L.) Hull
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Erica vulgaris|L.}}
| sn1     = {{../sn|1808|Calluna vulgaris|(L.) Hull}}
| sn2     = {{../sn|1906|Ericoides vulgaris|(L.) Merino}}
&lt;!-- -- --&gt;
| en1     = Heather
| en2     = Scotch heather
| en3     = Scots heather
| en4     = Common heather
| en5     = Ling
&lt;!-- -- --&gt;
| status  = Introduced
| from    = Eurasia
| from1   = northern Africa
| status1 = Naturalized
| c-rank  = X
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1337|X|}}
| usda    = {{../usda|CAVU|XX|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 8605
| fna-id  = 
| tro-id  = 12300012
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Calluna%20vulgaris
| cpc-id  = 
| cab-id  = 
| eol-id  =
| map     =  nymap.svg
| image1  = Calluna vulgaris sl3.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Rhodoreae====

=====''Rhododendron''=====

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Pentanthera''======

{{../txt
|img=Rhododendron periclymenoides WFNY-154.jpg
|cap=''Rhododendron periclymenoides''&lt;br&gt;pinxter flower
|''Rhododendron'' sect. ''Pentanthera'' contains the '''azaleas''', three of which are native to New York.
}}

{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pentanthera|Azalea|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron prinophyllum
| author  = (Small) Millais 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1914|Azalea prinophylla|Small}}
{{../sp-1|1917|R. prinophyllum|(Small) Millais}}
{{../sp-1|1921|R. roseum|(Loisel.) Rehder superfl.}}
{{../var1|1924|R. nudiflorum|Torr.|roseum|(Loisel.) Wiegand}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Early azalea|NYFA 2017 / FNA Ed. Comm., 2009}}
{{../vn1|Roseshell azalea|FNA Ed. Comm., 2009 / Castanea, 1994 per ARS-GRIN}}
{{../vn1|Woolly azalea|LBJ 2017}}
{{../vn1|Election-pink|Castanea, 1994 per ARS-GRIN / FNA Ed. Comm., 2009}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = Shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1339|5|Rhododendron prinophyllum (Small) Millais - early azalea - Dry to dry-mesic forests, forest edges, bluffs, hummocks and edges of swamps, and utility rights-of-way. Primarily a species of slightly open dry acidic oak-dominated forests but also somewhat frequent on hummocks in swamps.}}
| usda    = {{../usda|RHPR|N0|}}
| vascan  = 5562
| gobot   = rhododendron/prinophyllum
| its-id  = 
| ars-id  = 105058
| fna-id  = 242417131
| tro-id  = 12300652
| ipn-id  = 
| lbj-id  = RHPR
| nse-id  = 
| bna-id  = Rhododendron%20prinophyllum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron prinophyllum (Roseshell Azalea) (27254491900).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron periclymenoides
| author  = (Michx.) Shinners
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1762|Azalea nudiflora|L. illeg.}}
{{../sp-1|1803|A. periclymenoides|Michx.}}
{{../sp-1|1824|R. nudiflorum|(L.) Torr. illeg.}}
{{../for1|1941|R. nudiflorum|(L.) Torr.|glandiferum|(Porter) Fernald}}
{{../sp-1|1962|R. periclymenoides|(Michx.) Shinners}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Pinxter flower|NYFA}}
{{../vn1|Pinxter-flower|ARS-GRIN}}
{{../vn1|Pinkster|}}
{{../vn1|Pink azalea|}}
{{../vn1|Election-pink|Castanea 1994 per ARS-GRIN}}
{{../vn1|Pinxterbloom azalea|Castanea 1994 per ARS-GRIN}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = FAC
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1343|5|}}
| usda    = {{../usda|RHPE4|N0|}}
| vascan  = 
| gobot   = rhododendron/periclymenoides
| its-id  = 
| ars-id  = 316275
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron periclymenoides - Wild Pink Azalea 2.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron viscosum
| author  = (L.) Torr. 
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Azalea serrulata|Small|}}
{{../sp-1||Azalea viscosa|L.|}}
{{../sp-1|1824|R. viscosum|(L.) Torr.}}
{{../sp-1||R. serrulatum|(Small) Millais}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Swamp azalea|}}
{{../vn1|Clammy azalea |}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1352|5|}}
| usda    = {{../usda|RHVI2|N0|}}
| vascan  = 
| gobot   = rhododendron/viscosum
| its-id  = 
| ars-id  = 
| fna-id  = 250065656
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20viscosum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron viscosum (28332827000).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron calendulaceum
| author  = (Michx.) Torrey
| ---
| en1     = Flame azalea
| ---
| status  = Introduced
| from    = southeastern US
| status1 = N. America native
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1319|X|}}
| usda    = {{../usda|RHCA4|N0|}}
| vascan  = 
| gobot   = rhododendron/calendulaceum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20calendulaceum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron calendulaceum.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pentanthera|Azalea|228|X|txt=(excluded&amp;nbsp;taxa)|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron arborescens
| author  = (Pursh) Torr.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1813|Azalea arborescens|Pursh}}
{{../sp-1|1824|R. arborescens|(Pursh) Torr.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Smooth azalea|}}
{{../vn1|Sweet Azalea|}}
&lt;!-- -- --&gt;
| status  = N. America native
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|X|398|EXCLUDED}}
| usda    = {{../usda|RHAR3|N0|NY: Mitchell, R.S. (ed.). 1986.}}
| note    = 1986&lt;ref&gt;1986: Mitchell, R.S. (ed.). ''A checklist of New York State plants. Contributions of a Flora of New York State, Checklist III. New York State Bulletin No. 458.'' New York State Museum, Albany.&lt;/ref&gt;
| vascan  = 
| gobot   = rhododendron/arborescens
| its-id  = 
| ars-id  = 105055
| fna-id  = 
| tro-id  = 12300627
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20arborescens
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| map     =  nymap.svg
| image1  = Rhododendron arborescens - Sweet Azalea.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron canescens
| author  = (Michx.) Sweet
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1803|Azalea canescens|Michx.}}
{{../sp-1|1830|R. canescens|(Michx.) Sweet}}
{{../var1|1900|Azalea nudiflora|L.|canescens|Rehder in L.H.Bailey}}
{{../sp-1|1901|Azalea candida|Small}}
{{../sp-1|1916|R. candidum|(Small) Rehder}}
{{../var1|1921|R. canescens|(Michx.) Sweet|candidum|(Small) Rehder}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Mountain azalea|WFNY 1918}}
{{../vn1|Hoary azalea|WFNY 1918 / Tropicos 2017}}
{{../vn1|Wild azalea|}}
{{../vn1|Honeysuckle azalea|}}
{{../vn1|Piedmont azalea|Castanea 1994 per ARS-GRIN / Tropicos 2017}}
{{../vn1|Sweet azalea|}}
{{../vn1|Southern pinxterflower|Castanea 1994 per ARS-GRIN}}
{{../vn1|Sweet-Florida pinxter|Tropicos 2017}}
&lt;!-- -- --&gt;
| status  = N. America native
| status1 = Excluded
| c-rank  = 
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| usda    = {{../usda|RHCA7|N0|}}
| note    = 1918&lt;ref&gt;1918: Homer D. House, New York State Botanist, ''Wild Flowers of New York'', Vol 2, page 201, &quot;Mountain or Hoary Azalea, ''Azalea canescens'' Michaux ... eastern and southern New York....&quot; This appears to be a misidentification of the similar species ''Rhododendron periclymenoides'', which is common in that range.&lt;/ref&gt;
| vascan  = 
| gobot   = 0
| its-id  = 23712
| ars-id  = 5020
| fna-id  = 
| tro-id  = 12300635
| ipn-id  = 27637-2
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20canescens
| cpc-id  = 
| cab-id  = 
| eol-id  =
| kew-id  =
| map     =  nymap.svg
| image1  = Rhododendron canescens (Ericaceae).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Rhodora''======

{{../txt
|img=Rhododendron canadense flowers.JPG
|cap=''Rhododendron canadense''&lt;br&gt;rhodora
|''Rhododendron'' sect. ''Rhodora'' contains the single species ''Rhododendron canadense'' ('''rhodora'''), which is native to northeastern North America. 
}}
{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Rhodora|Rhodora|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron canadense
| author  = (L.) Torr.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1762|Rhodora canadensis|L.}}
{{../sp-1|1841|Rhododendron canadense|(L.) Torr.}}
{{../sp-1|1891|Azalea canadensis|(L.) Kuntze}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Rhodora|2017 NYFA}}
{{../vn1|Canada rosebay|2017 New England Wild Flower Society Go Botany}}
{{../vn1|Canada rhododendron|2017 VASCAN: Anions, M., Proposition de nom anglais (pers. comm.)}}
{{../vn1|Canadian rhododendron|2017 VASCAN: Darbyshire S.J., M. Favreau &amp; M. Murray (revu et augmenté par). 2000. Noms populaires et scientifiques des plantes nuisibles du Canada. Agriculture et Agroalimentaire Canada. Publication 1397. 132 pp.}}
&lt;!-- -- --&gt;
| fr-vns  = 
{{../vn1|Rhododendron du Canada|2017 VASCAN (accepted): Darbyshire S.J., M. Favreau &amp; M. Murray (revu et augmenté par). 2000. Noms populaires et scientifiques des plantes nuisibles du Canada. Agriculture et Agroalimentaire Canada. Publication 1397. 132 pp.}}
{{../vn1|Rhodora|2017 VASCAN: Marie-Victorin, Fr. 1995. Flore laurentienne. 3e éd. Mise à jour et annotée par L. Brouillet, S.G. Hay, I. Goulet, M. Blondeau, J. Cayouette et J. Labrecque. Gaétan Morin éditeur. 1093 pp.}}
{{../vn1|Rhodora du Canada|2017 VASCAN: Cunningham, G.C. 1958. Flore fiorestière du Canada. Ministère du Nord canadien et des Ressources nationales. 144 pp.}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Threatened
| c-rank  = 9
| nwi1    = FACW
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1311|2|	Rhododendron canadense (L.) Torr. - Rhodora - Bogs, edges of bogs, and wet thickets.}}
| usda    = {{../usda|RHCA6|NN|}}
| vascan  = 5556
| gobot   = rhododendron/canadense
| its-id  = 
| ars-id  = 5019
| fna-id  = 
| tro-id  = 50134272
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20canadense
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Rhododendron canadense (27196782616).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Hymenanthes'' sect. ''Pontica''======

{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Hymenanthes|sect=Pontica|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron maximum
| author  = L.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1753|Rhododendron maximum|L.}}
{{../sp-1|1935|Rhododendron ashleyi|Coker}}
{{../sp-1|1943|Hymenanthes maxima|(L.) H.F.Copel.}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Great rosebay|NYFA-1}}
{{../vn1|Great laurel|NYFA-2}}
{{../vn1|Great rhododendron|Terrell, E. E. et al. AH 505 1994 per ARS-GRIN}}
{{../vn1|Rose bay|Castanea 1994 per ARS-GRIN}}
{{../vn1|Wild rhododendron|}}
{{../vn1|Rosebay rhododendron|}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| c-rank  = 9
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1341|4|}}
| usda    = {{../usda|RHMA4|NN|}}
| vascan  = 
| gobot   = rhododendron/maximum
| its-id  = 
| ars-id  = 31441
| fna-id  = 
| tro-id  = 12300644
| ipn-id  = 
| lbj-id  = RHMA4
| nse-id  = 
| bna-id  = Rhododendron%20maximum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Budding (9249172175).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Rhododendron'' subg. ''Rhododendron'' sect. ''Rhododendron''======
{{../txt
|img=Lapland Rose-Bay (3815996423).jpg
|cap=''Rhododendron lapponicum''&lt;br&gt;Lapland rosebay
|''Rhododendron'' sect. ''Rhododendron'' contains the two New York natives '''Labrador tea''' (''R. groenlandicum'') and '''Lapland rosebay''' (''R. lapponicum'').
}}
{{../table|order=Ericales|Ericaceae|Ericoideae|Rhodoreae||Rhododendron|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Rhododendron|sect=Rhododendron|subsect=Ledum|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron groenlandicum
| author  = (Oeder) Kron &amp; Judd
&lt;!-- ========= --&gt;
| syns    = &lt;poem&gt;
1771. ''Ledum groenlandicum ''Oeder in G.C.Oeder &amp; al. (eds.), Fl. Dan. 4(10):5
1892. ''Ledum palustre      ''var. ''groenlandicum ''(Oeder) Rosenv. in Meddel. Grønland
1948. ''Ledum palustre      ''ssp. ''groenlandicum ''(Oeder) Hultén in Acta Univ. Lund.
1990. '' '''Rhododendron groenlandicum''' ''(Oeder) Kron &amp; Judd in  Syst. Bot. 15:67
&lt;/poem&gt;
&lt;!-- ========= --&gt;
| en1     = Labrador tea
| en2     = Bog Labrador tea
| en3     = Common Labrador tea
| ---
| status  = Native
| status1 = Likely secure
| c-rank  = 9
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1351|4-5|}}
| inatc   = {{../inat|180413-Rhododendron-groenlandicum||}}
| usda    = {{../usda|LEGR|NN|}}
| vascan  = 
| gobot   = rhododendron/groenlandicum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 50303842
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20groenlandicum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Ledem groenlandicum C.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Rhododendron|subg=Rhododendron|sect=Rhododendron|subsect=Lapponica|Rhododendron|228|8|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Rhododendron lapponicum
| author  = (L.) Wahlenb.
| sn0     = {{../sn||Azalea lapponica|L.}}
| ---
| en1     = Lapland rosebay
| en2     = Lapland azalea
| ---
| status  = Native
| status1 = Endangered
| c-rank  = 10
| nwi1    = 
| habit0  = 
| habit1  = 
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1353|1|}}
| usda    = {{../usda|RHLA2|NN|}}
| vascan  = 
| gobot   = rhododendron/lapponicum
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Rhododendron%20lapponicum
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Lapland Rose-Bay (3816823222).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Empetroideae===

'''Empetroideae''' Sweet, ''Hort. Brit.'': 491. Sep–Oct 1826.

====Tribe Empetreae====

Empetreae Horan., ''Char. Ess. Fam.'': 109. 17 Jun 1847.

=====''Empetrum''=====

{{../table|order=Ericales|Ericaceae|Empetroideae|Empetreae||Empetrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Empetrum|Crowberry|701|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Empetrum nigrum
| author  = L.
| sn0     = {{../sn|1753|Empetrum nigrum|L.}}
| sn1     = {{../sn|1913|E. eamesii|ssp=hermaphroditum|}}
| sn2     = {{../sn|1927|E. hermaphroditum|Lange ex Hagerup}}
| sn3     = {{../sn|1933|E. nigrum|var=hermaphroditum|}}
| sn4     = {{../sn|1952|E. nigrum|ssp=hermaphroditum|}}
| ---
| en1     = Black crowberry
| en2     = Crakeberry   
| en3     = Curlew-berry
| ---
| fr1     = Camarine noire
| status  = Native
| status1 = Rare
| nyfa    = {{../nyfa|1313|3|}}
| usda    = {{../usda|EMNI|NN|}}
| vascan  = 5514
| gobot   = empetrum/nigrum
| its-id  = 23743
| ars-id  = 15127
| fna-id  = 200012671
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Empetrum nigrum nymap.svg
| image1  = Empetrum nigrum (1).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Empetrum atropurpureum
| author  = Fernald &amp; Wiegand
| sn0     = {{../sn|1913|E. atropurpureum|Fernald &amp; Wiegand}}
| sn1     = {{../sn|1927|E. rubrum'' var. ''atropurpureum|}}
| sn2     = {{../sn|1960|E. eamesii'' ssp. ''atropurpureum|}}
| sn3     = {{../sn|1966|E. nigrum'' var. ''atropurpureum|}}&lt;ref&gt;USDA-NRCS-PLANTS and ITIS treat ''E. atropurpureum'' as a synonym of ''E. nigrum'', but NYFA, FNA, and ''Flora Novae Angliae'' all treat ''E. atropurpureum'' as a separate species.&lt;/ref&gt;
| ---
| en1     = Purple crowberry
| en2     = Red crowberry
| ---
| fr1     = Camarine noire-pouprée
| fr2     = Camarine atropourpre
| fr3     = Camarine pourpre
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1314|1|}}
| usda    = {{../usda|EMEAA|NN|as Empetrum eamesii Fernald &amp; Wiegand ssp. atropurpureum (Fernald &amp; Wiegand) D. Löve }}
| vascan  = 5511
| gobot   = empetrum/atropurpureum
| its-id  = 23743
| ars-id  = 
| fna-id  = 250065677
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Empetrum atropurpureum nymap.svg
| image1  = Empetrum nigrum (Crowberry) - unripe berries - Flickr - S. Rae.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Corema''=====

{{../table|order=Ericales|Ericaceae|Empetroideae|Empetreae||Corema|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Corema|Crowberry|992|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Corema conradii
| author  = (Torr.) Torr.
| sn0     = {{../sn|1837|Empetrum conradii|Torr.}}
| sn1     = {{../sn|1842|Corema conradii|(Torr.) Torr.}}
| ---
| en1     = Broom crowberry 
| en2     = Broom-crowberry 
| en3     = Poverty-grass 
| ---
| fr1     = Corème de Conrad
| fr2     = Camarine de Conrad
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1315|1|}}
| usda    = {{../usda|COCO9|NN|}}
| vascan  = 5509
| gobot   = corema/conradii
| its-id  = 23750
| ars-id  = 400579
| fna-id  = 250065678
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Corema conradii nymap.svg
| image1  = Corema conradii buds.jpeg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

===Subfamily Vaccinoideae===

====Tribe Oxydendreae====

=====''Oxydendrum''=====

{{../txt
|img=Oxydendrum arboreum 3zz.jpg
|cap=''Oxydendrum arboreum'' (L.) DC.&lt;br&gt;sourwood tree
|''Oxydendrum'' contains a single species, the '''sourwood''' tree (''O. arboreum''), which is endemic to the southeastern U.S., as far north as southwestern Pennsylvania. Its flowers grow in racemes and look similar to lily-of-the-valley flowers.  It is cultivated as an ornamental in southern New York, where it has escaped.}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Oxydendreae||Oxydendrum|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Oxydendrum|Sourwood|117|1|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Oxydendrum arboreum
| author  = (L.) DC.
| syns    = {{../snl
|{{../spa1|1753|Andromeda arborea|L.}}
|{{../spa1|1839|Oxydendrum arboreum|(L.) DC.}}
}}
&lt;!-- -- --&gt;
| en1     = Sourwood
| en2     = Sorreltree
| en3     = Lily-of-the-valley tree
| en4     = Swamp-cranberry
| fr1     = 
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| nyfa    = {{../nyfa|1333|X|State exotic or non-native}}
| usda    = {{../usda|OXAR|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 23690
| ars-id  = 311632
| fna-id  = 220009741
| tro-id  = 12300622
| nse-id  = Oxydendrum+arboreum
| ipn-id  = 1175876-2
| map     = Oxydendrum arboreum nymap.svg
| image1  = Sourwood leaves and flowers.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Oxydendreae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Lyonieae====

=====''Lyonia''=====

{{../txt
|img=Lyonia ligustrina 1120554.jpg
|cap=''Lyonia ligustrina'' - maleberry
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Lyonieae||Lyonia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Lyonia|Staggerbush|136|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lyonia ligustrina
| author  = (L.) DC.
| var     = ligustrina
| sn0     = {{../sn|1753|Vaccinium ligustrinum|L.}}
| sn1     = {{../sn|1813|Andromeda ligustrina|(L.) Muhl.}}
| sn2     = {{../sn|1839|Lyonia ligustrina|(L.) DC.}}
| sn3     = {{../sn|1894|Xolisma ligustrina|(L.) Britton}}
| sn4     = {{../sn|1913|Arsenococcus ligustrinus|(L.) Small}}
&lt;!-- -- --&gt;
| en1     = Maleberry
| en2     = Seedy buckberry
| en3     = Privet andromeda
| fr1     = Lyonie faux-troène
| fr2     = Lyonie ligustrine
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1331|5|}}
| usda    = {{../usda|LYLI|N0|}}
| vascan  = 23768
| gobot   = lyonia/ligustrina
| its-id  = 23559
| ars-id  = 22993
| fna-id  = 250065687
| tro-id  = 12300077
| nse-id  = Lyonia+ligustrina
| ipn-id  = 
| map     =  nymap.svg
| image1  = Lyonia ligustrina 2zz.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Lyonia mariana
| author  = (L.) D.Don
| sn0     = {{../sn|1753|Andromeda mariana|L.}}
| sn1     = {{../sn|1834|Lyonia mariana|(L.) D.Don}}
| sn2     = {{../sn|1876|Pieris mariana|(L.) Benth. &amp; Hook.f.}}
| sn3     = {{../sn|1913|Neopieris mariana|(L.) Britton}}
| sn4     = {{../sn|1924|Xolisma mariana|(L.) Rehder}}
&lt;!-- -- --&gt;
| en1     = Stagger-bush
| en2     = Piedmont staggerbush
| en3     = Maryland staggerbush
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1329|4|}}
| usda    = {{../usda|LYMA2|N0|}}
| vascan  = | south of Canada
| gobot   = lyonia/mariana
| its-id  = 23564
| ars-id  = 22995
| fna-id  = 242416810
| tro-id  = 12302715
| nse-id  = Lyonia+mariana
| ipn-id  = 331154-1
| map     =  nymap.svg
| image1  = Lyonia mariana WFNY-155B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Lyonieae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Andromedeae====

=====''Andromeda''=====

{{../txt
|img=Andromeda polifolia 2016-04-28 9221.jpg
|cap=''Andromeda polifolia''&lt;br&gt;bog rosemary
|''Andromeda polifolia'' ('''bog-rosemary''') is generally treated as the only member of its genus and is found in acidic bogs throughout the northern hemisphere.  Two varieties are considered to be present in North America, but only the '''glaucous-leaved bog rosemary''', ''A. polifolia'' var. ''latifolia'' (syn. ''A. glaucophylla'') has been reported in New York. 
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Andromedeae||Andromeda|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Andromeda|Bog-rosemary|824|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Andromeda polifolia
| author  = L.
| var     = latifolia
| var-au  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|A. polifolia|var=latifolia|var-au=Aiton}}
| sn1     = {{../sn|1821|A. glaucophylla|Link}}
| sn2     = {{../sn|1839|A. polifolia|var=glaucophylla|var-au=(Link) DC.}}
| sn3     = {{../sn|1914|A. canescens|Small}}
| sn4     = {{../sn|1916|A. glaucophylla|var=iodandra|var-au=Fernald}}
| sn5     = {{../sn|1924|A. glaucophylla|fo=latifolia|fo-au=(Aiton) Rehder}}
| sn6     = {{../sn|1927|A. glaucophylla|var=latifolia|var-au=(Aiton) Rehder}}
| sn7     = {{../sn|1948|A. polifolia|ssp=glaucophylla|ssp-au=(Link) Hultén}}
&lt;!-- -- --&gt;
| en1     = Bog rosemary
| en2     = Glaucous-leaved bog rosemary
| fr1     = Andromède glauque
&lt;!-- -- --&gt;
| status  = Native
| status1 = Secure
| c-rank  = 10
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1330|5|Andromeda polifolia L. var. latifolia Aiton - bog rosemary - Acidic bogs. Often growing mixed with other shrubs and forming a dense low shrub thicket in bogs. Also growing on hummocks or rises in bogs in small shrub ''islands''}}
| usda    = {{../usda|ANPOG|NN|}}
| vascan  = 5486
| gobot   = Andromeda/polifolia
| its-id  = 894555 
| ars-id  = 467871
| fna-id  = 250065693
| tro-id  = 50294868
| ipn-id  = 
| lbj-id  = ANPOG
| nse-id  = 
| bna-id  = 
| map     = Andromeda polifolia var glaucophylla NY-dist-map.png
| image1  = Andromeda polifolia var. latifolia WFNY-159A.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Andromedeae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

====Tribe Gaultherieae====

=====''Chamaedaphne''=====

{{../txt
|img=Chamaedaphne calyculata 1 (5097220701).jpg
|cap=''Chamaedaphne calyculata''&lt;br&gt;leatherleaf
|''Chamaedaphne calyculata'' ('''leatherleaf''') is the sole species of the genus and is native to much of the Northern Hemisphere, including New York State.
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Chamaedaphne|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Chamaedaphne|Leatherleaf|446|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Chamaedaphne calyculata
| author  = (L.) Moench
| ---
| en1     = Leatherleaf
| ---
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub
| light   = 
| chro-no = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1349|5|Chamaedaphne calyculata (L.) Moench - leatherleaf - Bogs, edges of ponds, and acidic peaty open sites. Mostly confined to acidic peatlands where it can form dense extensive monospecific stands or become mixed with other low shrubs to from dense shrub thickets.}}
| usda    = {{../usda|CHCA2|NN|}}
| vascan  = 
| gobot   = Chamaedaphne/calyculata
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| bbg-id  = 
| cpc-id  = 
| cab-id  = 
| eol-id  = 
| kew-id  = 
| red-id  = 
| mbg-id  = 
| adf-id  = 
| fed-id  = 
| map     = 
| image1  = Chamaedaphne calyculata WFNY-157A.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Gaultheria''=====
{{../txt
|img=Gaultheria procumbens.JPG
|cap=''Gaultheria procumbens''&lt;br&gt;wintergreen
|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Gaultheria|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gaultheria|Snowberry|160|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaultheria procumbens
| author  = L.
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Wintergreen|2018 New York Flora Atlas 1 / Flora of North America}}
{{../vn1|Teaberry|2018 New York Flora Atlas 2}}
{{../vn1|Eastern teaberry|2018 USDA NRCS National Plant Data Team / Flora of North America}}
{{../vn1|Checkerberry|2018 Flora of North America}}
{{../vn1|Eastern spicy-wintergreen|2018 New England Wild Flower Society, Go Botany}}
{{../vn1|Creeping wintergreen|2018 VASCAN Canadensys}}
{{../vn1|Spring wintergreen|2018 VASCAN Canadensys}}
{{../vn1|Mountain-tea|2018 ARS GRIN Huxley, A., ed. Dict. Gard 1994}}

| en3     = American wintergreen

| ---
| fr1     = Thé des bois
| fr2     = Thé rouge
| fr3     = Gaulthérie couchée
| status  = Native
| status1 = Secure
| habit0  = Perennial
| habit1  = Shrub, subshrub
| nyfa    = {{../nyfa|1310|5|}}
| usda    = {{../usda|GAPR2|NN|}}
| vascan  = 5520
| gobot   = Gaultheria/procumbens
| its-id  = 
| ars-id  = 360
| fna-id  = 220005469
| tro-id  = 12300028
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Golteria1-pl.JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaultheria hispidula
| author  = (L.) Muhl. ex Bigel.
| sn0     = Chiogenes hispidula
| ---
| en1     = Creeping snowberry
| ---
| status  = Native
| status1 = Secure
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1317|5|}}
| usda    = {{../usda|GAHI2|NN|}}
| vascan  = 
| gobot   = Gaultheria/hispidula
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Gaultheria hispidula 7847 (cropped).JPG
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Eubotrys''=====

{{:Flora of New York/txt
|img=Eubotrys racemosa - Swamp doghobble 2.jpg
|cap=''Eubotrys racemosa''
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Eubotrys|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Eubotrys|Fetterbush|1133|2|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Eubotrys racemosa
| author  = (L.) Nutt.
&lt;!-- -- --&gt;
| syns    = 
{{../sp-1|1842|Eubotrys racemosa|(L.) Nutt.}}
{{../sp-1|1856|Leucothoe racemosa|(L.) A. Gray}}
&lt;!-- -- --&gt;
| en-vns  = 
{{../vn1|Swamp fetterbush|2019 New York Flora Atlas}}
{{../vn1|Swamp doghobble|2019 Native Plant Trust, USDA NRCS National Plant Data Team}}
{{../vn1|Swamp deciduous dog-laurel|2019 Native Plant Trust}}
{{../vn1|Deciduous swamp fetterbush|Flora of North America}}
{{../vn1|Sweetbells|2019 Native Plant Trust}}
{{../vn1|Swamp sweetbells|2019 Native Plant Trust}}
&lt;!-- -- --&gt;
| status  = Native
| status1 = Likely secure
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1346|4|}}
| usda    = {{../usda|EURA5|N0|}}
| vascan  = 
| gobot   = Eubotrys/racemosa
| its-id  = 
| ars-id  = 
| fna-id  = 250065697
| tro-id  = 12302763
| ipn-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Leucothe racemosa (5625200643).gif
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Eubotrys recurva
| author  = (Buckley) Britton 
| ---
| en1     = Mountain fetterbush 
| en2     = Deciduous mountain fetterbush 
| en3     = Redtwig doghobble
| en4     = Red-twigged doghobble
| ---
| status  = Introduced
| status1 = US South native
| habit0  = Perennial
| habit1  = Shrub
| nyfa    = {{../nyfa|1342|X|}}
| usda    = {{../usda|EURE10|N0|}}
| vascan  = 
| gobot   = 
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Eubotrys%recurva
| map     =  nymap.svg
| image1  = Eubotrys recurva BB-1913.png
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Leucothoe''=====

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Gaultherieae||Leucothoe|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Leucothoe|Doghobble|991|1|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Leucothoe fontanesiana
| author  = (Steud.) Sleumer
| ---
| en1     = Highland dog-hobble
| ---
| status  = Introduced
| status1 = US South native
| status2 = Impersistent
| habit0  = 
| habit1  = 
| nyfa    = {{../nyfa|1344|Xm|}}
| usda    = {{../usda|LEFO|N0|}}
| vascan  = 
| gobot   = leucothoe/fontanesiana &lt;!--exotic in MA--&gt;
| its-id  = 
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = 
| bna-id  = Leucothoe%20fontanesiana
| map     =  nymap.svg
| image1  = Leucothe fontanesia.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Leucothoe axillaris
| author  = (Lam.) D.Don 
&lt;!-- -- --&gt;
| sn0     = {{../sn|1783|Andromeda axillaris|Lam.}}
| sn1     = {{../sn|1788|Andromeda catesbaei|Walter}}
| sn2     = {{../sn|1809|Andromeda walteri|Willd.}}
| sn3     = {{../sn|1834|Leucothoe axillaris|D.Don}}
| sn4     = {{../sn|1856|Leucothoe catesbaei|A.Gray|}}
&lt;!-- -- --&gt;
| en1     = Coastal doghobble
&lt;!-- -- --&gt;
| status  = Introduced
| status1 = US South native
| status2 = Cultivated
| status3 = No NY reports
&lt;!-- -- --&gt;
| habit0  = Perennial
| habit1  = Shrub
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|0|0|}}
| usda    = {{../usda|LEAX|N0|}}
| vascan  = 
| gobot   = &lt;!--Not listed in New Enland--&gt;
| its-id  = 
| ars-id  = 22006
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| nse-id  = Leucothoe+axillaris
| bna-id  = Leucothoe%20axillaris
| map     =  nymap.svg
| image1  = Leucothoe axillaris3.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table}}

====Tribe Vaccinieae====

{{../txt
|The Vaccinieae contain the morphologically similar '''huckleberries''' (''Gaylussacia'') and '''blueberries''' (''Vaccinium'').
}}

=====''Gaylussacia''=====

{{../txt
|img=Gaylussacia dumosa (6009920604).jpg
|cap=''Gaylussacia bigeloviana''&lt;br&gt;bog huckleberry
|
}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Gaylussacia|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Gaylussacia|Huckleberry|491|3|||
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia baccata
| author  = ({{../w|Wangenh.}}) {{../w|K.Koch}}
| sn0     = {{../sn|1787|Andromeda baccata|Wangenh.}}
| sn1     = {{../sn|1872|Gaylussacia baccata|K.Koch}}
| sn2     = {{../sn|1900|G. resinosa|var=glaucocarpa|B.L.Rob.}}
| sn3     = {{../sn|1907|G. baccata|var=glaucocarpa|Mack.}}
| sn4     = {{../sn|1933|Decachaena baccata|Small}}
| ---
| en1     = Black huckleberry
| ---
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|1307|5|}}
| usda    = {{../usda|GABA|NN|}}
| vascan  = 
| gobot   = gaylussacia/baccata
| its-id  = 
| ars-id  = 
| fna-id  = 242416580
| tro-id  = 12300598
| nse-id  = 
| ipn-id  = 
| map     =  
| image1  = Gaylussacia baccata 5497208.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia frondosa
| author  = (L.) Torr. &amp; A.Gray
| sn0     = {{../sn|1753|Vaccinium frondosum|L.}}
| sn1     = {{../sn|1843|Gaylussacia frondosa|Torr. &amp; A.Gray}}
| sn2     = {{../sn|1933|Decachaena frondosa|ex Small}}
| ---
| en1     = Blue huckleberry
| en2     = Dangleberry
| ---
| status  = Native
| status1 = Likely secure
| nyfa    = {{../nyfa|1308|4|Apparently secure in New York State.}}
| usda    = {{../usda|GAFR2|N0|}}
| vascan  = 
| gobot   = gaylussacia/frondosa
| its-id  = 
| ars-id  = 376
| fna-id  = 
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     =  
| image1  = Gaylussacia frondosa NRCS-01.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Gaylussacia bigeloviana
| author  = (Fernald) Sorrie &amp; Weakley
| sn0     = {{../sn|1911|'''{{../w|Gaylussacia dumosa}}'''|var1=bigeloviana|Fernald}}
| sn1     = {{../sn|1933|Lasiococcus dumosus|var=bigelovianus||Fernald}}
| sn2     = {{../sn|2007|Gaylussacia bigeloviana|Sorrie &amp; Weakley}}
| sn3     = {{../sn|auct|Gaylussacia dumosa|'''non''' (Andrews) A.Gray}}
| ---
| en1     = Dwarf huckleberry
| en2     = Bog huckleberry
| ---
| status  = Native
| status1 = Endangered
| nyfa    = {{../nyfa|1306|1-2|}}
| usda    = {{../usda|GADU|NN|as Gaylussacia dumosa (Andrews) Torr. &amp; A.Gray 1846}}
| vascan  = 
| gobot   = gaylussacia/bigeloviana
| its-id  = 894464
| ars-id  = 319642
| fna-id  = 250065725
| tro-id  = 
| nse-id  = 
| ipn-id  = 
| map     = Gaylussacia bigeloviana nymap.svg
| image1  = Gaylussacia bigeloviana WFNY-157B.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table|Ericaceae|Vaccinoideae|Vaccinieae|
}}&lt;!-- ------------------------------------------------------------ --&gt;

=====''Vaccinium''=====

{{../txt|The genus '''''Vaccinium''''' comprises the blueberries, cranberries, huckleberries and similar shrubs.  It is here broken down into sections as described in Flora of North America.&lt;ref&gt;[http://efloras.org/florataxon.aspx?flora_id=1&amp;taxon_id=134285 ''Vaccinium.'' Flora of North America Editorial Committee, eds. 1993+. Flora of North America North of Mexico.  20+ vols.  New York and Oxford.]&lt;/ref&gt;}}

======''Vaccinium'' sect. ''Cyanococcus''======

{{../txt|Section ''Cyanococcus'' was described by Asa Gray in ''Memoirs of the American Academy of Arts and Science'' in 1846.&lt;ref&gt;[http://www.tropicos.org/Name/50140387 ''Vaccinium'' sect. ''Cyanococcus'' A. Gray. Tropicos.org. Missouri Botanical Garden. 13 Oct 2016]&lt;/ref&gt; This section contains all of the New York '''blueberry''' species except the alpine blueberry.}}

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Cyanococcus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Blueberry|832|11|VACCI|14|sect=Cyanococcus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium angustifolium
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium angustifolium|Aiton}}
| sn1     = {{../sn|1861|V. pensylvanicum|var=nigrum|var-au=Alph. Wood}}
| sn2     = {{../sn|1894|V. nigrum|(Alph.Wood) Britton}}
| sn3     = {{../sn|1914|V. brittonii|Porter ex E. P. Bicknell}}
| sn4     = {{../sn|1931|Cyanococcus angustifolium|(Aiton) Rydb.}}
| sn5     = {{../sn|1943|V. lamarckii|Camp}}
&lt;!-- -- --&gt;
| en1     = Lowbush blueberry
| en2     = Late lowbush blueberry
| en3     = Low sweet blueberry
| en4     = Late sweet blueberry
| en5     = Sweet-hurts
&lt;!-- -- --&gt;
| fr1     = Bleuet à feuilles étroites
| fr2     = Airelle de Pennsylvanie
| status  = Native
| status1 = Secure
| c-rank  = 4
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = Sun - shade
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6470|5|Onon-Tomp-Cort-Oswe-Onei}}
| usda    = {{../usda|VAAN|NN|}}
| vascan  = 5566
| gobot   = Vaccinium/angustifolium
| its-id  = 23579
| ars-id  = 40981
| fna-id  = 250065717
| tro-id  = 12300044
| ipn-id  = 
| lbj-id  = VAAN
| nse-id  = 
| bna-id  = Vaccinium%20angustifolium
| map     =  nymap.svg
| image1  = 2016-09 Monts Valin 23.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium corymbosum
| author  = L.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Vaccinium corymbosum|L.}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Highbush blueberry
| en2     = Swamp blueberry
| en3     = Whortleberry
| en4     = New Jersey blueberry
| en5     = Southern blueberry
| ---
| fr1     = Bleuet en corymbe 
| status  = Native
| status1 = Secure
| c-rank  = 6
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6472|5|}}
| usda    = {{../usda|VACO|NN|}}
| vascan  = 5569
| gobot   = Vaccinium/corymbosum
| its-id  = 23573
| ars-id  = 41002
| fna-id  = 242417401
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20corymbosum
| map     =  nymap.svg
| image1  = 2013-08-25 14 02 03 Closeup of blueberries along the shore of Spring Lake at 88 Lake Road in Berlin, New York.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium × atlanticum
| author  = E.P.Bicknell (pro sp.)
| hp1     = Vaccinium angustifolium
| hp2     = Vaccinium corymbosum
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Atlantic huckleberry
| en2     = {{../hybrid-of|Lowbush blueberry|Highbush blueberry}}
| ---
| status  = Native
| status1 = Unranked
| c-rank  = 
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1345|U|}}
| usda    = {{../usda|VAAT3|N0|×atlanticum NY &amp; CT only}}
| vascan  = 
| gobot   = 
| its-id  = 505629
| ars-id  = 
| fna-id  = 
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20X%20atlanticum
| map     =  nymap.svg
| image1  = Ny hybrid.svg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium myrtilloides
| author  = Michx.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1803|Vaccinium myrtilloides|Michx.}}
| sn1     = {{../sn|1823|V. canadense|Kalm ex Richardson}}
| sn2     = {{../sn|1917|Cyanococcus canadensis|(Kalm ex Richardson) Rydb.}}
| sn3     = {{../sn|1921|V. angustifolium|var=myrtilloides|var-au=(Michx.) House}}
| sn4     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Velvet-leaved blueberry
| en2     = Velvet-leaved huckleberry
| en3     = Velvetleaf huckleberry
| en4     = Sour-top blueberry
| en5     = Sourtop
| en6     = Canada blueberry
| fr1     = Bleuet fausse-myrtille
| fr2     = Bleuet rameau-velouté
| fr3     = Bleuet du Canada
| fr4     = Bleuets
| fr5     = Airelle fausse-myrtille
| fr6     = Airelle faux-myrtille
| fr7     = Airelle du Canada
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|6471|5|Hummocks in swamps, edges of swamps, cool northern forests, edges of forests, forest openings, barrens, and bluffs. More common in the northern and cooler parts of New York.}}
| usda    = {{../usda|VAMY|NN|}}
| vascan  = 5573
| gobot   = vaccinium/myrtilloides
| its-id  = 
| ars-id  = 41039
| fna-id  = 250065719
| tro-id  = 12300054
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20myrtilloides
| map     =  nymap.svg
| image1  = Vaccinium myrtilloides berries.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium pallidum
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium pallidum|Aiton}}
| sn1     = {{../sn|1856|V. corymbosum|var=pallidum|var-au=(Aiton) A. Gray}}
| sn2     = {{../sn|1933|Cyanococcus pallidus|(Aiton) Small}}
&lt;!-- -- --&gt;
| en1     = Early lowbush blueberry
| en2     = Late lowbush blueberry
| en3     = Blue Ridge blueberry
| en4     = Hillside blueberry
| ---
| status  = Native
| status1 = Secure
| c-rank  = 7
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1320|5|}}
| usda    = {{../usda|VAPA4|NN|}}
| vascan  = 
| gobot   = Vaccinium/pallidum
| its-id  = 
| ars-id  = 41049
| fna-id  = 242417402
| tro-id  = 12300678
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20pallidum
| map     =  nymap.svg
| image1  = Vaccinium pallidum - Blue Ridge Blueberry.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium boreale
| author  = I.V.Hall &amp; Aalders
| sn0     = {{../sn|1848|V. pensylvanicum|var=angustifolium|var-au=(Aiton) A. Gray}}
| sn1     = {{../sn|1861|V. pensylvanicum|var=alpinum|var-au=Wood}}
| sn2     = {{../sn|1961|V. boreale|I.V.Hall &amp; Aalders}}
| ---
| en1     = Northern blueberry
| en2     = High-mountain blueberry
| en3     = Sweet hurts
| ---
| fr1     = Bleuet boréal
| status  = Native
| status1 = Threatened
| c-rank  = 10
| nwi1    = 
| habit0  = Perennial
| habit1  = Shrub
| light   = 
&lt;!-- -- --&gt;
| nynhp   = 2&lt;ref&gt;{{../nynhp-ref|9021|Vaccinium boreale|Threatened|S2|G4}}&lt;/ref&gt;
| nyfa    = {{../nyfa|1309|2|}}
| usda    = {{../usda|VABO|NN|}}
| vascan  = 5567
| gobot   = vaccinium/boreale
| its-id  = 23583 
| ars-id  = 315259
| fna-id  = 250065716
| tro-id  = 50077834
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = Vaccinium%20boreale
| map     =  nymap.svg
| image1  = 
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Vaccinium''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Vaccinium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|sect=Vaccinium|Blueberry|832|11|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium uliginosum
| author  = L. (1753)
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Alpine blueberry
| en2     = Bog blueberry
| en3     = Bog bilberry
| en4     = Bog whortleberry
| en5     = Northern bilberry
| en6     = Airelle de marécages
| status  = Native
| status1 = Rare
| nyfa    = {{../nyfa|6478|3|}}
| usda    = {{../usda|VAUL|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23574
| ars-id  = 41063
| fna-id  = 200016726
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Ripe bilberries, Dipton Wood - geograph.org.uk - 1472503.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Myrtillus''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Myrtillus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Dwarf blueberry|832|11|VACCI|14|sect=Myrtillus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium cespitosum
| author  = Michx.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1803|V. caespitosum|Michx.}}
| sn1     = {{../sn|1818|V. geminiflorum|Kunth}}
| sn2     = {{../sn|1899|V. arbuscula|(A.Gray) Merriam|}}
| sn3     = {{../sn|1942|V. nivictum|Camp|}}
| sn4     = {{../sn|1942|V. paludicola|Camp|}}
&lt;!-- -- --&gt;
| en1     = Dwarf bilberry
| en2     = Dwarf blueberry
| fr1     = Airelle gazonnante
&lt;!-- -- --&gt;
| status  = Native
| status1 = Endangered
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1340|1|}}
| usda    = {{../usda|VACE|NN|}}
| vascan  = 
| gobot   = Vaccinium/cespitosum
| its-id  = 23587
| ars-id  = 40996
| fna-id  = 250065710
| tro-id  = 12300045
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium caespitosum 3-eheep (5097507665).jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Oxycoccus''======

'''Cranberries''' (''Vaccinium'' sect. ''Oxycoccus'') can be found in acidic bogs throughout the state.

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Oxycoccus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Cranberry|832|11|VACCI|14|sect=Oxycoccus|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium macrocarpon
| author  = Aiton
&lt;!-- -- --&gt;
| sn0     = {{../sn|1789|Vaccinium macrocarpon|Aiton}}
| sn1     = {{../sn|1803|V. oxycoccos|var=oblongifolium|var-au=Michx.}}
| sn2     = {{../sn|1805|Oxycoccus macrocarpos|(Aiton) Pers.}}
| sn3     = {{../sn|1805|O. palustris|var=macrocarpos|var-au=(Aiton) Pers.}}
| sn4     = {{../sn|1894|Schollera macrocarpos|(Aiton) Britton}}
&lt;!-- -- --&gt;
| en1     = Cranberry
| en2     = Large cranberry
| en3     = American cranberry
| fr1     = Canneberge à gros fruits
| fr2     = Ronce d'Amérique
| status  = Native
| status1 = Secure
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub, subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1305|5|}}
| usda    = {{../usda|VAMA|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23599
| ars-id  = 41030
| fna-id  = 250065705
| tro-id  = 12300051
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium macrocarpon01.jpg|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium oxycoccos
| author  = L.
&lt;!-- -- --&gt;
| sn0     = {{../sn|1753|Vaccinium oxycoccos|L.}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Cranberry
| en2     = Small cranberry
| en3     = European cranberry
| fr1     = Canneberge commune
| status  = Native
| status1 = Secure
| status2 = (circumboreal)
| c-rank  = 8
| nwi1    = OBL
| habit0  = Perennial
| habit1  = Shrub, subshrub
| light   = 
&lt;!-- -- --&gt;
| nyfa    = {{../nyfa|1321|5|}}
| usda    = {{../usda|VAOX|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 505635
| ars-id  = 41047
| fna-id  = 200016697
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccinium oxycoccos 09072.JPG|width1=120
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

======''Vaccinium'' sect. ''Polycodium''======

{{../table|order=Ericales|Ericaceae|Vaccinoideae|Vaccinieae||Vaccinium||Polycodium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../genus|Vaccinium|Deerberry|832|11|VACCI|14|sect=Polycodium|
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../taxon
| species = Vaccinium stamineum
| author  = L. (1753)
&lt;!-- -- --&gt;
| sn0     = {{../sn|||}}
| sn1     = {{../sn|||}}
| sn2     = {{../sn|||}}
&lt;!-- -- --&gt;
| en1     = Deerberry
| en2     = Southern-gooseberry
| status  = Native
| status1 = Secure
| nyfa    = {{../nyfa|6477|5|}}
| usda    = {{../usda|VAST|NN|}}
| vascan  = 
| gobot   = 
| its-id  = 23615
| ars-id  = 41061
| fna-id  = 242417403
| tro-id  = 
| ipn-id  = 
| lbj-id  = 
| nse-id  = 
| bna-id  = 
| map     =  nymap.svg
| image1  = Vaccineum stamineum 1120600.jpg
}}&lt;!-- ------------------------------------------------------------ --&gt;
{{../end table
}}&lt;!-- ------------------------------------------------------------ --&gt;

{{BookCat}}

==References==
{{Reflist}}</text>
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        <username>Ziv</username>
        <id>3267536</id>
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      <minor />
      <comment>([[c:GR|GR]])  [[File:Azul.png]] → [[File:Azul square.svg]] → File replacement: jpg/png/gif to svg vector image ([[c::c:GR]])</comment>
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My eyes are the colour blue.&lt;ref&gt;Gaynor, V (2013) The guide: &lt;/ref&gt; I LOVE P!nk!


[[File:PinkInVienna.JPG|thumbnail]]

[[File:Azul square.svg|thumbnail|Blue]]

References
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    <title>Lentis/Law Enforcement Access to Encrypted Data</title>
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        <username>AJMelton1105</username>
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      <comment>Finished Social Analysis sections for Digital Autonomy and Public Trust social values; adjusted some citations</comment>
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      <text bytes="59768" sha1="fuj33g0xu6mb5y8qzxsm85ytkqcnbh3" xml:space="preserve">Though once considered a munition&lt;ref&gt;US Department of State. (1992). Code of Federal Regulations. https://epic.org/crypto/export_controls/itar.html&lt;/ref&gt;, [[w: Encryption software | encrypted communications]] have become commonplace in commercially available software. [[w:Apple Inc. | Apple’s]] [[w: IMessage | iMessage]] system, for instance, is [[w: end-to-end encryption | end-to-end encrypted]], meaning that no-one (not even Apple) can read a message except the sender and the recipient&lt;ref&gt;Apple Corporation. (2011). iOS 5 Press Release. http://www.apple.com/pr/library/2011/06/06New-Version-of-iOS-Includes-Notification-Center-iMessage-Newsstand-Twitter-Integration-Among-200-New-Features.html&lt;/ref&gt;. Apple now also encrypts all data on its [[w:IOS | iOS]] devices,&lt;ref&gt;Apple Corporation. (2014). iOS Security Guide, September 2014. https://www.documentcloud.org/documents/1302613-ios-security-guide-sept-2014.html&lt;/ref&gt; and [[w:Google | Google]] does the same on its [[w: Android (operating system) | Android operating system]].&lt;ref&gt;Android Project. (2015). Full Disk Encryption. https://source.android.com/security/encryption/&lt;/ref&gt; These systems prevent anyone besides the device owner from accessing the data on the device. Law enforcement agencies have had problems with this technology, however, because it also prevents them from viewing the data on phones, even when they have a search warrant. In one case, Apple refused to open a locked [[w: IPhone | iPhone]] for the [[w: Federal Bureau of Investigation | FBI]] in a drug investigation,&lt;ref&gt;Appuzo, M., Sanger, D., Schmidt, M. (2015). Apple and Other Tech Companies Tangle with US Over Data Access. New York Times, September 8, 2015. http://www.nytimes.com/2015/09/08/us/politics/apple-and-other-tech-companies-tangle-with-us-over-access-to-data.html?_r=0 &lt;/ref&gt; stating that doing so was impossible. Law enforcement agencies have argued that tech companies should provide them with a [[w: Backdoor (computing) | “backdoor,”]] a special way to decrypt messages that can only be used by law enforcement and which would require a search warrant.&lt;ref&gt;Comey, J. (2015). http://www.theguardian.com/technology/2015/jul/08/fbi-chief-backdoor-access-encryption-isis&lt;/ref&gt; In this chapter, we examine the sociotechnical forces surrounding this proposal in the United States and its allies.

==Background==
Although relevant today, law enforcement's access to encrypted communications is not a new issue in the United States. As personal digital communication grew more common in the early 1990s, the Clinton Administration began to push for a way to &quot;preserve a government capability to conduct electronic surveillance in furtherance of legitimate law enforcement and national security interests.&quot;&lt;ref&gt;White House. (1993). Public encryption management. https://fas.org/irp/offdocs/pdd5.htm&lt;/ref&gt; There have been several subsequent proposals for policies directly addressing the issue, though a variety of technical and social issues have prevented any such policy from taking effect.

===Clipper Chip===
[[File:MYK-78 Clipper chip markings.jpg|thumb|Clipper Chip that could be included in the hardware of any personal computer or communication device]]
The Clinton Administration's first proposal was the Clipper Chip, a hardware device designed by the United States government that manufacturers would build into their computers. This used a system called [[w: Key escrow | &quot;key escrow&quot;]] where each chip would be given a corresponding key that could decrypt any message encrypted by the chip. The key would be held by the United States government and released to law enforcement when legally required.&lt;ref name=&quot;Blaze94&quot;&gt;Blaze, M. (1994). Protocol failure in the escrowed encryption standard. http://www.crypto.com/papers/eesproto.pdf&lt;/ref&gt; A variety of social and technical problems, such as the high monetary and economic costs of implementation&lt;ref name=&quot;Deutch96&quot;&gt;Deutch, J. (1996). Memorandum for the president. https://web.archive.org/web/20121015182952/http://www.foia.cia.gov/docs/DOC_0000239468/DOC_0000239468.pdf&lt;/ref&gt; and weaknesses in the encryption algorithm&lt;ref name=&quot;Blaze94 /&gt; meant that the Clipper Chip was never adopted.

===Key Recovery===
Following the Clipper Chip's failure, the Clinton Administration proposed a system it called &quot;key recovery.&quot; In this system, trusted third party organizations would hold users' keys and provide them to law enforcement upon request.&lt;ref name=&quot;Deutch96&quot; /&gt; At the time of this proposal encryption technology was still classified as a munition, making it illegal to export from the United States. With the backing of the CIA, the Clinton Administration proposed weakening these restrictions as long as companies agreed to develop key recovery technology.&lt;ref name=&quot;Deutch96&quot; /&gt; By doing this, the Administration hoped to make key recovery the standard domestically and globally, giving law enforcement and intelligence agencies access to encrypted data worldwide. 

Organized groups within the Clinton Administration had opposing views on this issue. Though the Administration, with the internal backing of the CIA, publicly supported this key recovery plan, the Justice Department internally opposed it, believing that the plan did not go far enough and proposing that the law be modified to only allow the export and import of encryption technology that worked with key recovery systems.&lt;ref name=&quot;Deutch96&quot; /&gt;

Outside of government, many prominent groups opposed the plan for largely the same reasons that they had opposed the Clipper Chip. Trade groups such as the [[w:Institute of Electrical and Electronics Engineers|Institute of Electrical and Electronics Engineers (IEEE)]] and [[w:Association for Computing Machinery|Association for Computing Machinery (ACM)]] opposed the plan on economic, technical, and moral grounds. They believed that these restrictions would put American companies at an economic disadvantage while ultimately proving insecure and threatening civil freedoms.&lt;ref&gt;IEEE and ACM. (1997). Letter to Senator John McCain. http://usacm.acm.org/images/documents/spna_letter.pdf&lt;/ref&gt; Ultimately legislation implementing the proposal never passed the Senate.&lt;ref&gt;McCain, J. (1997). Secure Public Networks Act. http://thomas.loc.gov/cgi-bin/bdquery/z?d105:s909:&lt;/ref&gt;

===Current Status===
Though there have been many proposals for extending law enforcement's access to encrypted data in the United States, no proposals the scale of the Clipper Chip or key recovery have come into effect. Companies and individuals can generally develop and use cryptography systems freely.

Recent criminal investigations where the United States Justice Department was unable to retrieve encrypted information from phones, along with concerns over terrorist attacks, have led policymakers such as Senate Judiciary Chair [[w:Chuck Grassley|Chuck Grassley]] and Deputy Attorney General [[w:Sally Yates|Sally Yates]] to renew calls for expanding law enforcement's access to encrypted data,&lt;ref&gt;Grassley, C. (2015). Grassley questions White House commitment to going dark solution. http://www.grassley.senate.gov/news/news-releases/grassley-questions-white-house-commitment-going-dark-solution&lt;/ref&gt; bringing the issue back into the public eye.

==Support for Expanded Law Enforcement Access==
In the United States and abroad, many organized groups want to give government agencies expanded power to read encrypted data in order to prevent and prosecute criminal activity. Government and law enforcement organizations are the most prominent of these groups.

=== The Obama Administration ===
In the aftermath of the [[w: November_2015_Paris_attacks|terrorist attacks in Paris]] and [[w: 2015_San_Bernardino_shooting|San Bernardino]], President Obama gave a national address about responding to future threats.&lt;ref&gt; White House. (2015). Address to the Nation by the President. https://www.whitehouse.gov/the-press-office/2015/12/06/address-nation-president &lt;/ref&gt;  One step in the plan is to prevent criminal activity on encrypted channels.&lt;ref&gt;Rampton, R. (2015). Obama appeals to Silicon Valley for help with online anti-extremist campaign. http://www.reuters.com/article/california-shooting-cyber-idUSKBN0TQ0A320151207&lt;/ref&gt;  However, it is unclear by what means the administration intends this to be accomplished.  Earlier in the year, the Obama administration conceded its fight to make tech companies add a [[w: Backdoor_(computing)|backdoor]] to encryption, agreeing that doing so would weaken defenses against foreign governments and cybercriminals.&lt;ref name=&quot;obama&quot;&gt;Perlroth, N. &amp; Sanger, D. (2015). Obama Won’t Seek Access to Encrypted User Data. http://www.nytimes.com/2015/10/11/us/politics/obama-wont-seek-access-to-encrypted-user-data.html&lt;/ref&gt;  One of the Administration’s fears is that requiring a backdoor for companies in the United States would set a similar precedent for American companies in other countries, such as China.&lt;ref name=&quot;obama&quot;&gt;&lt;/ref&gt;

=== Federal Bureau of Investigation ===
The Federal Bureau of Investigation (FBI) has recently fought with Apple over encryption in iMessage.&lt;ref&gt;Hern, A. (2015). Apple's Encryption Means It Can't Comply with US Court Order. http://www.theguardian.com/technology/2015/sep/08/apple-encryption-comply-us-court-order-iphone-imessage-justice&lt;/ref&gt;  In the newest versions of iOS, iMessage uses [[w: end-to-end encryption|end-to-end encryption]] so that only the sender and recipient can read messages&lt;ref name = &quot;Apple privacy&quot;&gt;Apple. (2015). Our Approach to Privacy. http://www.apple.com/privacy/approach-to-privacy/&lt;/ref&gt;, making it impossible for law enforcement to access encrypted content.  FBI Director [[w: James Comey|James Comey]] compared this to an unlockable door and said that a warrant should be all that is required to access this data&lt;ref&gt;Nakashima, E. (2015). With Court Order, Federal Judge Seeks to Fuel Debate About Data Encryption. https://www.washingtonpost.com/world/national-security/federal-judge-stokes-debate-about-data-encryption/2015/10/10/c75da20e-6f6f-11e5-9bfe-e59f5e244f92_story.html&lt;/ref&gt;.  Comey noted how a terrorist who attempted to attack an event in Texas in May 2015 had been communicating with other terrorists abroad using encrypted messages&lt;ref&gt;Harte, J. &amp; Volz, D. (2015). Shooter at Texas 'Draw Mohammed' Contest Messaged Foreign Militants: FBI. http://www.reuters.com/article/us-texas-shooting-comey-idUSKBN0TS2FU20151209&lt;/ref&gt;, implying that these messages would have given authorities advance notice of the attempt had they been intercepted and read.  The United States has no [[w: key disclosure law|key disclosure law]], unlike Canada, the UK, and France, effectively keeping these encrypted messages a secret.  While key disclosure allows agencies to access encrypted data, it only helps when the suspect is known.  It is useful for gathering evidence, but not for intelligence, when fast response times are necessary, especially in preventing terrorist attacks.

=== United Kingdom's Cameron Administration ===
Similar battles are happening outside the United States.  In the United Kingdom, Prime Minister [[w:David Cameron|David Cameron]] has spoken out against apps that send encrypted messages because most of them cannot be read by government.&lt;ref name=&quot;cameron&quot;&gt;Bienkov, A. (2015). David Cameron: Twitter and Facebook Privacy is Unsustainable. http://www.politics.co.uk/news/2015/06/30/david-cameron-twitter-and-facebook-privacy-is-unsustainable&lt;/ref&gt;  Cameron argued that, in the past, government agents could get a warrant and intercept phone calls or mail, but now there are channels that government cannot access, creating a “safe space” for criminals and terrorists.&lt;ref name=&quot;cameron&quot; /&gt;  Cameron would like legislation that bans apps that use encryption unless the app has a government backdoor.&lt;ref name=&quot;cameron&quot;&gt;&lt;/ref&gt;

==Opposition to Expanded Law Enforcement Access==

Not everyone is in favor of expanding the government's access to private encrypted data. According to a 2014 poll, a majority of Americans (54%) “disapprove” of bulk government collection of phone and internet records,&lt;ref&gt; Pew Research Center. (2014). Beyond Red vs. Blue: The Political Typology. http://www.people-press.org/files/2014/06/6-26-14-Political-Typology-release1.pdf &lt;/ref&gt; and presidential candidates as diverse as [[w: Bernie Sanders | Bernie Sanders]]&lt;ref&gt;https://berniesanders.com/issues/war-and-peace/&lt;/ref&gt; and [[w: Rand Paul | Rand Paul]]&lt;ref&gt;https://www.randpaul.com/issue/ending-nsa-spying&lt;/ref&gt; have spoken in favor of reining in the [[w: National Security Agency | National Security Agency’s ]] [[w: PRISM (surveillance program) | data collection programs]] during the 2016 election cycle. Most opposition comes from participants with concerns over privacy or technical issues, though other factors such as economic issues are also relevant. 

===Participants with Privacy Concerns===

Many opponents of government access to data do so on privacy grounds. For instance, [[w : Tim Cook | Apple CEO Tim Cook]], speaking about his company’s [[w: end-to-end encryption | end-to-end encrypted]] [[w: IMessage | iMessage]] service, said “we believe the contents of your text messages and your video chats is none of our business.”&lt;ref&gt;Cook, T. (2015). Speech at Electronic Privacy Information Center. http://techcrunch.com/2015/06/02/apples-tim-cook-delivers-blistering-speech-on-encryption-privacy/#.ulz3r9:kVGu&lt;/ref&gt; Groups like the [[w: Electronic Frontier Foundation | Electronic Frontier Foundation]]&lt;ref&gt;Electronic Frontier Foundation. (2015). https://www.eff.org&lt;/ref&gt; advocate for reducing government access to data on privacy grounds and enjoy reasonable support, including a budget in the tens of millions of dollars.&lt;ref&gt; Electronic Frontier Foundation. (2015). Annual Report. https://www.eff.org/about/annual-reports-and-financials &lt;/ref&gt;

Privacy concerns are not a new topic in the social sciences, but researchers have struggled even to define &quot;privacy.&quot;&lt;ref&gt;Solove, D. J. (2008). Understanding privacy.&lt;/ref&gt; Despite the difficulty in defining it, researchers have found that people place significant value on keeping their data private,&lt;ref&gt;Acquisti, A., John, L. K., &amp; Loewenstein, G. (2013). What is privacy worth?. The Journal of Legal Studies, 42(2), 249-274.&lt;/ref&gt; and concern over privacy is likely at least partially motivated by economic factors; companies like Apple even expressly use their protection of consumer privacy as a selling point.&lt;ref name = &quot;Apple privacy&quot;&gt;&lt;/ref&gt;

===Participants with Technical Concerns===

Many participants oppose weakening encryption on technical grounds. The basic argument is that a[[w: Backdoor (computing) | backdoor]] for law enforcement means that the encryption cannot be perfectly secure, because an attacker could use the backdoor to gain entry. As the[[w:  Information Technology Industry Council | Information Technology Industry Council]] said in a statement following calls to add backdoors after the [[w: November 2015 Paris attacks | Paris attacks ]] in November 2015, weakening encryption by adding backdoors “simply doesn’t make sense.”&lt;ref&gt;Information Technology Industry Council. (2015). http://www.reuters.com/article/us-tech-encryption-idUSKCN0T82SS20151119&lt;/ref&gt; This is the same argument that security experts used when the[[w: Clipper Chip | Clipper Chip]] was proposed, and which ultimately helped defeat it.&lt;ref&gt;Abelson, H., Anderson, R. N., Bellovin, S. M., Benaloh, J., Blaze, M., Diffie, W., ... &amp; Schneier, B. (1997). The risks of key recovery, key escrow, and trusted third-party encryption.&lt;/ref&gt;

Opponents also contend that adding backdoors to encryption systems will not actually help law enforcement defeat threats. Nate Cardozo, staff attorney for the Electronic Frontier Foundation, has said that &quot;intel agencies are drowning in data... It's not about having enough data; it's a matter of not knowing what to do with the data they already have.&quot;&lt;ref&gt;Cardozo, N. (2015). http://www.wired.com/2015/11/paris-attacks-cia-director-john-brennan-what-he-gets-wrong-about-encryption-backdoors/ &lt;/ref&gt; Reports that the 2015 Paris attacks (which have been cited in proposals for backdoors, including by[[w: Central Intelligence Agency | CIA]] director[[w: John O. Brennan | John Brennan]])&lt;ref&gt; Brennan, J. (2015). Speech at Center for Strategic International Studies. http://www.defenseone.com/technology/2015/11/brennan-paris-wakeup-call-europe-encryption/123732/ &lt;/ref&gt; were planned using unencrypted communications only make this claim stronger.&lt;ref&gt; Froomkin, D. (2015). Signs point to unencrypted communications between terror suspects. https://theintercept.com/2015/11/18/signs-point-to-unencrypted-communications-between-terror-suspects/ &lt;/ref&gt;

== Legal Framework and the Battle of Social Values ==
In legal settings, social values often come under pressure when two or more groups hold competing interests. Several court cases have influenced controversial interpretations of widespread key social values, adding legal, political, and social complexity to the question of how much access law enforcement should have to encrypted data - and when that access becomes too much.

=== &lt;u&gt;United States Constitution: The Foundation of Our Rights&lt;/u&gt; ===
Almost every court case listed in this chapter either cites, is influenced by, or challenges two fundamental constitutional amendments. Before the interpretations of these amendments are discussed and analyzed, it is important to clearly establish what is currently written to understand how each argument may or may not align with our [[wikipedia:List_of_amendments_to_the_Constitution_of_the_United_States|constitutional rights]]:

==== 4th Amendment ====
The 4th Amendment of the United States Constitution states:

&quot;The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.&quot;&lt;ref&gt;{{Cite web |title=The 4th Amendment of the U.S. Constitution |url=https://constitutioncenter.org/the-constitution/amendments/amendment-iv |website=National Constitution Center – constitutioncenter.org |language=en}}&lt;/ref&gt;

In several court cases, the 4th Amendment was cited for its violation in law enforcement seizing several &quot;things&quot; being seized or searched, typically warrantless. The verdicts for these court cases typically protected individuals and established the legal need for a warrant to seize and search encrypted devices and data. Over the past 50 years, the amendment and its interpretation has remained consistent, protecting individuals and groups' dignity, privacy, and autonomy.

==== 5th Amendment ====
The 5th Amendment of the United States Constitution states:

&quot;No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.&quot;&lt;ref&gt;{{Cite web |title=The 5th Amendment of the U.S. Constitution |url=https://constitutioncenter.org/the-constitution/amendments/amendment-v |website=National Constitution Center – constitutioncenter.org |language=en}}&lt;/ref&gt;

The 5th Amendment protects individuals from being compelled to provide testimonial evidence against themselves ([[wikipedia:Self-incrimination|self-incrimination)]] a protection that becomes particularly important in cases involving encrypted devices and data. It is also well known for its guarantee of [[wikipedia:Due_Process_Clause|proper due process]], which underpins how searches, seizures, and arrests must be conducted. Courts have repeatedly examined whether entering a passcode or decrypting data communicates knowledge, such as ownership, control, or the existence of specific files, and therefore qualifies as a testimonial. When decryption would reveal new information to law enforcement, the 5th Amendment applies; when the government can already describe the evidence with reasonable particularity, courts have sometimes permitted compelled access.

=== &lt;u&gt;Cases on Personal Privacy vs. Public Safety&lt;/u&gt; ===
The following court cases ''Smith v. Maryland, U.S. v. Cotterman, Riley v. California,'' and ''Carpenter v. U.S.'' highlight the evolving legal battle over how far the government may go in accessing digital information in the name of public safety and justice. Together, they illustrate how constitutional protections written in the 18th century are continually reinterpreted to address current technologies, from cell phone applications to location tracking. These cases also go beyond legal bounds and address competing social values, where on one side is law enforcement's interest in preventing harm and investigating crime while on the other side is society's growing expectation of personal privacy extends to a digital space. As each case redefines the balance between these values, it also influences how U.S. citizens understand their legal rights and the conditions under which those rights can be limited or lost.

==== ''Smith v. Maryland'' (1979) ====
In [[wikipedia:Smith_v._Maryland|''Smith v. Maryland'']], Michael Lee Smith was arrested and indicted for robbing Patricia McDonough. To confirm Smith's identity after cross-referencing against the victim's description, the police asked the telephone company to install a [[wikipedia:Pen_register|pen register]]. However, they did so without a warrant or court order, since they believed Smith was the source of threatening calls made to McDonough. Once the pen register confirmed outgoing numbers, police obtained a warrant to search Smith's residence, where they seized the phonebook and identified Smith to be the robber. At trial, Smith sought to suppress the evidence collected by the pen register since the police did not obtain a warrant to seize his telecommunication information &quot;prior to its installation,&quot; which he claimed violated his 4th Amendment protection against unreasonable searches and seizures. On June 20, 1979, the court ruled in favor of Maryland, holding that Smith had no &quot;legitimate expectation of privacy&quot; in the numbers he dialed because he voluntarily conveyed that information to the phone company, which they recorded as a part of their regular business practices. &lt;ref&gt;{{Cite web |title=Smith v. Maryland, 442 U.S. 735 (1979) |url=https://supreme.justia.com/cases/federal/us/442/735/ |website=Justia Law |language=en}}&lt;/ref&gt;

==== ''U.S. v. Cotterman'' (2013) ====
''[[wikipedia:United_States_v._Cotterman|U.S. v. Cotterman]]'' began when Howard Cotterman returned from Mexico and was flagged by the [[wikipedia:TECS|Treasury Enforcement Communications System (TECS)]] database due to prior sexual offenses. At Lukeville, Arizona port of entry, border patrol officers conducted two searches (the second after the TECS notification), finding two laptops and 3 cameras, which [[wikipedia:United_States_Immigration_and_Customs_Enforcement|Immigration and Customs Enforcement (ICE)]] later came and seized.&lt;ref&gt;{{Cite web |last=hlr |date=2014-01-17 |title=United States v. Cotterman |url=https://harvardlawreview.org/print/vol-127/ninth-circuit-holds-forensic-search-of-laptop-seized-at-border-requires-showing-of-reasonable-suspicion-ae-united-states-v-cotterman-709-f-3d-952-9th-cir-2013-en-banc/ |access-date=2026-04-15 |website=Harvard Law Review |language=en-US}}&lt;/ref&gt; After conducting an off-site digital forensic search, the government found hundreds of files containing child pornography on Cotterman's devices, and they indicted him for possession of such digital content. Cotterman motioned to suppress the evidence gained from the devices, claiming the forensic search violated the 4th Amendment. The United States argued that there are exceptions to the 4th Amendment regarding border searches. After being granted then reversed, the case was reheard [https://www.law.cornell.edu/wex/en_banc en banc]. The Ninth Circuit en banc finally held that the United States violated Cotterman's 4th Amendment rights, saying that forensic, comprehensive searches of digital devices at the border require reasonable suspicion, recognizing the heightened privacy interests in electronic data.

==== ''Riley v. California'' (2014) ====
In ''[[wikipedia:Riley_v._California|Riley v. California]],'' David Riley was stopped for a traffic violation, arrested on weapons charges, and had his cell phone seized and searched without a warrant, linking Riley to gang affiliation and involvement in a prior shooting. Riley moved to suppress the phone evidence, but the lower court denied his motion. A similar case, ''U.S. v. Wurie'' (2014), involved a similar warrantless search of a flip phone that linked Wurie to a drug stash in his apartment.&lt;ref&gt;{{Cite web |title=United States v. Wurie |url=https://www.law.cornell.edu/supct/cert/13-212 |website=LII / Legal Information Institute |language=en}}&lt;/ref&gt; The Supreme Court reviewed both cases together and ultimately decided that &quot;the police generally may not, without a warrant, search digital information on a cell phone seized from an individual who has been arrested&quot;&lt;ref&gt;{{Cite web |title=Riley v. California, 573 U.S. 373 (2014) |url=https://supreme.justia.com/cases/federal/us/573/373/ |website=Justia Law |language=en}}&lt;/ref&gt;. As a result, Riley's conviction was reversed while Wurie's case ruling was sustained.

==== ''Carpenter v. U.S.'' (2018) ====
''[[wikipedia:Carpenter_v._United_States|Carpenter v. U.S.]]'' is similar to ''Riley v. California,'' in regard to how both involved data associated with cell phones. However, in the Carpenter case, police arrested four men with connection to an armed robbery. One of the men confessed and gave his number and all of the numbers of the other participants to the FBI, which they used to obtain orders to gain transactional records on each phone number. This was granted under the [[wikipedia:Stored_Communications_Act|Stored Communications Act, 18 U.S.C. 2703(d)]], which provides the government the ability to &quot;require the disclosure of certain telecommunication records when 'specific and articulable facts show[] that there are reasonable grounds to believe that the contents of a wire or electronic communication, or the records or other information sought, are relevant and material to an ongoing criminal investigation.'&quot;&lt;ref&gt;Carpenter v. United States. (n.d.). ''Oyez''. &lt;nowiki&gt;https://www.oyez.org/cases/2017/16-402&lt;/nowiki&gt;&lt;/ref&gt; The transactional records included evidential data about phone call location data and other [[wikipedia:Cell_site|cell site]] location information (CSLI). Timothy Carpenter moved to suppress the evidence gained from the CSLI data, arguing it violated the 4th Amendment. Ultimately, the Supreme Court narrowly sided with Carpenter, stating the warrantless search and seizure of phone users' location data violated the 4th Amendment, but this case also extended the [[wikipedia:Third-party_doctrine|third-party doctrine]], which was originally established in ''Smith v. Maryland'' and ''U.S. v. Miller'' (1976).

==== Social Analysis: Privilege of Privacy Amongst Individuals ====
While ''Smith v. Maryland'' established early boundaries for digital privacy through the [[wikipedia:Third-party_doctrine|third-party doctrine]], the struggle between digital privacy and the government's duty to ensure public safety and promote the general welfare remains foundationally contested. Later cases ''U.S. v. Cotterman, Riley v. California, and Carpenter v. U.S.'' shifted the legal landscape by recognizing that digital privacy is not merely limited to or defined by location, device, or type of data. Further, these decisions affirm that modern technology and encrypted data function as extensions of one's identity: &quot;phones are no longer merely phones, but important communication ...[and] data centers, and perhaps most significantly, loci for the digital culture.&quot;&lt;ref&gt;Steven I. Friedland, Riley v. California and the Stickiness Principle, 14 ''Duke Law &amp; Technology Review'' 121-139 (2016)&lt;/ref&gt; In a world where personal identity, social life, and daily conduct are increasingly mediated through digital platforms, individuals carry with them a reasonable expectation of privacy that is qualitatively different from what was considered in the past.

However, this expectation is not universally accepted or consistently interpreted. The ability to protect one's digital privacy, through encryption, secure devices, and knowledge of digital rights, often underlines developing social privileges. As governments and courts continue to define the limitations and applications of privacy in a rapidly evolving cyber-physical space, tensions will continue to emerge between the trend of expanding privacy protections and ensuring that privacy is not viewed as a hindrance to justice, safety, or accountability. These tensions have been heightened by the recent rise in AI surveillance and automated data decryption services, where, as the [[wikipedia:American_Civil_Liberties_Union|ACLU]] notes, &quot;the automated nature of these systems means that traditional limitations on police resources—once a vital check on government overreach—have been effectively eliminated.&quot;&lt;ref&gt;{{Cite web |date=February 12, 2026 |title=AI-Powered Surveillance Is Turning the United States into a Digital Police State. Now is the Time to Stop It. |url=https://www.aclum.org/publications/ai-powered-surveillance-is-turning-the-united-states-into-a-digital-police-state-now-is-the-time-to-stop-it/ |website=ACLU of Massachusetts |language=en-US}}&lt;/ref&gt; These cases reveal that privacy is not only a legal right but a social resource - one that is unevenly accessible, persistently contested, and potentially constrained depending on the level of influence and position one has within society, public institutions, and digital systems.

=== &lt;u&gt;Cases on Digital Autonomy vs. Law Enforcement's Authority&lt;/u&gt; ===
The following court cases highlight a nearly twenty-year trend in how courts navigate the tension between personal digital autonomy and expanding authority of law enforcement. Central to this trend is the forgone conclusion doctrine, a limited exception to the 5th Amendment's Self-Incrimination Clause that can compel an individual to release encrypted data and passcodes if three factors are met: 

# The Government has knowledge of the existence of the evidence demanded
# The defendant possessed or controlled the evidence, and
# The evidence is authentic.

The government must also be able to describe the evidence with &quot;reasonable particularity.&quot; &lt;ref&gt;{{Cite web |title=Foregone Conclusion Doctrine Allows Government to Make Criminal Defendant Disclose Computer Password |url=https://goldsteinmehta.com/blog/foregone-conclusion-doctrine |website=Goldstein Mehta LLC |language=en-US}}&lt;/ref&gt; Although this doctrine is framed as a narrow case-by-case rule, its application in the digital era effectively shifts power toward law enforcement in situations where devices and encrypted data act as extensions of personal identity, memory, and autonomy. As encryption continues to become a default feature of everyday technology, compelled-decryption cases raise questions about how much control individuals retain over their private data and whether constitutional protections can keep up with rapidly evolving digital systems. While many courts, often citing ''In re Boucher'', have supported decryption under the doctrine, cases such as ''U.S. v. Doe'' and the ''Oakland Biometrics Case'' emphasizes that society still recognizes meaningful limits on government authority and continue to affirm the social and legal value of personal digital autonomy.

==== ''In re Boucher Case'' (2009) ====
The [[wikipedia:In_re_Boucher|''In re Grand Jury Subpoena to Boucher'' case]] arose when Sebastien Boucher was arrested at the border between Canada and Vermont, where an officer's secondary inspection led to the discovery of child pornography on Boucher's computer. However, after seizing the laptop and once the computer was shut down, the hard drive's content could not be viewed again without entering in a passphrase. A Secret Service computer forensics expert testified that the drive had been encrypted using [[wikipedia:Pretty_Good_Privacy|Pretty Good Privacy (PGP)]] software and there was no backdoor for this type of encryption software.&lt;ref&gt;Pfefferkorn, Riana, In Re Boucher: Hard Drives Make Pretty Good Law? Encryption Passphrases as Testimonial Evidence Under the Fifth Amendment (February 1, 2009). Available at SSRN: &lt;nowiki&gt;https://ssrn.com/abstract=1883697&lt;/nowiki&gt; or &lt;nowiki&gt;http://dx.doi.org/10.2139/ssrn.1883697&lt;/nowiki&gt;&lt;/ref&gt; In response, the grand jury subpoenaed Boucher to release the passphrases to the government, in which Boucher successfully moved to quash the subpoena, initially. Boucher argued that this violated his 5th Amendment rights against self-incrimination, but after further review the U.S. District Court for Vermont reversed the initial grant. The reasoning for this reversal is that since the files sought allegedly contain child pornographic content, the entry of the password is incriminating and therefore the real question is whether the &quot;subpoena seeks testimonial communication.&quot; Also, both parties agreed that the content in the files did not share the same 5th Amendment privileges as revealing the password, as the files were &quot;voluntarily prepared&quot; and not testimonial.&lt;ref&gt;In Re Boucher (United States District Court for the District of Vermont November 29, 2009). &lt;nowiki&gt;https://www.volokh.com/files/Boucher.pdf&lt;/nowiki&gt;.&lt;/ref&gt; The real question lies in whether entry of the password, and ultimately handing the files over to the government, would be testimonial and a violation of the 5th Amendment. The final verdict states that &quot;Boucher has no act of production privilege to refuse to provide the grand jury with an unencrypted version of the Z drive of his computer,&quot; and that &quot;the government may not make use of Boucher's act of production to authenticate the unencrypted Z drive or its contents either before the grand jury or a petit jury,&quot; ultimately legally compelling Boucher to decrypt his device and hand the files to the government.&lt;ref&gt;In re Grand Jury Subpoena to Sebastien Boucher, &lt;nowiki&gt;https://yalelawtech.org/wp-content/uploads/2010/02/boucher_-_sessions_-_appeal.pdf&lt;/nowiki&gt; (United States District Court for the District of Vermont. February 19, 2009).&lt;/ref&gt; This case became a significant landmark of [[wikipedia:Key_disclosure_law|key disclosure law]] through being regarded as [[wikipedia:Case_law|case law]], and its influence is significant in several other court cases discussed in the battle of self autonomy and authority.

==== ''U.S. v. Doe'' (Dated March 25, 2011) ====
In the case of ''U.S. v. Doe,'' FBI agents seized two laptops and five external hard drives from a man under investigation, but the data was stored behind an encryption program assumed to be [[wikipedia:TrueCrypt|TrueCrypt]]. When ordered by a grand jury to produce unencrypted contents of the hard drives, he invoked the 5th Amendment, claiming a breach of protections against self-incrimination. Specifically, Doe argued that &quot;the Government’s use of the decrypted contents of the hard drives would constitute derivative use of his immunized testimony, use not protected by the district court’s grant of immunity.&quot;&lt;ref&gt;{{Cite web |title=In Re: Grand Jury Subpoena Duces Tecum Dated March 25, 2011, USA v. John Doe, No. 11-12268 (11th Cir. 2012) |url=https://law.justia.com/cases/federal/appellate-courts/ca11/11-12268/11-12268-2012-02-23.html |website=Justia Law |language=en}}&lt;/ref&gt; Doe also explained he was unable to decrypt the hard drive (which was disregarded in this case as well as the appeal case), but ultimately the court held Doe in contempt and jailed him. After appealing the decision, the circuit court ruled that decrypting the data is testimonial and therefore protected by the 5th Amendment (in favor of Doe), a comparably different outcome than seen in ''In re Boucher'' and ''U.S. v. Fricosu,'' and a significant limitation on the foregone conclusion doctrine.

==== ''U.S. v. Fricosu'' (2012) ====
In ''[[wikipedia:United_States_v._Fricosu|U.S. v. Fricosu]],'' the FBI obtained a search warrant for Colorado resident Ramona Fricosu under suspicion of mortgage fraud. During the search, agents seized six computers. One of the six computers seized was password-encrypted by a program called PGP Desktop, which &quot;provides easy to use and secure encryption to protect sensitive data on your laptop or desktop computers.&quot; &lt;ref&gt;{{Cite web |title=What is Symantec Encryption Desktop? (formerly known as PGP Desktop Encryption) |url=https://help.uillinois.edu/TDClient/37/uic/KB/PrintArticle?ID=776 |website=help.uillinois.edu}}&lt;/ref&gt; The day after the execution of the search warrant, Fricosu made a call to her incarcerated divorced ex-husband, and the recorded conversation revealed that both her and her ex-husband were aware of the content in her laptop, as well as the encryption program and her legal instruction to not provide any passwords, as protected by the 5th Amendment. In closing, several facts of the case (including file names, location of laptop, and connection to her ex-husband and his convictions) led the court to hold that Fricosu is legally compelled to provide an unencrypted copy of the files to the government as evidence, aligning with the ''In re Boucher'' case and applying the forgone conclusion doctrine.&lt;ref&gt;{{Cite web |title=United States v. Fricosu, Criminal Case No. 10–cr–00509–REB–02. {{!}} D. Colo., Judgment, Law, casemine.com |url=https://www.casemine.com/judgement/us/5914e216add7b049348ee62f |website=https://www.casemine.com |language=en}}&lt;/ref&gt;

==== ''Commonwealth v. Gelfgatt'' (2012) ====
''Commonwealth v. Gelfgatt'' is another case regarding compelled decryption via [[wikipedia:Key_disclosure_law|key disclosure laws]] influenced by ''In re Boucher''. Leon I. Gelfgatt, an attorney, was indicted with seventeen counts of forgery of a document, seventeen counts of [[wikipedia:Uttering|uttering]] a forged instrument, and three counts of attempting to commit larceny. The charges came from allegations against Gelfgatt that he conducted a scheme, using several computers, to receive ill-obtained funds that he used to pay mortgage loans for residential properties. The Commonwealth asked the judge to grant legal compulsion of Gelfgatt to decrypt his computers, in which the judge denied originally but asked for higher-level advisement on the question:&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Commonwealth v. Gelfgatt |url=https://law.justia.com/cases/massachusetts/supreme-court/2014/sjc-11358.html |website=Justia Law |language=en}}&lt;/ref&gt; 

&quot;Can the defendant be compelled pursuant to the Commonwealth's proposed protocol to provide his key to seized encrypted digital evidence despite the rights and protections provided by the Fifth Amendment to the United States Constitution and Article Twelve of the Massachusetts Declaration of Rights?&quot;&lt;ref name=&quot;:0&quot; /&gt;

The Supreme Judicial Court, after further review of the law in question, reversed the judges denial to the Commonwealth's request for Gelfgatt's compulsion, implying in their verdict that &quot;the government can ... compel the defendant to produce it [digital evidence] where it already knows what the information is, and the defendant’s disclosure would add little or nothing to what the government’s knowledge,&quot; another application of the forgone conclusion doctrine.&lt;ref&gt;{{Cite web |last=Fernandez |first=Zoraida |date=2019-03-20 |title=Can the Commonwealth Compel You to Unlock Your Cell Phone? SJC Clarifies Certain Circumstances in Which it Can |url=https://www.bostonlawyerblog.com/can-the-commonwealth-compel-you-to-unlock-your-cell-phone-sjc-clarifies-certain-circumstances-in-which-it-can/ |website=Boston Lawyer Blog |language=en-US}}&lt;/ref&gt;

==== ''Oakland Biometrics Case'' (2019) ====
The ''In re: Search of A Residence in Oakland, California'' case presents a notably different application of the 4th and 5th Amendments in a cyber-physical system, compared to cases like ''In re Boucher'', ''U.S. v. Fricosu'', and ''Commonwealth v. Gelfgatt.'' In this case, the government was investigating two individuals suspected of committing extortion scheme conducted through Facebook Messenger. The government's warrant application asked for the authority to search the Oakland residence and seize all digital devices on the premise, and to compel any individual present to unlock the devices through [[wikipedia:Biometrics|biometric]] features, even if they were not the subjects of the warrant. The magistrate judge denied the application, finding that &quot;while the Government had probable cause to search the premises under the Fourth Amendment, it did not have probable cause to search and seize ''all'' digital devices nor compel their unlocking using biometric features because the search warrant application was not limited to digital devices owned or controlled by the suspects.&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite news |last=Nowell D. Bamberger, Melissa Gohlke, Sameer Jaywant |date=2019-01-23 |title=Court Holds That 5th Amendment Self-Incrimination Privilege Precludes Compelling Fingerprint or Facial Recognition Access to Digital Devices {{!}} Cleary Enforcement Watch |language=en-US |work=Cleary Enforcement Watch |url=https://www.clearyenforcementwatch.com/2019/01/court-holds-5th-amendment-self-incrimination-privilege-precludes-compelling-fingerprint-facial-recognition-access-digital-devices/}}&lt;/ref&gt;

Citing ''Carpenter v. U.S.'' in 2018, the court emphasized how &quot;the advancement of technology should not be accompanied by an erosion in the constitutional rights of the people, and that courts should vigilantly guard against such erosion.&quot;&lt;ref name=&quot;:1&quot; /&gt; In its final reasoning, the court held that biometric data cannot be compelled to unlock devices, including &quot;fingers, thumbs, facial recognition, optical/iris, or any other biometric feature to unlock electronic devices.&quot; &lt;ref&gt;{{Cite web |last=Baron |first=Ethan |last2=News |first2=The Mercury |date=2019-01-16 |title=Biometrics Ruling Limits How Police Can Access a Smartphone |url=https://www.govtech.com/products/Biometrics-Ruling-Limits-How-Police-Can-Access-a-Smartphone.html |website=GovTech |language=en}}&lt;/ref&gt;

==== ''State v. Andrews'' (2020) ====
''State v. Andrews'' began with the arrest of Quincy Lowery for suspected involvement in a New Jersey drug-trafficking ring. However, Lowery confessed officer Robert Andrews helped him evade detection by warning Lowery of the wiretap on his phone and the GPS tracker on his Jeep. Lowery even showed the investigators the text messages to prove such evidence, but because he reset his phone a month earlier, most of the text messages were wiped. Officers requested Andrews to hand over his two iPhones, in which Andrews did so voluntarily without giving out the passwords to unlock them. Without a way to effectively decrypt the passwords, the State obtained warrants for the phones' contents and motioned for Andrews to reveal the passwords. Andrews refused citing the 5th Amendment's Self-Incrimination Clause. Once the trial court supported the State's motion and Andrews appealed the courts decision, the appeals court upheld the trial court's decision, stating &quot;because the State had already established that Andrews owned, possessed, and controlled his iPhones, his act of producing their passcodes, while 'testimonial,' fell under this 'foregone conclusion' exception.&quot;&lt;ref name=&quot;:2&quot;&gt;{{Cite web |last= |date=2021-04-12 |title=State v. Andrews |url=https://harvardlawreview.org/print/vol-134/state-v-andrews/ |website=Harvard Law Review |language=en-US}}&lt;/ref&gt; Andrews appealed the decision again, and the New Jersey Supreme Court affirmed both lower courts, holding &quot;that the Fifth Amendment privilege was overcome by the government’s knowledge of three key facts: the passcodes’ existence, their possession by the defendant, and their authenticity,&quot;&lt;ref name=&quot;:2&quot; /&gt; ultimately compelling Andrews to reveal the passwords to the iPhones.

==== Social Analysis: Increasing Authority at the Cost of Digital Autonomy ====
The current U.S. legal landscape underlines a shift in how society balances personal digital autonomy against government authority. As courts increasingly apply the forgone conclusion doctrine to encrypted devices and data, they reshape social expectations about privacy, identity, and governmental power. With digital devices now functioning as extensions of our being and lives, compelled decryption has social implications far beyond traditional evidence production. Cases such as ''In re Boucher, U.S. v. Fricosu, Commonwealth v. Gelfgatt,'' and ''State v. Andrews'' have normalized broader government access to encrypted data, strengthening a social narrative that national security interests outweigh individual control over digital identity. This trend can erode trust and accountability, particularly in communities already skeptical of law enforcement, and may create chilling effects on expression, association, and political participation. As compelled access becomes more routine, society risks normalizing a surveillance-based relationship between citizens and the government.

However, cases like ''U.S. v. Doe'' and the ''Oakland Biometrics Case'' demonstrate that courts still recognize the social value of digital autonomy and the importance of limiting the government's power when compelled decryption becomes indistinguishable from compelled testimony, or otherwise self-incrimination. Advocacies, professional organizations, and non-profits also play a crucial role in shaping both the public's view and judicial reasoning. For example, the [[wikipedia:Electronic_Frontier_Foundation|Electric Frontier Foundation (EFF)]] wrote amicus briefs (a legal document providing advice, information, or arguments related to a case) for both ''U.S. v. Fricosu'' and ''U.S. v. Doe'', consistently arguing that the 5th Amendment protected individuals from all forms of compelled decryption. They emphasized that the different outcomes in these cases resulted from not doctrinal inconsistency but from the defendant's own disclosures, underlining how &quot;silence is golden.&quot; Noting how &quot;Boucher talked to law enforcement&quot; and &quot;Fricosu talked to her ex-husband and co-defendant in jail,&quot; the EFF claims &quot;it was this talking that defeated their Fifth Amendment privilege through the foregone conclusion doctrine.&quot;&lt;ref&gt;{{Cite web |last=Fakhoury |first=Hanni |date=2012-03-07 |title=A Tale of Two Encryption Cases |url=https://www.eff.org/deeplinks/2012/03/tale-two-encryption-cases |website=Electronic Frontier Foundation |language=en}}&lt;/ref&gt; Together, these cases and the various proponents of conflicting civil liberties, often driven by strong social values, illustrate a significant moment in the definition of digital autonomy, one which courts, civil liberties organizations, and the public are actively shaping the future boundaries of state authority in an increasingly data-driven society.

=== &lt;u&gt;Cases on Managing Public Trust &amp; Accountability&lt;/u&gt; ===
The final cases in this chapter addresses the critical but often overlooked social values of public trust and accountability, particularly in the management of personal data. As emerging technologies such as [[wikipedia:Quantum_computing|quantum computing]], [[wikipedia:Artificial_intelligence|AI]], and [[wikipedia:Edge_computing|edge computing]] reshape how information is stored and accessed, public trust in institutions responsible for protecting digital rights has become increasingly fragile. People generally rely on governments, courts, and large organizations to act as stewards of their data and to ensure that justice is upheld when digital rights are threatened. However, the following cases reveal a series of paradoxes, where entities display conflicting interests in data management and the legal framework governing their practices lag behind the change in technology. These tensions propose significant questions in who truly manages data and, by extension, who the public can trust to enact accountability in an increasingly digital world.

==== ''U.S. v. Microsoft'' (2018) ====
[[File:BalticServers data center.jpg|thumb|Data centers hold vast amounts of information about millions of individuals and organizations, making them a significant source of digital evidence.]]
In ''[[wikipedia:Microsoft_Corp._v._United_States|U.S. v. Microsoft]],'' the FBI in 2013 obtained a warrant under the [[wikipedia:Stored_Communications_Act|Stored Communications Act]] requiring Microsoft to release encrypted email and other information connected to one of their customers that was under suspicion of being involved in illegal drug trafficking. Microsoft refused to comply, arguing these emails were stored in its Dublin, Ireland data center and that the SCA did not authorize U.S. courts to compel production of data outside of its borders. Microsoft moved to quash the warrant, questioning how a United States federal law can be enforced outside of U.S. borders. While the move to quash the warrant was denied by the district court, Microsoft appealed the ruling and was granted their motion to quash it by the Second Circuit court. The U.S. Department of Justice petitioned to review the new decision, claiming that the warrant was &quot;a domestic application of the SCA,&quot; and that the appealed decision &quot;would be 'impractical and detrimental to law enforcement' and that the use of the SCA to obtain user data stored abroad would respect 'the United States’ international obligations.'&quot;&lt;ref&gt;{{Cite web |title=United States v. Microsoft |url=https://epic.org/documents/united-states-v-microsoft/ |website=EPIC - Electronic Privacy Information Center |language=en-US}}&lt;/ref&gt; After the Supreme Court granted the petition and planned to hold another hearing, the federal government enacted the [[wikipedia:CLOUD_Act|Clarifying Lawful Overseas Use of Data Act (CLOUD Act)]], which in essence established that U.S. service providers must produce data within their possession, custody, or control regardless of where the data is physically stored. This ultimately led to the case being dismissed as unnecessary and Microsoft being legally compelled by the CLOUD Act to obtain and provide the data from its Irish datacenter, due to Microsoft being a U.S. company.&lt;ref&gt;{{Cite web |title=United States v. Microsoft Corp., 584 U.S. ___ (2018) |url=https://supreme.justia.com/cases/federal/us/584/17-2/ |website=Justia Law |language=en}}&lt;/ref&gt;&lt;ref&gt;United States v. Microsoft Corporation. (n.d.). ''Oyez''. &lt;nowiki&gt;https://www.oyez.org/cases/2017/17-2&lt;/nowiki&gt;&lt;/ref&gt; 

==== ''Van Buren v. U.S.'' (2021) ====
''[[wikipedia:Van_Buren_v._United_States|Van Buren v. U.S.]]'' addresses access to encrypted data very differently from ''U.S. v. Microsoft''. Georgia police sergeant Nathan Van Buren was bribed by Andrew Albo, who was cooperating with an FBI sting operation, to run a license plate search on a possible undercover cop. Van Buren complied to the request, and through the [[wikipedia:Crime_information_center|Georgia Crime Information Center (GCIC)]] he used his authorized credentials to look up the provided license plate to confirm the identity of an undercover cop. Although officers are trained that GCIC access is strictly limited to law enforcement purposes specifically, Van Buren technically accessed only information he was already authorized to view. He was charged with violating the [[wikipedia:Computer_Fraud_and_Abuse_Act|Computer Fraud &amp; Abuse Act (CFAA)]], which he petitioned for review claiming that he technically did not exceed access authority as a law enforcement official with proper credentials to access the GCIC.&lt;ref&gt;{{Cite web |title=Van Buren v. United States |url=https://epic.org/documents/van-buren-v-united-states/ |website=EPIC - Electronic Privacy Information Center |language=en-US}}&lt;/ref&gt; The court argued that &quot;the purpose in using the system was immaterial,&quot; and with there being no purpose limitation cited in the CFAA, &quot;Van Buren was authorized to access the database and di not violate the CFAA.&quot;&lt;ref name=&quot;:3&quot;&gt;Scott T. Lashway and Matthew M.K. Stein, ''Signs Inscribed on a Gate: The Impact of Van Buren v. United States on Civil Claims Under the Computer Fraud and Abuse Act'', 44 W. New Eng. L. Rev. 109 (2022), &lt;nowiki&gt;https://digitalcommons.law.wne.edu/lawreview/vol44/iss1/5&lt;/nowiki&gt;&lt;/ref&gt; 

The court also made three important notes regarding the outcome and implications of this case and the CFAA, stating as follows:

# Amending the CFAA to include a purpose-based limitation criminalizes ordinary conduct, such as when an employee uses a business computer to send and check personal emails.
# Purpose-based limitations are unstable and potentially unclear, blurring what is considered &quot;normal&quot; and &quot;criminal&quot; conduct.
# The CFAA was only concerned in Van Buren's credentials, not his purpose for accessing GCIC records. Future disputes may address how contracts, policies, and terms of service interact with the statute's access-based limitations.

==== Social Analysis: Continuously Balancing Restriction and Freedoms ====
As [[wikipedia:Thomas_Jefferson|Thomas Jefferson]] once stated, &quot;when a man assumes a public trust, he should consider himself as public property.&quot;&lt;ref&gt;{{Cite web |date=2015-07-01 |title=Basic Obligation of Public Service {{!}} U.S. Department of the Interior |url=https://www.doi.gov/ethics/basic-obligations-of-public-service |website=www.doi.gov |language=en}}&lt;/ref&gt; 

The principle behind this quote is that public institutions derive their legitimacy from the people and must remain accountable to them. Yet the modern legal framework surrounding encrypted data often moves in the opposite direction. In cases such as ''U.S. v. Microsoft,'' the government asserted authority beyond U.S. borders, raising important questions about which institutions, domestic or foreign, citizens are expected to trust with their digital lives. Microsoft and privacy advocates warned that if the U.S. can issue a warrant that legally compels decryption abroad, &quot;other countries will in turn use their own laws to seize data stored in the U.S&quot; which &quot;could put Americans’ emails (and other data) at risk of seizure by foreign governments.&quot;&lt;ref&gt;Kruse, N. (2018, March 1). ''U.S. v. Microsoft: Is Your Data and Privacy at Risk?''. Byte Back. &lt;nowiki&gt;https://www.bytebacklaw.com/2018/03/u-s-v-microsoft-is-your-data-and-privacy-at-risk/&lt;/nowiki&gt;&lt;/ref&gt; The CLOUD Act further complicates this landscape by shifting practical control of data from the government to private, for-profit corporations. Even when data is physically stored in another country, ownership and decision-making authority often remain with companies like Microsoft, Facebook, and Apple, who all supported the CLOUD Act and whose incentives are shaped not by democratic accountability but by reputation management, market pressures, and financial interests. This raises two currently unanswered questions of public trust: first, how can citizens trust governmental systems when jurisdictional boundaries blur and government authority extends beyond its traditional limits, and second, how can they trust corporations to act as responsible stewards of their data when their obligations to shareholders may conflict with the public's interest in privacy and autonomy?

Furthermore, ''Van Buren v. U.S.'' produces a paradoxical outcome, where individuals protected from being criminalized for everyday digital behavior while simultaneously exposing the public to greater risks of insider misuses of encrypted data by law enforcement. The CFAA was originally designed to prevent unauthorized access to sensitive information, but Van Buren narrowed its scope by rejecting any inquiry into a person's purpose for accessing data. In doing so, the court prevented the CFAA from becoming overly broad &quot;terms-of-service crime,&quot; yet it also removed a potential safeguard against officers who misuse their legitimate credentials to access personal information for improper reasons. As scholars note, the CFAA retains its force, &quot;even with the purpose inquiry foreclosed: access may not be authorization, and authorization or its limits could be inferred from context and terms of use.&quot;&lt;ref name=&quot;:3&quot; /&gt; Regardless, this case “does not definitively resolve the question of whether unauthorized access must be barred by a hardware or software gateway or if activity can become ‘unauthorized’ by a contractual ban&quot; and “largely for that reason, the practical effects on the CFAA’s civil enforcement provisions remain to be seen.”&lt;ref&gt;Melanie Assad, Van Buren v. United States: An Employer Defeat or Hacker’s Victory – Or Something in Between, 21 UIC REV. INTELL. PROP. L. 166 (2022).&lt;/ref&gt; From a public trust perspective, this creates a double-edged sword: the decision limits government overreach in one sense, yet it also leaves the public more vulnerable to the very actors entrusted with sensitive data, raising the difficult questions about accountability, oversight, and the social value of trust in law enforcement. 

Both ''U.S. v Microsoft'' and ''Van Buren v. U.S.'' offer different stories connected by their influence over public trust and accountability; one raises concerns about how far a government's authority extends and the inherent ownership of encrypted data by companies driven by the market and productivity, while the other is seen as a win for protecting individuals from accidental criminalization of normal activities and a loss for holding law enforcement accountable by removing an extra layer of protection through authorization and credentials. As technology changes and evolves, shaping the application of data encryption, management, and access, future litigations may arise, and the question of who will maintain the public's trust in data privacy and digital autonomy will most likely continue to be overlooked and potentially mishandled. All of these court cases point toward a future that will quickly challenge long-held and accepted social values, especially in regard to extending those values to encrypted digital or cyber-physical platforms, and it is up to all participants and interested parties to make these social values known and considered, whether it is noted by an advocacy, non-profit organization, for-profit corporation, law enforcement agencies, regular citizens, or the courts of law.

==Conclusion==

The struggle over encryption backdoors is far from over, and there will almost certainly be more developments on this topic in the future as new solutions are proposed and enacted or defeated. Although we have covered significant parts of the debate in the United States and the United Kingdom, participants in other places have different perspectives. Interesting additions to this chapter could include updates covering future developments and expansion to cover encryption worldwide.

==References==
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      <text bytes="60020" sha1="jk35aj0ztabfpgqfuojicgmj7vc554s" xml:space="preserve">Though once considered a munition&lt;ref&gt;US Department of State. (1992). Code of Federal Regulations. https://epic.org/crypto/export_controls/itar.html&lt;/ref&gt;, [[w: Encryption software | encrypted communications]] have become commonplace in commercially available software. [[w:Apple Inc. | Apple’s]] [[w: IMessage | iMessage]] system, for instance, is [[w: end-to-end encryption | end-to-end encrypted]], meaning that no-one (not even Apple) can read a message except the sender and the recipient&lt;ref&gt;Apple Corporation. (2011). iOS 5 Press Release. http://www.apple.com/pr/library/2011/06/06New-Version-of-iOS-Includes-Notification-Center-iMessage-Newsstand-Twitter-Integration-Among-200-New-Features.html&lt;/ref&gt;. Apple now also encrypts all data on its [[w:IOS | iOS]] devices,&lt;ref&gt;Apple Corporation. (2014). iOS Security Guide, September 2014. https://www.documentcloud.org/documents/1302613-ios-security-guide-sept-2014.html&lt;/ref&gt; and [[w:Google | Google]] does the same on its [[w: Android (operating system) | Android operating system]].&lt;ref&gt;Android Project. (2015). Full Disk Encryption. https://source.android.com/security/encryption/&lt;/ref&gt; These systems prevent anyone besides the device owner from accessing the data on the device. Law enforcement agencies have had problems with this technology, however, because it also prevents them from viewing the data on phones, even when they have a search warrant. In one case, Apple refused to open a locked [[w: IPhone | iPhone]] for the [[w: Federal Bureau of Investigation | FBI]] in a drug investigation,&lt;ref&gt;Appuzo, M., Sanger, D., Schmidt, M. (2015). Apple and Other Tech Companies Tangle with US Over Data Access. New York Times, September 8, 2015. http://www.nytimes.com/2015/09/08/us/politics/apple-and-other-tech-companies-tangle-with-us-over-access-to-data.html?_r=0 &lt;/ref&gt; stating that doing so was impossible. Law enforcement agencies have argued that tech companies should provide them with a [[w: Backdoor (computing) | “backdoor,”]] a special way to decrypt messages that can only be used by law enforcement and which would require a search warrant.&lt;ref&gt;Comey, J. (2015). http://www.theguardian.com/technology/2015/jul/08/fbi-chief-backdoor-access-encryption-isis&lt;/ref&gt; In this chapter, we examine the sociotechnical forces surrounding this proposal in the United States and its allies.

==Background==
Although relevant today, law enforcement's access to encrypted communications is not a new issue in the United States. As personal digital communication grew more common in the early 1990s, the Clinton Administration began to push for a way to &quot;preserve a government capability to conduct electronic surveillance in furtherance of legitimate law enforcement and national security interests.&quot;&lt;ref&gt;White House. (1993). Public encryption management. https://fas.org/irp/offdocs/pdd5.htm&lt;/ref&gt; There have been several subsequent proposals for policies directly addressing the issue, though a variety of technical and social issues have prevented any such policy from taking effect.

===Clipper Chip===
[[File:MYK-78 Clipper chip markings.jpg|thumb|Clipper Chip that could be included in the hardware of any personal computer or communication device]]
The Clinton Administration's first proposal was the Clipper Chip, a hardware device designed by the United States government that manufacturers would build into their computers. This used a system called [[w: Key escrow | &quot;key escrow&quot;]] where each chip would be given a corresponding key that could decrypt any message encrypted by the chip. The key would be held by the United States government and released to law enforcement when legally required.&lt;ref name=&quot;Blaze94&quot;&gt;Blaze, M. (1994). Protocol failure in the escrowed encryption standard. http://www.crypto.com/papers/eesproto.pdf&lt;/ref&gt; A variety of social and technical problems, such as the high monetary and economic costs of implementation&lt;ref name=&quot;Deutch96&quot;&gt;Deutch, J. (1996). Memorandum for the president. https://web.archive.org/web/20121015182952/http://www.foia.cia.gov/docs/DOC_0000239468/DOC_0000239468.pdf&lt;/ref&gt; and weaknesses in the encryption algorithm&lt;ref name=&quot;Blaze94 /&gt; meant that the Clipper Chip was never adopted.

===Key Recovery===
Following the Clipper Chip's failure, the Clinton Administration proposed a system it called &quot;key recovery.&quot; In this system, trusted third party organizations would hold users' keys and provide them to law enforcement upon request.&lt;ref name=&quot;Deutch96&quot; /&gt; At the time of this proposal encryption technology was still classified as a munition, making it illegal to export from the United States. With the backing of the CIA, the Clinton Administration proposed weakening these restrictions as long as companies agreed to develop key recovery technology.&lt;ref name=&quot;Deutch96&quot; /&gt; By doing this, the Administration hoped to make key recovery the standard domestically and globally, giving law enforcement and intelligence agencies access to encrypted data worldwide. 

Organized groups within the Clinton Administration had opposing views on this issue. Though the Administration, with the internal backing of the CIA, publicly supported this key recovery plan, the Justice Department internally opposed it, believing that the plan did not go far enough and proposing that the law be modified to only allow the export and import of encryption technology that worked with key recovery systems.&lt;ref name=&quot;Deutch96&quot; /&gt;

Outside of government, many prominent groups opposed the plan for largely the same reasons that they had opposed the Clipper Chip. Trade groups such as the [[w:Institute of Electrical and Electronics Engineers|Institute of Electrical and Electronics Engineers (IEEE)]] and [[w:Association for Computing Machinery|Association for Computing Machinery (ACM)]] opposed the plan on economic, technical, and moral grounds. They believed that these restrictions would put American companies at an economic disadvantage while ultimately proving insecure and threatening civil freedoms.&lt;ref&gt;IEEE and ACM. (1997). Letter to Senator John McCain. http://usacm.acm.org/images/documents/spna_letter.pdf&lt;/ref&gt; Ultimately legislation implementing the proposal never passed the Senate.&lt;ref&gt;McCain, J. (1997). Secure Public Networks Act. http://thomas.loc.gov/cgi-bin/bdquery/z?d105:s909:&lt;/ref&gt;

===Current Status===
Though there have been many proposals for extending law enforcement's access to encrypted data in the United States, no proposals the scale of the Clipper Chip or key recovery have come into effect. Companies and individuals can generally develop and use cryptography systems freely.

Recent criminal investigations where the United States Justice Department was unable to retrieve encrypted information from phones, along with concerns over terrorist attacks, have led policymakers such as Senate Judiciary Chair [[w:Chuck Grassley|Chuck Grassley]] and Deputy Attorney General [[w:Sally Yates|Sally Yates]] to renew calls for expanding law enforcement's access to encrypted data,&lt;ref&gt;Grassley, C. (2015). Grassley questions White House commitment to going dark solution. http://www.grassley.senate.gov/news/news-releases/grassley-questions-white-house-commitment-going-dark-solution&lt;/ref&gt; bringing the issue back into the public eye.

==Support for Expanded Law Enforcement Access==
In the United States and abroad, many organized groups want to give government agencies expanded power to read encrypted data in order to prevent and prosecute criminal activity. Government and law enforcement organizations are the most prominent of these groups.

=== The Obama Administration ===
In the aftermath of the [[w: November_2015_Paris_attacks|terrorist attacks in Paris]] and [[w: 2015_San_Bernardino_shooting|San Bernardino]], President Obama gave a national address about responding to future threats.&lt;ref&gt; White House. (2015). Address to the Nation by the President. https://www.whitehouse.gov/the-press-office/2015/12/06/address-nation-president &lt;/ref&gt;  One step in the plan is to prevent criminal activity on encrypted channels.&lt;ref&gt;Rampton, R. (2015). Obama appeals to Silicon Valley for help with online anti-extremist campaign. http://www.reuters.com/article/california-shooting-cyber-idUSKBN0TQ0A320151207&lt;/ref&gt;  However, it is unclear by what means the administration intends this to be accomplished.  Earlier in the year, the Obama administration conceded its fight to make tech companies add a [[w: Backdoor_(computing)|backdoor]] to encryption, agreeing that doing so would weaken defenses against foreign governments and cybercriminals.&lt;ref name=&quot;obama&quot;&gt;Perlroth, N. &amp; Sanger, D. (2015). Obama Won’t Seek Access to Encrypted User Data. http://www.nytimes.com/2015/10/11/us/politics/obama-wont-seek-access-to-encrypted-user-data.html&lt;/ref&gt;  One of the Administration’s fears is that requiring a backdoor for companies in the United States would set a similar precedent for American companies in other countries, such as China.&lt;ref name=&quot;obama&quot;&gt;&lt;/ref&gt;

=== Federal Bureau of Investigation ===
The Federal Bureau of Investigation (FBI) has recently fought with Apple over encryption in iMessage.&lt;ref&gt;Hern, A. (2015). Apple's Encryption Means It Can't Comply with US Court Order. http://www.theguardian.com/technology/2015/sep/08/apple-encryption-comply-us-court-order-iphone-imessage-justice&lt;/ref&gt;  In the newest versions of iOS, iMessage uses [[w: end-to-end encryption|end-to-end encryption]] so that only the sender and recipient can read messages&lt;ref name = &quot;Apple privacy&quot;&gt;Apple. (2015). Our Approach to Privacy. http://www.apple.com/privacy/approach-to-privacy/&lt;/ref&gt;, making it impossible for law enforcement to access encrypted content.  FBI Director [[w: James Comey|James Comey]] compared this to an unlockable door and said that a warrant should be all that is required to access this data&lt;ref&gt;Nakashima, E. (2015). With Court Order, Federal Judge Seeks to Fuel Debate About Data Encryption. https://www.washingtonpost.com/world/national-security/federal-judge-stokes-debate-about-data-encryption/2015/10/10/c75da20e-6f6f-11e5-9bfe-e59f5e244f92_story.html&lt;/ref&gt;.  Comey noted how a terrorist who attempted to attack an event in Texas in May 2015 had been communicating with other terrorists abroad using encrypted messages&lt;ref&gt;Harte, J. &amp; Volz, D. (2015). Shooter at Texas 'Draw Mohammed' Contest Messaged Foreign Militants: FBI. http://www.reuters.com/article/us-texas-shooting-comey-idUSKBN0TS2FU20151209&lt;/ref&gt;, implying that these messages would have given authorities advance notice of the attempt had they been intercepted and read.  The United States has no [[w: key disclosure law|key disclosure law]], unlike Canada, the UK, and France, effectively keeping these encrypted messages a secret.  While key disclosure allows agencies to access encrypted data, it only helps when the suspect is known.  It is useful for gathering evidence, but not for intelligence, when fast response times are necessary, especially in preventing terrorist attacks.

=== United Kingdom's Cameron Administration ===
Similar battles are happening outside the United States.  In the United Kingdom, Prime Minister [[w:David Cameron|David Cameron]] has spoken out against apps that send encrypted messages because most of them cannot be read by government.&lt;ref name=&quot;cameron&quot;&gt;Bienkov, A. (2015). David Cameron: Twitter and Facebook Privacy is Unsustainable. http://www.politics.co.uk/news/2015/06/30/david-cameron-twitter-and-facebook-privacy-is-unsustainable&lt;/ref&gt;  Cameron argued that, in the past, government agents could get a warrant and intercept phone calls or mail, but now there are channels that government cannot access, creating a “safe space” for criminals and terrorists.&lt;ref name=&quot;cameron&quot; /&gt;  Cameron would like legislation that bans apps that use encryption unless the app has a government backdoor.&lt;ref name=&quot;cameron&quot;&gt;&lt;/ref&gt;

==Opposition to Expanded Law Enforcement Access==

Not everyone is in favor of expanding the government's access to private encrypted data. According to a 2014 poll, a majority of Americans (54%) “disapprove” of bulk government collection of phone and internet records,&lt;ref&gt; Pew Research Center. (2014). Beyond Red vs. Blue: The Political Typology. http://www.people-press.org/files/2014/06/6-26-14-Political-Typology-release1.pdf &lt;/ref&gt; and presidential candidates as diverse as [[w: Bernie Sanders | Bernie Sanders]]&lt;ref&gt;https://berniesanders.com/issues/war-and-peace/&lt;/ref&gt; and [[w: Rand Paul | Rand Paul]]&lt;ref&gt;https://www.randpaul.com/issue/ending-nsa-spying&lt;/ref&gt; have spoken in favor of reining in the [[w: National Security Agency | National Security Agency’s ]] [[w: PRISM (surveillance program) | data collection programs]] during the 2016 election cycle. Most opposition comes from participants with concerns over privacy or technical issues, though other factors such as economic issues are also relevant. 

===Participants with Privacy Concerns===

Many opponents of government access to data do so on privacy grounds. For instance, [[w : Tim Cook | Apple CEO Tim Cook]], speaking about his company’s [[w: end-to-end encryption | end-to-end encrypted]] [[w: IMessage | iMessage]] service, said “we believe the contents of your text messages and your video chats is none of our business.”&lt;ref&gt;Cook, T. (2015). Speech at Electronic Privacy Information Center. http://techcrunch.com/2015/06/02/apples-tim-cook-delivers-blistering-speech-on-encryption-privacy/#.ulz3r9:kVGu&lt;/ref&gt; Groups like the [[w: Electronic Frontier Foundation | Electronic Frontier Foundation]]&lt;ref&gt;Electronic Frontier Foundation. (2015). https://www.eff.org&lt;/ref&gt; advocate for reducing government access to data on privacy grounds and enjoy reasonable support, including a budget in the tens of millions of dollars.&lt;ref&gt; Electronic Frontier Foundation. (2015). Annual Report. https://www.eff.org/about/annual-reports-and-financials &lt;/ref&gt;

Privacy concerns are not a new topic in the social sciences, but researchers have struggled even to define &quot;privacy.&quot;&lt;ref&gt;Solove, D. J. (2008). Understanding privacy.&lt;/ref&gt; Despite the difficulty in defining it, researchers have found that people place significant value on keeping their data private,&lt;ref&gt;Acquisti, A., John, L. K., &amp; Loewenstein, G. (2013). What is privacy worth?. The Journal of Legal Studies, 42(2), 249-274.&lt;/ref&gt; and concern over privacy is likely at least partially motivated by economic factors; companies like Apple even expressly use their protection of consumer privacy as a selling point.&lt;ref name = &quot;Apple privacy&quot;&gt;&lt;/ref&gt;

===Participants with Technical Concerns===

Many participants oppose weakening encryption on technical grounds. The basic argument is that a[[w: Backdoor (computing) | backdoor]] for law enforcement means that the encryption cannot be perfectly secure, because an attacker could use the backdoor to gain entry. As the[[w:  Information Technology Industry Council | Information Technology Industry Council]] said in a statement following calls to add backdoors after the [[w: November 2015 Paris attacks | Paris attacks ]] in November 2015, weakening encryption by adding backdoors “simply doesn’t make sense.”&lt;ref&gt;Information Technology Industry Council. (2015). http://www.reuters.com/article/us-tech-encryption-idUSKCN0T82SS20151119&lt;/ref&gt; This is the same argument that security experts used when the[[w: Clipper Chip | Clipper Chip]] was proposed, and which ultimately helped defeat it.&lt;ref&gt;Abelson, H., Anderson, R. N., Bellovin, S. M., Benaloh, J., Blaze, M., Diffie, W., ... &amp; Schneier, B. (1997). The risks of key recovery, key escrow, and trusted third-party encryption.&lt;/ref&gt;

Opponents also contend that adding backdoors to encryption systems will not actually help law enforcement defeat threats. Nate Cardozo, staff attorney for the Electronic Frontier Foundation, has said that &quot;intel agencies are drowning in data... It's not about having enough data; it's a matter of not knowing what to do with the data they already have.&quot;&lt;ref&gt;Cardozo, N. (2015). http://www.wired.com/2015/11/paris-attacks-cia-director-john-brennan-what-he-gets-wrong-about-encryption-backdoors/ &lt;/ref&gt; Reports that the 2015 Paris attacks (which have been cited in proposals for backdoors, including by[[w: Central Intelligence Agency | CIA]] director[[w: John O. Brennan | John Brennan]])&lt;ref&gt; Brennan, J. (2015). Speech at Center for Strategic International Studies. http://www.defenseone.com/technology/2015/11/brennan-paris-wakeup-call-europe-encryption/123732/ &lt;/ref&gt; were planned using unencrypted communications only make this claim stronger.&lt;ref&gt; Froomkin, D. (2015). Signs point to unencrypted communications between terror suspects. https://theintercept.com/2015/11/18/signs-point-to-unencrypted-communications-between-terror-suspects/ &lt;/ref&gt;

== Legal Framework and the Battle of Social Values ==
In legal settings, social values often come under pressure when two or more groups hold competing interests. Several court cases have influenced controversial interpretations of widespread key social values, adding legal, political, and social complexity to the question of how much access law enforcement should have to encrypted data - and when that access becomes too much.

=== &lt;u&gt;United States Constitution: The Foundation of Our Rights&lt;/u&gt; ===
Almost every court case listed in this chapter either cites, is influenced by, or challenges two fundamental constitutional amendments. Before the interpretations of these amendments are discussed and analyzed, it is important to clearly establish what is currently written to understand how each argument may or may not align with our [[wikipedia:List_of_amendments_to_the_Constitution_of_the_United_States|constitutional rights]]:

==== 4th Amendment ====
The 4th Amendment of the United States Constitution states:

&quot;The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.&quot;&lt;ref&gt;{{Cite web |title=The 4th Amendment of the U.S. Constitution |url=https://constitutioncenter.org/the-constitution/amendments/amendment-iv |website=National Constitution Center – constitutioncenter.org |language=en}}&lt;/ref&gt;

In several court cases, the 4th Amendment was cited for its violation in law enforcement seizing several &quot;things&quot; being seized or searched, typically warrantless. The verdicts for these court cases typically protected individuals and established the legal need for a warrant to seize and search encrypted devices and data. Over the past 50 years, the amendment and its interpretation has remained consistent, protecting individuals and groups' dignity, privacy, and autonomy.

==== 5th Amendment ====
The 5th Amendment of the United States Constitution states:

&quot;No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.&quot;&lt;ref&gt;{{Cite web |title=The 5th Amendment of the U.S. Constitution |url=https://constitutioncenter.org/the-constitution/amendments/amendment-v |website=National Constitution Center – constitutioncenter.org |language=en}}&lt;/ref&gt;

The 5th Amendment protects individuals from being compelled to provide testimonial evidence against themselves ([[wikipedia:Self-incrimination|self-incrimination)]] a protection that becomes particularly important in cases involving encrypted devices and data. It is also well known for its guarantee of [[wikipedia:Due_Process_Clause|proper due process]], which underpins how searches, seizures, and arrests must be conducted. Courts have repeatedly examined whether entering a passcode or decrypting data communicates knowledge, such as ownership, control, or the existence of specific files, and therefore qualifies as a testimonial. When decryption would reveal new information to law enforcement, the 5th Amendment applies; when the government can already describe the evidence with reasonable particularity, courts have sometimes permitted compelled access.

=== &lt;u&gt;Cases on Personal Privacy vs. Public Safety&lt;/u&gt; ===
The following court cases ''Smith v. Maryland, U.S. v. Cotterman, Riley v. California,'' and ''Carpenter v. U.S.'' highlight the evolving legal battle over how far the government may go in accessing digital information in the name of public safety and justice. Together, they illustrate how constitutional protections written in the 18th century are continually reinterpreted to address current technologies, from cell phone applications to location tracking. These cases also go beyond legal bounds and address competing social values, where on one side is law enforcement's interest in preventing harm and investigating crime while on the other side is society's growing expectation of personal privacy extends to a digital space. As each case redefines the balance between these values, it also influences how U.S. citizens understand their legal rights and the conditions under which those rights can be limited or lost.

==== ''Smith v. Maryland'' (1979) ====
In [[wikipedia:Smith_v._Maryland|''Smith v. Maryland'']], Michael Lee Smith was arrested and indicted for robbing Patricia McDonough. To confirm Smith's identity after cross-referencing against the victim's description, the police asked the telephone company to install a [[wikipedia:Pen_register|pen register]]. However, they did so without a warrant or court order, since they believed Smith was the source of threatening calls made to McDonough. Once the pen register confirmed outgoing numbers, police obtained a warrant to search Smith's residence, where they seized the phonebook and identified Smith to be the robber. At trial, Smith sought to suppress the evidence collected by the pen register since the police did not obtain a warrant to seize his telecommunication information &quot;prior to its installation,&quot; which he claimed violated his 4th Amendment protection against unreasonable searches and seizures. On June 20, 1979, the court ruled in favor of Maryland, holding that Smith had no &quot;legitimate expectation of privacy&quot; in the numbers he dialed because he voluntarily conveyed that information to the phone company, which they recorded as a part of their regular business practices. &lt;ref&gt;{{Cite web |title=Smith v. Maryland, 442 U.S. 735 (1979) |url=https://supreme.justia.com/cases/federal/us/442/735/ |website=Justia Law |language=en}}&lt;/ref&gt;

==== ''U.S. v. Cotterman'' (2013) ====
''[[wikipedia:United_States_v._Cotterman|U.S. v. Cotterman]]'' began when Howard Cotterman returned from Mexico and was flagged by the [[wikipedia:TECS|Treasury Enforcement Communications System (TECS)]] database due to prior sexual offenses. At Lukeville, Arizona port of entry, border patrol officers conducted two searches (the second after the TECS notification), finding two laptops and 3 cameras, which [[wikipedia:United_States_Immigration_and_Customs_Enforcement|Immigration and Customs Enforcement (ICE)]] later came and seized.&lt;ref&gt;{{Cite web |last=hlr |date=2014-01-17 |title=United States v. Cotterman |url=https://harvardlawreview.org/print/vol-127/ninth-circuit-holds-forensic-search-of-laptop-seized-at-border-requires-showing-of-reasonable-suspicion-ae-united-states-v-cotterman-709-f-3d-952-9th-cir-2013-en-banc/ |access-date=2026-04-15 |website=Harvard Law Review |language=en-US}}&lt;/ref&gt; After conducting an off-site digital forensic search, the government found hundreds of files containing child pornography on Cotterman's devices, and they indicted him for possession of such digital content. Cotterman motioned to suppress the evidence gained from the devices, claiming the forensic search violated the 4th Amendment. The United States argued that there are exceptions to the 4th Amendment regarding border searches. After being granted then reversed, the case was reheard [https://www.law.cornell.edu/wex/en_banc en banc]. The Ninth Circuit en banc finally held that the United States violated Cotterman's 4th Amendment rights, saying that forensic, comprehensive searches of digital devices at the border require reasonable suspicion, recognizing the heightened privacy interests in electronic data.

==== ''Riley v. California'' (2014) ====
In ''[[wikipedia:Riley_v._California|Riley v. California]],'' David Riley was stopped for a traffic violation, arrested on weapons charges, and had his cell phone seized and searched without a warrant, linking Riley to gang affiliation and involvement in a prior shooting. Riley moved to suppress the phone evidence, but the lower court denied his motion. A similar case, ''U.S. v. Wurie'' (2014), involved a similar warrantless search of a flip phone that linked Wurie to a drug stash in his apartment.&lt;ref&gt;{{Cite web |title=United States v. Wurie |url=https://www.law.cornell.edu/supct/cert/13-212 |website=LII / Legal Information Institute |language=en}}&lt;/ref&gt; The Supreme Court reviewed both cases together and ultimately decided that &quot;the police generally may not, without a warrant, search digital information on a cell phone seized from an individual who has been arrested&quot;&lt;ref&gt;{{Cite web |title=Riley v. California, 573 U.S. 373 (2014) |url=https://supreme.justia.com/cases/federal/us/573/373/ |website=Justia Law |language=en}}&lt;/ref&gt;. As a result, Riley's conviction was reversed while Wurie's case ruling was sustained.

==== ''Carpenter v. U.S.'' (2018) ====
''[[wikipedia:Carpenter_v._United_States|Carpenter v. U.S.]]'' is similar to ''Riley v. California,'' in regard to how both involved data associated with cell phones. However, in the Carpenter case, police arrested four men with connection to an armed robbery. One of the men confessed and gave his number and all of the numbers of the other participants to the FBI, which they used to obtain orders to gain transactional records on each phone number. This was granted under the [[wikipedia:Stored_Communications_Act|Stored Communications Act, 18 U.S.C. 2703(d)]], which provides the government the ability to &quot;require the disclosure of certain telecommunication records when 'specific and articulable facts show[] that there are reasonable grounds to believe that the contents of a wire or electronic communication, or the records or other information sought, are relevant and material to an ongoing criminal investigation.'&quot;&lt;ref&gt;Carpenter v. United States. (n.d.). ''Oyez''. &lt;nowiki&gt;https://www.oyez.org/cases/2017/16-402&lt;/nowiki&gt;&lt;/ref&gt; The transactional records included evidential data about phone call location data and other [[wikipedia:Cell_site|cell site]] location information (CSLI). Timothy Carpenter moved to suppress the evidence gained from the CSLI data, arguing it violated the 4th Amendment. Ultimately, the Supreme Court narrowly sided with Carpenter, stating the warrantless search and seizure of phone users' location data violated the 4th Amendment, but this case also extended the [[wikipedia:Third-party_doctrine|third-party doctrine]], which was originally established in ''Smith v. Maryland'' and ''U.S. v. Miller'' (1976).

==== Social Analysis: Privilege of Privacy Amongst Individuals ====
While ''Smith v. Maryland'' established early boundaries for digital privacy through the [[wikipedia:Third-party_doctrine|third-party doctrine]], the struggle between digital privacy and the government's duty to ensure public safety and promote the general welfare remains foundationally contested. Later cases ''U.S. v. Cotterman, Riley v. California, and Carpenter v. U.S.'' shifted the legal landscape by recognizing that digital privacy is not merely limited to or defined by location, device, or type of data. Further, these decisions affirm that modern technology and encrypted data function as extensions of one's identity: &quot;phones are no longer merely phones, but important communication ...[and] data centers, and perhaps most significantly, loci for the digital culture.&quot;&lt;ref&gt;Steven I. Friedland, Riley v. California and the Stickiness Principle, 14 ''Duke Law &amp; Technology Review'' 121-139 (2016)&lt;/ref&gt; In a world where personal identity, social life, and daily conduct are increasingly mediated through digital platforms, individuals carry with them a reasonable expectation of privacy that is qualitatively different from what was considered in the past.

However, this expectation is not universally accepted or consistently interpreted. The ability to protect one's digital privacy, through encryption, secure devices, and knowledge of digital rights, often underlines developing social privileges. As governments and courts continue to define the limitations and applications of privacy in a rapidly evolving cyber-physical space, tensions will continue to emerge between the trend of expanding privacy protections and ensuring that privacy is not viewed as a hindrance to justice, safety, or accountability. These tensions have been heightened by the recent rise in AI surveillance and automated data decryption services, where, as the [[wikipedia:American_Civil_Liberties_Union|ACLU]] notes, &quot;the automated nature of these systems means that traditional limitations on police resources—once a vital check on government overreach—have been effectively eliminated.&quot;&lt;ref&gt;{{Cite web |date=February 12, 2026 |title=AI-Powered Surveillance Is Turning the United States into a Digital Police State. Now is the Time to Stop It. |url=https://www.aclum.org/publications/ai-powered-surveillance-is-turning-the-united-states-into-a-digital-police-state-now-is-the-time-to-stop-it/ |website=ACLU of Massachusetts |language=en-US}}&lt;/ref&gt; These cases reveal that privacy is not only a legal right but a social resource - one that is unevenly accessible, persistently contested, and potentially constrained depending on the level of influence and position one has within society, public institutions, and digital systems.

=== &lt;u&gt;Cases on Digital Autonomy vs. Law Enforcement's Authority&lt;/u&gt; ===
The following court cases highlight a nearly twenty-year trend in how courts navigate the tension between personal digital autonomy and expanding authority of law enforcement. Central to this trend is the forgone conclusion doctrine, a limited exception to the 5th Amendment's Self-Incrimination Clause that can compel an individual to release encrypted data and passcodes if three factors are met: 

# The Government has knowledge of the existence of the evidence demanded
# The defendant possessed or controlled the evidence, and
# The evidence is authentic.

The government must also be able to describe the evidence with &quot;reasonable particularity.&quot; &lt;ref&gt;{{Cite web |title=Foregone Conclusion Doctrine Allows Government to Make Criminal Defendant Disclose Computer Password |url=https://goldsteinmehta.com/blog/foregone-conclusion-doctrine |website=Goldstein Mehta LLC |language=en-US}}&lt;/ref&gt; Although this doctrine is framed as a narrow case-by-case rule, its application in the digital era effectively shifts power toward law enforcement in situations where devices and encrypted data act as extensions of personal identity, memory, and autonomy. As encryption continues to become a default feature of everyday technology, compelled-decryption cases raise questions about how much control individuals retain over their private data and whether constitutional protections can keep up with rapidly evolving digital systems. While many courts, often citing ''In re Boucher'', have supported decryption under the doctrine, cases such as ''U.S. v. Doe'' and the ''Oakland Biometrics Case'' emphasizes that society still recognizes meaningful limits on government authority and continue to affirm the social and legal value of personal digital autonomy.

==== ''In re Boucher Case'' (2009) ====
The [[wikipedia:In_re_Boucher|''In re Grand Jury Subpoena to Boucher'' case]] arose when Sebastien Boucher was arrested at the border between Canada and Vermont, where an officer's secondary inspection led to the discovery of child pornography on Boucher's computer. However, after seizing the laptop and once the computer was shut down, the hard drive's content could not be viewed again without entering in a passphrase. A Secret Service computer forensics expert testified that the drive had been encrypted using [[wikipedia:Pretty_Good_Privacy|Pretty Good Privacy (PGP)]] software and there was no backdoor for this type of encryption software.&lt;ref&gt;Pfefferkorn, Riana, In Re Boucher: Hard Drives Make Pretty Good Law? Encryption Passphrases as Testimonial Evidence Under the Fifth Amendment (February 1, 2009). Available at SSRN: &lt;nowiki&gt;https://ssrn.com/abstract=1883697&lt;/nowiki&gt; or &lt;nowiki&gt;http://dx.doi.org/10.2139/ssrn.1883697&lt;/nowiki&gt;&lt;/ref&gt; In response, the grand jury subpoenaed Boucher to release the passphrases to the government, in which Boucher successfully moved to quash the subpoena, initially. Boucher argued that this violated his 5th Amendment rights against self-incrimination, but after further review the U.S. District Court for Vermont reversed the initial grant. The reasoning for this reversal is that since the files sought allegedly contain child pornographic content, the entry of the password is incriminating and therefore the real question is whether the &quot;subpoena seeks testimonial communication.&quot; Also, both parties agreed that the content in the files did not share the same 5th Amendment privileges as revealing the password, as the files were &quot;voluntarily prepared&quot; and not testimonial.&lt;ref&gt;In Re Boucher (United States District Court for the District of Vermont November 29, 2009). &lt;nowiki&gt;https://www.volokh.com/files/Boucher.pdf&lt;/nowiki&gt;.&lt;/ref&gt; The real question lies in whether entry of the password, and ultimately handing the files over to the government, would be testimonial and a violation of the 5th Amendment. The final verdict states that &quot;Boucher has no act of production privilege to refuse to provide the grand jury with an unencrypted version of the Z drive of his computer,&quot; and that &quot;the government may not make use of Boucher's act of production to authenticate the unencrypted Z drive or its contents either before the grand jury or a petit jury,&quot; ultimately legally compelling Boucher to decrypt his device and hand the files to the government.&lt;ref&gt;In re Grand Jury Subpoena to Sebastien Boucher, &lt;nowiki&gt;https://yalelawtech.org/wp-content/uploads/2010/02/boucher_-_sessions_-_appeal.pdf&lt;/nowiki&gt; (United States District Court for the District of Vermont. February 19, 2009).&lt;/ref&gt; This case became a significant landmark of [[wikipedia:Key_disclosure_law|key disclosure law]] through being regarded as [[wikipedia:Case_law|case law]], and its influence is significant in several other court cases discussed in the battle of self autonomy and authority.

==== ''U.S. v. Doe'' (Dated March 25, 2011) ====
In the case of ''U.S. v. Doe,'' FBI agents seized two laptops and five external hard drives from a man under investigation, but the data was stored behind an encryption program assumed to be [[wikipedia:TrueCrypt|TrueCrypt]]. When ordered by a grand jury to produce unencrypted contents of the hard drives, he invoked the 5th Amendment, claiming a breach of protections against self-incrimination. Specifically, Doe argued that &quot;the Government’s use of the decrypted contents of the hard drives would constitute derivative use of his immunized testimony, use not protected by the district court’s grant of immunity.&quot;&lt;ref&gt;{{Cite web |title=In Re: Grand Jury Subpoena Duces Tecum Dated March 25, 2011, USA v. John Doe, No. 11-12268 (11th Cir. 2012) |url=https://law.justia.com/cases/federal/appellate-courts/ca11/11-12268/11-12268-2012-02-23.html |website=Justia Law |language=en}}&lt;/ref&gt; Doe also explained he was unable to decrypt the hard drive (which was disregarded in this case as well as the appeal case), but ultimately the court held Doe in contempt and jailed him. After appealing the decision, the circuit court ruled that decrypting the data is testimonial and therefore protected by the 5th Amendment (in favor of Doe), a comparably different outcome than seen in ''In re Boucher'' and ''U.S. v. Fricosu,'' and a significant limitation on the foregone conclusion doctrine.

==== ''U.S. v. Fricosu'' (2012) ====
In ''[[wikipedia:United_States_v._Fricosu|U.S. v. Fricosu]],'' the FBI obtained a search warrant for Colorado resident Ramona Fricosu under suspicion of mortgage fraud. During the search, agents seized six computers. One of the six computers seized was password-encrypted by a program called PGP Desktop, which &quot;provides easy to use and secure encryption to protect sensitive data on your laptop or desktop computers.&quot; &lt;ref&gt;{{Cite web |title=What is Symantec Encryption Desktop? (formerly known as PGP Desktop Encryption) |url=https://help.uillinois.edu/TDClient/37/uic/KB/PrintArticle?ID=776 |website=help.uillinois.edu}}&lt;/ref&gt; The day after the execution of the search warrant, Fricosu made a call to her incarcerated divorced ex-husband, and the recorded conversation revealed that both her and her ex-husband were aware of the content in her laptop, as well as the encryption program and her legal instruction to not provide any passwords, as protected by the 5th Amendment. In closing, several facts of the case (including file names, location of laptop, and connection to her ex-husband and his convictions) led the court to hold that Fricosu is legally compelled to provide an unencrypted copy of the files to the government as evidence, aligning with the ''In re Boucher'' case and applying the forgone conclusion doctrine.&lt;ref&gt;{{Cite web |title=United States v. Fricosu, Criminal Case No. 10–cr–00509–REB–02. {{!}} D. Colo., Judgment, Law, casemine.com |url=https://www.casemine.com/judgement/us/5914e216add7b049348ee62f |website=https://www.casemine.com |language=en}}&lt;/ref&gt;

==== ''Commonwealth v. Gelfgatt'' (2012) ====
''Commonwealth v. Gelfgatt'' is another case regarding compelled decryption via [[wikipedia:Key_disclosure_law|key disclosure laws]] influenced by ''In re Boucher''. Leon I. Gelfgatt, an attorney, was indicted with seventeen counts of forgery of a document, seventeen counts of [[wikipedia:Uttering|uttering]] a forged instrument, and three counts of attempting to commit larceny. The charges came from allegations against Gelfgatt that he conducted a scheme, using several computers, to receive ill-obtained funds that he used to pay mortgage loans for residential properties. The Commonwealth asked the judge to grant legal compulsion of Gelfgatt to decrypt his computers, in which the judge denied originally but asked for higher-level advisement on the question:&lt;ref name=&quot;:0&quot;&gt;{{Cite web |title=Commonwealth v. Gelfgatt |url=https://law.justia.com/cases/massachusetts/supreme-court/2014/sjc-11358.html |website=Justia Law |language=en}}&lt;/ref&gt; 

&quot;Can the defendant be compelled pursuant to the Commonwealth's proposed protocol to provide his key to seized encrypted digital evidence despite the rights and protections provided by the Fifth Amendment to the United States Constitution and Article Twelve of the Massachusetts Declaration of Rights?&quot;&lt;ref name=&quot;:0&quot; /&gt;

The Supreme Judicial Court, after further review of the law in question, reversed the judges denial to the Commonwealth's request for Gelfgatt's compulsion, implying in their verdict that &quot;the government can ... compel the defendant to produce it [digital evidence] where it already knows what the information is, and the defendant’s disclosure would add little or nothing to what the government’s knowledge,&quot; another application of the forgone conclusion doctrine.&lt;ref&gt;{{Cite web |last=Fernandez |first=Zoraida |date=2019-03-20 |title=Can the Commonwealth Compel You to Unlock Your Cell Phone? SJC Clarifies Certain Circumstances in Which it Can |url=https://www.bostonlawyerblog.com/can-the-commonwealth-compel-you-to-unlock-your-cell-phone-sjc-clarifies-certain-circumstances-in-which-it-can/ |website=Boston Lawyer Blog |language=en-US}}&lt;/ref&gt;

==== ''Oakland Biometrics Case'' (2019) ====
The ''In re: Search of A Residence in Oakland, California'' case presents a notably different application of the 4th and 5th Amendments in a cyber-physical system, compared to cases like ''In re Boucher'', ''U.S. v. Fricosu'', and ''Commonwealth v. Gelfgatt.'' In this case, the government was investigating two individuals suspected of committing extortion scheme conducted through Facebook Messenger. The government's warrant application asked for the authority to search the Oakland residence and seize all digital devices on the premise, and to compel any individual present to unlock the devices through [[wikipedia:Biometrics|biometric]] features, even if they were not the subjects of the warrant. The magistrate judge denied the application, finding that &quot;while the Government had probable cause to search the premises under the Fourth Amendment, it did not have probable cause to search and seize ''all'' digital devices nor compel their unlocking using biometric features because the search warrant application was not limited to digital devices owned or controlled by the suspects.&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite news |last=Nowell D. Bamberger, Melissa Gohlke, Sameer Jaywant |date=2019-01-23 |title=Court Holds That 5th Amendment Self-Incrimination Privilege Precludes Compelling Fingerprint or Facial Recognition Access to Digital Devices {{!}} Cleary Enforcement Watch |language=en-US |work=Cleary Enforcement Watch |url=https://www.clearyenforcementwatch.com/2019/01/court-holds-5th-amendment-self-incrimination-privilege-precludes-compelling-fingerprint-facial-recognition-access-digital-devices/}}&lt;/ref&gt;

Citing ''Carpenter v. U.S.'' in 2018, the court emphasized how &quot;the advancement of technology should not be accompanied by an erosion in the constitutional rights of the people, and that courts should vigilantly guard against such erosion.&quot;&lt;ref name=&quot;:1&quot; /&gt; In its final reasoning, the court held that biometric data cannot be compelled to unlock devices, including &quot;fingers, thumbs, facial recognition, optical/iris, or any other biometric feature to unlock electronic devices.&quot; &lt;ref&gt;{{Cite web |last=Baron |first=Ethan |last2=News |first2=The Mercury |date=2019-01-16 |title=Biometrics Ruling Limits How Police Can Access a Smartphone |url=https://www.govtech.com/products/Biometrics-Ruling-Limits-How-Police-Can-Access-a-Smartphone.html |website=GovTech |language=en}}&lt;/ref&gt;

==== ''State v. Andrews'' (2020) ====
''State v. Andrews'' began with the arrest of Quincy Lowery for suspected involvement in a New Jersey drug-trafficking ring. However, Lowery confessed officer Robert Andrews helped him evade detection by warning Lowery of the wiretap on his phone and the GPS tracker on his Jeep. Lowery even showed the investigators the text messages to prove such evidence, but because he reset his phone a month earlier, most of the text messages were wiped. Officers requested Andrews to hand over his two iPhones, in which Andrews did so voluntarily without giving out the passwords to unlock them. Without a way to effectively decrypt the passwords, the State obtained warrants for the phones' contents and motioned for Andrews to reveal the passwords. Andrews refused citing the 5th Amendment's Self-Incrimination Clause. Once the trial court supported the State's motion and Andrews appealed the courts decision, the appeals court upheld the trial court's decision, stating &quot;because the State had already established that Andrews owned, possessed, and controlled his iPhones, his act of producing their passcodes, while 'testimonial,' fell under this 'foregone conclusion' exception.&quot;&lt;ref name=&quot;:2&quot;&gt;{{Cite web |last= |date=2021-04-12 |title=State v. Andrews |url=https://harvardlawreview.org/print/vol-134/state-v-andrews/ |website=Harvard Law Review |language=en-US}}&lt;/ref&gt; Andrews appealed the decision again, and the New Jersey Supreme Court affirmed both lower courts, holding &quot;that the Fifth Amendment privilege was overcome by the government’s knowledge of three key facts: the passcodes’ existence, their possession by the defendant, and their authenticity,&quot;&lt;ref name=&quot;:2&quot; /&gt; ultimately compelling Andrews to reveal the passwords to the iPhones.

==== Social Analysis: Increasing Authority at the Cost of Digital Autonomy ====
The current U.S. legal landscape underlines a shift in how society balances personal digital autonomy against government authority. As courts increasingly apply the forgone conclusion doctrine to encrypted devices and data, they reshape social expectations about privacy, identity, and governmental power. With digital devices now functioning as extensions of our being and lives, compelled decryption has social implications far beyond traditional evidence production. Cases such as ''In re Boucher, U.S. v. Fricosu, Commonwealth v. Gelfgatt,'' and ''State v. Andrews'' have normalized broader government access to encrypted data, strengthening a social narrative that national security interests outweigh individual control over digital identity. This trend can erode trust and accountability, particularly in communities already skeptical of law enforcement, and may create chilling effects on expression, association, and political participation. As compelled access becomes more routine, society risks normalizing a surveillance-based relationship between citizens and the government.

However, cases like ''U.S. v. Doe'' and the ''Oakland Biometrics Case'' demonstrate that courts still recognize the social value of digital autonomy and the importance of limiting the government's power when compelled decryption becomes indistinguishable from compelled testimony, or otherwise self-incrimination. Advocacies, professional organizations, and non-profits also play a crucial role in shaping both the public's view and judicial reasoning. For example, the [[wikipedia:Electronic_Frontier_Foundation|Electric Frontier Foundation (EFF)]] wrote amicus briefs (a legal document providing advice, information, or arguments related to a case) for both ''U.S. v. Fricosu'' and ''U.S. v. Doe'', consistently arguing that the 5th Amendment protected individuals from all forms of compelled decryption. They emphasized that the different outcomes in these cases resulted from not doctrinal inconsistency but from the defendant's own disclosures, underlining how &quot;silence is golden.&quot; Noting how &quot;Boucher talked to law enforcement&quot; and &quot;Fricosu talked to her ex-husband and co-defendant in jail,&quot; the EFF claims &quot;it was this talking that defeated their Fifth Amendment privilege through the foregone conclusion doctrine.&quot;&lt;ref&gt;{{Cite web |last=Fakhoury |first=Hanni |date=2012-03-07 |title=A Tale of Two Encryption Cases |url=https://www.eff.org/deeplinks/2012/03/tale-two-encryption-cases |website=Electronic Frontier Foundation |language=en}}&lt;/ref&gt; Together, these cases and the various proponents of conflicting civil liberties, often driven by strong social values, illustrate a significant moment in the definition of digital autonomy, one which courts, civil liberties organizations, and the public are actively shaping the future boundaries of state authority in an increasingly data-driven society.

=== &lt;u&gt;Cases on Managing Public Trust &amp; Accountability&lt;/u&gt; ===
The final cases in this chapter addresses the critical but often overlooked social values of public trust and accountability, particularly in the management of personal data. As emerging technologies such as [[wikipedia:Quantum_computing|quantum computing]], [[wikipedia:Artificial_intelligence|AI]], and [[wikipedia:Edge_computing|edge computing]] reshape how information is stored and accessed, public trust in institutions responsible for protecting digital rights has become increasingly fragile. People generally rely on governments, courts, and large organizations to act as stewards of their data and to ensure that justice is upheld when digital rights are threatened. However, the following cases reveal a series of paradoxes, where entities display conflicting interests in data management and the legal framework governing their practices lag behind the change in technology. These tensions propose significant questions in who truly manages data and, by extension, who the public can trust to enact accountability in an increasingly digital world.

==== ''U.S. v. Microsoft'' (2018) ====
[[File:BalticServers data center.jpg|thumb|Data centers hold vast amounts of information about millions of individuals and organizations, making them a significant source of digital evidence.]]
In ''[[wikipedia:Microsoft_Corp._v._United_States|U.S. v. Microsoft]],'' the FBI in 2013 obtained a warrant under the [[wikipedia:Stored_Communications_Act|Stored Communications Act]] requiring Microsoft to release encrypted email and other information connected to one of their customers that was under suspicion of being involved in illegal drug trafficking. Microsoft refused to comply, arguing these emails were stored in its Dublin, Ireland data center and that the SCA did not authorize U.S. courts to compel production of data outside of its borders. Microsoft moved to quash the warrant, questioning how a United States federal law can be enforced outside of U.S. borders. While the move to quash the warrant was denied by the district court, Microsoft appealed the ruling and was granted their motion to quash it by the Second Circuit court. The U.S. Department of Justice petitioned to review the new decision, claiming that the warrant was &quot;a domestic application of the SCA,&quot; and that the appealed decision &quot;would be 'impractical and detrimental to law enforcement' and that the use of the SCA to obtain user data stored abroad would respect 'the United States’ international obligations.'&quot;&lt;ref&gt;{{Cite web |title=United States v. Microsoft |url=https://epic.org/documents/united-states-v-microsoft/ |website=EPIC - Electronic Privacy Information Center |language=en-US}}&lt;/ref&gt; After the Supreme Court granted the petition and planned to hold another hearing, the federal government enacted the [[wikipedia:CLOUD_Act|Clarifying Lawful Overseas Use of Data Act (CLOUD Act)]], which in essence established that U.S. service providers must produce data within their possession, custody, or control regardless of where the data is physically stored. This ultimately led to the case being dismissed as unnecessary and Microsoft being legally compelled by the CLOUD Act to obtain and provide the data from its Irish datacenter, due to Microsoft being a U.S. company.&lt;ref&gt;{{Cite web |title=United States v. Microsoft Corp., 584 U.S. ___ (2018) |url=https://supreme.justia.com/cases/federal/us/584/17-2/ |website=Justia Law |language=en}}&lt;/ref&gt;&lt;ref&gt;United States v. Microsoft Corporation. (n.d.). ''Oyez''. &lt;nowiki&gt;https://www.oyez.org/cases/2017/17-2&lt;/nowiki&gt;&lt;/ref&gt; 

==== ''Van Buren v. U.S.'' (2021) ====
''[[wikipedia:Van_Buren_v._United_States|Van Buren v. U.S.]]'' addresses access to encrypted data very differently from ''U.S. v. Microsoft''. Georgia police sergeant Nathan Van Buren was bribed by Andrew Albo, who was cooperating with an FBI sting operation, to run a license plate search on a possible undercover cop. Van Buren complied to the request, and through the [[wikipedia:Crime_information_center|Georgia Crime Information Center (GCIC)]] he used his authorized credentials to look up the provided license plate to confirm the identity of an undercover cop. Although officers are trained that GCIC access is strictly limited to law enforcement purposes specifically, Van Buren technically accessed only information he was already authorized to view. He was charged with violating the [[wikipedia:Computer_Fraud_and_Abuse_Act|Computer Fraud &amp; Abuse Act (CFAA)]], which he petitioned for review claiming that he technically did not exceed access authority as a law enforcement official with proper credentials to access the GCIC.&lt;ref&gt;{{Cite web |title=Van Buren v. United States |url=https://epic.org/documents/van-buren-v-united-states/ |website=EPIC - Electronic Privacy Information Center |language=en-US}}&lt;/ref&gt; The court argued that &quot;the purpose in using the system was immaterial,&quot; and with there being no purpose limitation cited in the CFAA, &quot;Van Buren was authorized to access the database and di not violate the CFAA.&quot;&lt;ref name=&quot;:3&quot;&gt;Scott T. Lashway and Matthew M.K. Stein, ''Signs Inscribed on a Gate: The Impact of Van Buren v. United States on Civil Claims Under the Computer Fraud and Abuse Act'', 44 W. New Eng. L. Rev. 109 (2022), &lt;nowiki&gt;https://digitalcommons.law.wne.edu/lawreview/vol44/iss1/5&lt;/nowiki&gt;&lt;/ref&gt; 

The court also made three important notes regarding the outcome and implications of this case and the CFAA, stating as follows:

# Amending the CFAA to include a purpose-based limitation criminalizes ordinary conduct, such as when an employee uses a business computer to send and check personal emails.
# Purpose-based limitations are unstable and potentially unclear, blurring what is considered &quot;normal&quot; and &quot;criminal&quot; conduct.
# The CFAA was only concerned in Van Buren's credentials, not his purpose for accessing GCIC records. Future disputes may address how contracts, policies, and terms of service interact with the statute's access-based limitations.

==== Social Analysis: Continuously Balancing Restriction and Freedoms ====
[[File:Official Presidential portrait of Thomas Jefferson (by Rembrandt Peale, 1800)(cropped).jpg|thumb|Thomas Jefferson recognized the power the public provides to its government, and the importance of the government maintaining the public's trust.]]
As [[wikipedia:Thomas_Jefferson|Thomas Jefferson]] once stated, &quot;when a man assumes a public trust, he should consider himself as public property.&quot;&lt;ref&gt;{{Cite web |date=2015-07-01 |title=Basic Obligation of Public Service {{!}} U.S. Department of the Interior |url=https://www.doi.gov/ethics/basic-obligations-of-public-service |website=www.doi.gov |language=en}}&lt;/ref&gt; 

The principle behind this quote is that public institutions derive their legitimacy from the people and must remain accountable to them. Yet the modern legal framework surrounding encrypted data often moves in the opposite direction. In cases such as ''U.S. v. Microsoft,'' the government asserted authority beyond U.S. borders, raising important questions about which institutions, domestic or foreign, citizens are expected to trust with their digital lives. Microsoft and privacy advocates warned that if the U.S. can issue a warrant that legally compels decryption abroad, &quot;other countries will in turn use their own laws to seize data stored in the U.S&quot; which &quot;could put Americans’ emails (and other data) at risk of seizure by foreign governments.&quot;&lt;ref&gt;Kruse, N. (2018, March 1). ''U.S. v. Microsoft: Is Your Data and Privacy at Risk?''. Byte Back. &lt;nowiki&gt;https://www.bytebacklaw.com/2018/03/u-s-v-microsoft-is-your-data-and-privacy-at-risk/&lt;/nowiki&gt;&lt;/ref&gt; The CLOUD Act further complicates this landscape by shifting practical control of data from the government to private, for-profit corporations. Even when data is physically stored in another country, ownership and decision-making authority often remain with companies like Microsoft, Facebook, and Apple, who all supported the CLOUD Act and whose incentives are shaped not by democratic accountability but by reputation management, market pressures, and financial interests. This raises two currently unanswered questions of public trust: first, how can citizens trust governmental systems when jurisdictional boundaries blur and government authority extends beyond its traditional limits, and second, how can they trust corporations to act as responsible stewards of their data when their obligations to shareholders may conflict with the public's interest in privacy and autonomy?

Furthermore, ''Van Buren v. U.S.'' produces a paradoxical outcome, where individuals protected from being criminalized for everyday digital behavior while simultaneously exposing the public to greater risks of insider misuses of encrypted data by law enforcement. The CFAA was originally designed to prevent unauthorized access to sensitive information, but Van Buren narrowed its scope by rejecting any inquiry into a person's purpose for accessing data. In doing so, the court prevented the CFAA from becoming overly broad &quot;terms-of-service crime,&quot; yet it also removed a potential safeguard against officers who misuse their legitimate credentials to access personal information for improper reasons. As scholars note, the CFAA retains its force, &quot;even with the purpose inquiry foreclosed: access may not be authorization, and authorization or its limits could be inferred from context and terms of use.&quot;&lt;ref name=&quot;:3&quot; /&gt; Regardless, this case “does not definitively resolve the question of whether unauthorized access must be barred by a hardware or software gateway or if activity can become ‘unauthorized’ by a contractual ban&quot; and “largely for that reason, the practical effects on the CFAA’s civil enforcement provisions remain to be seen.”&lt;ref&gt;Melanie Assad, Van Buren v. United States: An Employer Defeat or Hacker’s Victory – Or Something in Between, 21 UIC REV. INTELL. PROP. L. 166 (2022).&lt;/ref&gt; From a public trust perspective, this creates a double-edged sword: the decision limits government overreach in one sense, yet it also leaves the public more vulnerable to the very actors entrusted with sensitive data, raising the difficult questions about accountability, oversight, and the social value of trust in law enforcement. 

Both ''U.S. v Microsoft'' and ''Van Buren v. U.S.'' offer different stories connected by their influence over public trust and accountability; one raises concerns about how far a government's authority extends and the inherent ownership of encrypted data by companies driven by the market and productivity, while the other is seen as a win for protecting individuals from accidental criminalization of normal activities and a loss for holding law enforcement accountable by removing an extra layer of protection through authorization and credentials. As technology changes and evolves, shaping the application of data encryption, management, and access, future litigations may arise, and the question of who will maintain the public's trust in data privacy and digital autonomy will most likely continue to be overlooked and potentially mishandled. All of these court cases point toward a future that will quickly challenge long-held and accepted social values, especially in regard to extending those values to encrypted digital or cyber-physical platforms, and it is up to all participants and interested parties to make these social values known and considered, whether it is noted by an advocacy, non-profit organization, for-profit corporation, law enforcement agencies, regular citizens, or the courts of law.

==Conclusion==

The struggle over encryption backdoors is far from over, and there will almost certainly be more developments on this topic in the future as new solutions are proposed and enacted or defeated. Although we have covered significant parts of the debate in the United States and the United Kingdom, participants in other places have different perspectives. Interesting additions to this chapter could include updates covering future developments and expansion to cover encryption worldwide.

==References==
{{reflist}}

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    <title>User:PeterEasthope/sandbox</title>
    <ns>2</ns>
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      <contributor>
        <username>PeterEasthope</username>
        <id>660399</id>
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      <comment>Trying an alternative presentation of the system Variants as suggested by Steve Litt.</comment>
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      <text bytes="1930" sha1="j84u4i7833vy1p8wjyjo1bd7s0ex9jz" xml:space="preserve">&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;André Fischer, afi&lt;/span&gt;
&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;[[w:Zürich|Zürich&lt;/span&gt;]]
&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;[[w:Switzerland|Swiss Confederation]]

&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;[https://github.com/andreaspirklbauer Andreas Pirklbauer]&lt;/span&gt;
&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;[[w:Zürich|Zürich&lt;/span&gt;]]
&lt;span class=&quot;mw-ui-button&quot; style=&quot;border-style: solid; border-width: 1px; display: inline-block; margin: auto; width: 13em; text-align: center; Background-color:#F0FFFF;&quot;&gt;[[w:Switzerland|Swiss Confederation]]

{{plainlist |
* &lt;span style=&quot;background-color:#f0fff0;&quot;&gt;André Fischer, afi&lt;/span&gt;,&lt;ref name=&quot;afi&quot;/&gt; [[w:Zürich|Zürich]], [[w:Switzerland|Swiss Confederation]].
* &lt;span style=&quot;background-color:#f0fff0;&quot;&gt;[https://github.com/andreaspirklbauer Andreas Pirklbauer]&lt;/span&gt;, [[w:Zürich|Zürich]], [[w:Switzerland|Swiss Confederation]].
}}

{{unbulleted list
| &lt;span style=&quot;background-color:#f0fff0;&quot;&gt;André Fischer, afi&lt;/span&gt;,&lt;ref name=&quot;afi&quot;/&gt; [[w:Zürich|Zürich]], [[w:Switzerland|Swiss Confederation]].
| &lt;span style=&quot;background-color:#f0fff0;&quot;&gt;[https://github.com/andreaspirklbauer Andreas Pirklbauer]&lt;/span&gt;, [[w:Zürich|Zürich]], [[w:Switzerland|Swiss Confederation]].
}}</text>
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  <page>
    <title>Associative Composition Algebra/Split-quaternions</title>
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        <username>Rgdboer</username>
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      <comment>/* Computations */ mv one up, expand final</comment>
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      <text bytes="9206" sha1="64ifmg4zlprbq6ec52qisx5179yy1jz" xml:space="preserve">[[File:Hyperboloid2.png|thumb|right|250px|The imaginary units ''v'', such that 
''v''&lt;sup&gt; 2&lt;/sup&gt; = −1, lie on a two-sheeted hyperboloid in split quaternions]]
There are at least three portals leading to split quaternions: the dihedral group of a square, [[Abstract Algebra/2x2 real matrices|matrix products in M(2,R)]], and the modified Cayley-Dickson construction. The work of [[w: Max Zorn|Max Zorn]] on split octonions showed the necessity of including split real AC algebras in the aufbau of the category. 

The development through the dihedral group was started with a lemma in the [[Associative Composition Algebra/Introduction|Introduction]], and is completed with exercises below. 

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, &lt;math&gt;\begin{pmatrix}0 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}&lt;/math&gt; is j, &lt;math&gt;\begin{pmatrix}0 &amp; 1 \\- 1 &amp; 0 \end{pmatrix}&lt;/math&gt; is i, and &lt;math&gt;\begin{pmatrix}1 &amp; 0 \\ 0 &amp; -1 \end{pmatrix}&lt;/math&gt; is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:
: j&lt;sup&gt;2&lt;/sup&gt; = +1 = k&lt;sup&gt;2&lt;/sup&gt;, &amp;nbsp; j k = − i .

Then the real AC algebra of split-quaternions uses coefficients ''w, x, y, z'' &amp;isin; R to express an element, its conjugate, and the quadratic form N:
:&lt;math&gt;q = w + xi + yj + zk,\quad q^* = w - xi - yj - zk, \quad N(q) = w^2 + x^2 - y^2 - z^2 .&lt;/math&gt;

===Exercises===
1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

==Computations==
Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j&lt;sup&gt;2&lt;/sup&gt; = +1 = k&lt;sup&gt;2&lt;/sup&gt; and jk = &amp;minus;i, contrary to the ''quaternion group'', which is expressed with the same letters i, j, k, but which here refer to the ''dihedral group of a square'' instead.
# For r = j cos &amp;theta; + k sin &amp;theta;, show that r&lt;sup&gt;2&lt;/sup&gt; = +1 = &amp;minus;r r*.
# Show that &lt;math&gt;i r = j \cos (\theta + \pi/2) + k \sin (\theta + \pi/2) &lt;/math&gt;.
# Recall  that &lt;''q, t''&gt; = (''q t''* + ''t q''*)/2. Show &lt;''q, t''&gt; = real part of ''q t''*
# If ''u'' is a unit, &lt;''qu, tu''&gt; = ''uu''* &lt;''q, t''&gt;.
# '''Definition''': ''q'' and ''t'' are orthogonal when &lt;''q, t''&gt; = 0.
# Let ''p'' = i sinh ''a'' + ''r'' cosh ''a''. Show that ''p''&lt;sup&gt;2&lt;/sup&gt; = +1 for any ''a'' and ''r''.
# Let ''v'' = i cosh ''a'' + ''r'' sinh ''a''. Show that ''v''&lt;sup&gt;2&lt;/sup&gt; = &amp;minus;1.
# For a given ''a'' and ''r'', show that ''p'' and ''v'' are orthogonal.
# Let ''m'' = ''p'' exp(''bp'') = sinh ''b'' + ''p'' cosh ''b''. Show that ''m m''* = &amp;minus;1.
# Let ''w'' = exp(''bp'') = cosh ''b'' + ''p'' sinh ''b''. Show that ''m'' is orthogonal to ''w''.
# Show that ''m'' is orthogonal to ''v''.
# For any &amp;theta;, ''a'', and ''b'' defining ''r, p, w, v'', and ''m'', the set {''m, w, v'', i''r''} has these properties: the elements are pair-wise orthogonal and each has a norm of +1 or &amp;minus;1.

== Division-binarion representation ==
Split-quaternions are represented by matrices of the form &lt;math&gt;\begin{pmatrix} w &amp; z \\ z^* &amp; w^* \end{pmatrix}&lt;/math&gt; where ''w'' and ''z'' are division binarions. These matrices form a subalgebra of M(2,C) with determinant w w* &amp;minus; z z*. The group of units is obtained with non-zero determinant. When the determinant is equal to one, the subgroup is called the ''special unitary group'' and is written SU(1,1), expressing signature (1,1) with respect to the division binarions C.

Verification of the representation follows from the multiplicative properties of basis matrices:
:&lt;math&gt;i = \begin{pmatrix}i &amp; 0 \\ 0 &amp; -i \end{pmatrix}, \quad j = \begin{pmatrix}0 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}, \quad k = \begin{pmatrix}0 &amp; i \\ -i &amp; 0 \end{pmatrix}, \quad .&lt;/math&gt;
Thus, the squares of j and k are the identity matrix, and the square of i is negative identity.

Exercise: Verify the other identities involving i, j, and k for this representation.

==Vortex==
[[File:Cyclone_Catarina_from_the_ISS_on_March_26_2004.JPG|thumb|right|100px|Cyclone Catarina, March 26, 2004]]

The natural phenomena of hurricanes, tornados, waterspouts, and dust devils illustrate the vortex motion. They have in common a circular motion and upwelling in the center. The split-quaternions contribute to representation of such a velocity field when a certain differential operator is introduced.
===Toy vortex===
[[File:CouetteTaylorSystem.svg|thumb|right|200px|Fluid responds to differential rotation of cylinders]]
Concentric rotating cylinders have been used to study fluid motion in response to differential rotation of the cylinders. With a translucent outer cylinder, and bits of pepper to exhibit the fluid motion, there are various states of the system. With a small differential between rates of rotation of the inner and outer cylinder the state of laminar flow occurs. Each increment of radius is a lamina rotating in a circle. With a somewhat higher differential a secondary motion appears in the various meridians of the model. This motion corresponds to the upwelling seen in the natural phenomena. A third state of motion in the model has waves forming in the circulating mass. The principal investigators in this experimental setup were [[w:Arnulph Mallock]], [[w:Maurice Couette]], and [[w:G. I. Taylor]].

===Zabla===
The differential operator &quot;nabla&quot; or &quot;del&quot; is &lt;math&gt;\nabla = i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z} &lt;/math&gt; . A velocity potential is a function &lt;math&gt;f: R^3 \to R&lt;/math&gt;.

When nabla is applied to a velocity potential it gives a velocity field, a vector-valued function.

Zabla is the same operator as in quaternion case, but there is a difference when it is squared:
:&lt;math&gt;\nabla^2 = - \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} &lt;/math&gt; .

===Zarmonic functions===
[[File:Taylor-Couette_Streamlines_Re%3D950.png |thumb| right | 100px | Streamlines in x-r plane, theta partial zero]]

A ''harmonic function f'' has &lt;math&gt;\nabla^2 f = 0&lt;/math&gt; with the nabla convention, while a function will be called ''zarmonic'' when the zabla convention holds, as here with split-quaternions where the square of j or k is opposite the square of i.

Let &lt;math&gt;y^2 + z^2 = r^2&lt;/math&gt; with the y-z plane described in polar coordinates (r, &amp;theta;). Then
:&lt;math&gt;\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac {\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2}.&lt;/math&gt;

In the plane x = 0, a zarmonic function is harmonic. When the partial with respect to theta is zero, a zarmonic function satisfies the wave equation in an x-r plane.

The three states of motion in the toy vortex arise as terms differ from zero. The laminar flow, such as &lt;math&gt;f(r) = \log r&lt;/math&gt;, corresponds to partials with respect to ''x'' and &amp;theta; as zero. The meridian flow starts when the partial with respect to ''x'' departs from zero. The wavy characteristic show that partial with respect to theta is non-zero.

==Comparison to Minkowski space==
Split-quaternions can be compared to Minkowski space since both are four-dimensional real vector spaces admitting transformation by circular and hyperbolic rotations. In fact, split-quaternion spacetime has signature (2,2) and has unit sphere corresponding to Lie group SU(1,1) which frequently is applied as motions of the Poincare disk model of the hyperbolic plane.

The spacetime interval &lt;math&gt;t^2 - x^2 - y^2 - z^2&lt;/math&gt; is characteristic of Minkowski space &lt;math&gt;M = \{ (t, x, y, z) : t, x, y, z \in \Reals \} ,&lt;/math&gt; which has been taken for the canonical empty spacetime, and is used as the tangent space to curved spacetime with matter. The null cone in M is &lt;math&gt;K = \{ (t, x, y, z) : t^2 - x^2 - y^2 - z^2 = 0 \}&lt;/math&gt;, which is used to carry light in M as cosmos, hence K is called the light cone.
===Light torus===
[[File:Torus_cycles2.svg|right|thumb|200px|A torus with intersecting cycles]]
Using the division-binarion representation, the set of null split-quaternions is N(q) = |w|&lt;sup&gt;2&lt;/sup&gt; &amp;minus; |z|&lt;sup&gt;2&lt;/sup&gt; = 0. Thus, the null elements satisfy |w| = |z| = t. In the polar form, 
:&lt;math&gt;w = t \exp (i \theta), \quad z = t \exp(i \phi) .&lt;/math&gt; This null manifold corresponds to a parameter space
:&lt;math&gt;(t, \theta, \phi) \in \Reals_{&gt;0} \times S^1 \times S^1 .&lt;/math&gt;
So instead of the null cone there is a product space of positive real numbers with the null torus as the locus of light, which might be called the '''light torus'''. 

Let V be the null manifold and (V, x) the associative magma with matrix multiplication. This magma is non-commutative and has no identity element.

The change of signature from (1,3) to (2,2) augments light geometry and cuts down space, for instance, conceiving longitude to be a temporal variable, which is consistent with the development (1730 to 1761) of the marine chronometer by [[w:John Harrison]].


{{Chapter navigation|Quaternions|Homographies}}
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    <title>Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nf6/3. Nxe5/3...Nc6/4. Nxc6/4...dxc6</title>
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      <comment>Wrong move: should be Qh5 (Qa5 is not possible)</comment>
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|Stafford gambit
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|parent=[[../|Stafford gambit]]
}}
== 4...dxc6 ==

Black recaptures with the d-pawn. This opens the d-file for Black's queen and the diagonal for Black's light square bishop. The g4-square is now covered by Black's light square bishop. Black is prepared for a lighting-fast attack on White's position. They want to play ...Bc5 and pressure f2 with ...Ng4 or ...Nxe4, or by forming a battery with the bishop with ...Qd4. To prove White's advantage, they must keep their head and play solidly, stalling out Black's momentum. 

White's priorities are, in order:

# Find a way to protect their e-pawn to hold on to their material lead. 
# Find a way to prevent the knight from coming to g4. 

=== Defend with a pawn ===

The first option is [[/5. d3|'''5. d3''']] to defend e4. Though this appears to limit's White's light-square bishop, White plans to follow 5...Bc5 with 6. Be2 and control g4 that way, then build a strong centre with c3 and d4. 

[[/5. f3|'''5. f3''']] is an interesting try that derails the usual Stafford line. Ignoring common wisdom to &quot;never play f3&quot;, White defends e4 and g4 at the same time. Now 5...Bc5 runs into 6. c3 and 7. d4, and Black has a hard time making progress. The best move is instead 5...Be6. 

=== Defend while developing a piece ===

[[/5. Nc3|'''5. Nc3''']] is the most natural move, to develop a piece while defending e4, and staying flexible regarding the light-square bishop. This is a sharper, counterattacking alternative that generates tactics for both sides: playable but double-edged. 

White may hope to play d4 in one move, but will find it hard to do so as they no longer have a knight on f3 to support it, and White's knight on c3 prevents the c3 and d4 plan to shutdown Black's bishop. It is important in the 5. Nc3 line that White then controls g4 with 6. h3, and not try the combination 5. Nc3 and 6. Be2?? together. This is because White's bishop prevents Qf3 (see explanation in sample game). 

[[/5. Qe2|'''5. Qe2''']] can defend the e-pawn, and though it looks unnatural, it is playable so long as White avoids the trap of playing 5...Bc5 6. e5?? (6...Ng4 and a crushing attack on f2 follows, e.g. 7. f3 Nf2 8. Rg1 Nd3+ 9. Qxd3 Qxd3 10. Bxd3 Bxg1{{Chess/not|---}}). 6. h3 h5 7. d3 instead, and Black's initiative fizzles out.

=== Move the pawn ===

[[/5. e5|'''5. e5!?''']], kicking the knight, is another interesting try for White. After 5...Ne4, White must play 6. d4 to disallow 6...Bc5. 6. d3??, to kick the knight again, allows 6...Bc5 as in the [[Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nf6/3. Nxe5/3...Nc6#History|Lowens vs Stafford 1950 game]], where Black threatens mate in 2 (...Bxf2+ Ke2 ...Bg4#); 7. dxe4 and Black wins White's queen, 7. Be3 and Black wins an exchange.

== Sample game ==
{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6|caption=The position after 4...dxc6. White has accepted the Stafford gambit.|frame=1|float=right}}
=== Sergey Erenburg v Eric Rosen (2024) ===

This miniature played at the FIDE World Blitz championship illustrates the sharpness of the gambit and the precariousness of inaccurate play by White.

'''1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6'''

White accepts the Stafford gambit and this position is reached. 

'''5. Nc3'''

White chooses to defend the e4 pawn with their knight. 5. d3 is a sound alternative. 

'''5...Bc5'''

Black develops their bishop and pressures f2. Black threatens ...Ng4, attacking f2 twice and threatening ...Bxf2+. If White does not react to this threat they lose their advantage and more, e.g. 6. d3? Ng4 7. Be3 Nxe3 8. fxe3 Bxe3 {{Chess/not|--}}. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2|caption=Position after 6. Be2. White attempts defence with both Nc3 and Be2. This allows Black to win back the pawn, but Black chooses to push for a larger advantage.|frame=1|float=right}}

'''6. Be2?'''

Be2 protects the g4 square, but it is a mistake in the 5. Nc3 line. Black can win back the pawn with 6...Qd4!, forming a battery against f2 and forking the e4 pawn. 7. O-O Nxe4 8. Nxe4 Qxe4 {{Chess/not}}.

Be2 works in the 5. d3 line because the e4 pawn is guarded with a pawn. 5. d3 Bc5 6. Be2 Qd4? 7. O-O Nxe4?? 8. dxe4 Qxe4 9. Nc3 {{Chess/not|+++}}. Black is down an knight and a pawn for two pawns. 

'''6...h5!?''' 

Black chooses not to play 6...Qd4 (despite what the commentators may have thought, Rosen ''was'' familiar with the idea but opted for this tricky line instead).&lt;ref name=&quot;rosen_youtube&quot;&gt;[https://www.youtube.com/watch?v=qJdrMhfk5XA  Stafford Gambit Crushes Grandmaster at the World Championship - Eric Rosen (Youtube)]&lt;/ref&gt;

6...h5 supports the g4 square so that Black may play ...Ng4. Black's real intention is to use trading their knight on g4 to open the h file: ...Ng4 Bxg4 hxg4!

'''7. h3?'''

White's move seems to add control over g4 and so prevent the knight from coming in. However, Black is interested in prying open the h file and this will help them to do so. 

An improvement would be 7. d3. Black can half-open the h file with 7...Ng4 8. Bxg4 hxg4 but this is not so dangerous if White avoids short-castling. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2 h5 7. h3 Qd4|caption=Position after 7...Qd4. White must deal with the threat of Qxf2.|frame=1|float=right}}

'''7...Qd4''' 

''Now'' Black threatens mate on f2, encouraging White to castle into the danger. 

'''8. O-O Ng4 9. hxg4?? hxg4'''

An improvement would be 8. Rf1 to prevent mate.

Black sacrifices the knight to pry open the h file. Even here White had a choice: 9. Bxg4 hxg4 and White needn't take back on g4. 10. d3 and if 10...gxh3 11. g3 h2+ 12. Kh1 and the king is quite safe. White is only slightly worse. 

'''10. g3'''

N.B. 10. d3?? Qe5 (mate in 4) 11. g3?? Qxg3#. 

'''10...Qe5'''

Threatens mate in 1 (f-pawn is pinned).

'''11. Kg2'''

Unpins f-pawn. 

'''11...Bxf2'''

11...Qh5? fails to Rh1. Bxf2 deflects the rook from the back rank. 

'''12. Rxf2 Qh5 13. Bf3'''

If 12. Kxf2 Rh2+ 13. Ke3 (Ke1?? Qxg3#) Qxg3+ 14. Bf3{{Chess/not|---}} 

13. Bf3 creates an escape route for the king via f1 and e2.

'''13...gxf3+ 14. Qxf3?? Qh1# 0-1'''

White blunders mate in one in time pressure. 14. Rxf3 Qh2+ 15. Kf1{{Chess/not|---}} allows the game to go on.

== Theory table ==

{{ChessMid}}
==References==
{{reflist}}
{{wikipedia|Petrov Defence}}
{{Chess Opening Theory/Footer}}</text>
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        <username>Rsteadman17</username>
        <id>3577540</id>
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      <comment>Moved the note about move 12 after move 12.</comment>
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      <text bytes="6507" sha1="sjmngb1z5enfybt4eaa2gdkij1ksyzg" xml:space="preserve">{{Chess Opening Theory/Position|=
|Stafford gambit
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|parent=[[../|Stafford gambit]]
}}
== 4...dxc6 ==

Black recaptures with the d-pawn. This opens the d-file for Black's queen and the diagonal for Black's light square bishop. The g4-square is now covered by Black's light square bishop. Black is prepared for a lighting-fast attack on White's position. They want to play ...Bc5 and pressure f2 with ...Ng4 or ...Nxe4, or by forming a battery with the bishop with ...Qd4. To prove White's advantage, they must keep their head and play solidly, stalling out Black's momentum. 

White's priorities are, in order:

# Find a way to protect their e-pawn to hold on to their material lead. 
# Find a way to prevent the knight from coming to g4. 

=== Defend with a pawn ===

The first option is [[/5. d3|'''5. d3''']] to defend e4. Though this appears to limit's White's light-square bishop, White plans to follow 5...Bc5 with 6. Be2 and control g4 that way, then build a strong centre with c3 and d4. 

[[/5. f3|'''5. f3''']] is an interesting try that derails the usual Stafford line. Ignoring common wisdom to &quot;never play f3&quot;, White defends e4 and g4 at the same time. Now 5...Bc5 runs into 6. c3 and 7. d4, and Black has a hard time making progress. The best move is instead 5...Be6. 

=== Defend while developing a piece ===

[[/5. Nc3|'''5. Nc3''']] is the most natural move, to develop a piece while defending e4, and staying flexible regarding the light-square bishop. This is a sharper, counterattacking alternative that generates tactics for both sides: playable but double-edged. 

White may hope to play d4 in one move, but will find it hard to do so as they no longer have a knight on f3 to support it, and White's knight on c3 prevents the c3 and d4 plan to shutdown Black's bishop. It is important in the 5. Nc3 line that White then controls g4 with 6. h3, and not try the combination 5. Nc3 and 6. Be2?? together. This is because White's bishop prevents Qf3 (see explanation in sample game). 

[[/5. Qe2|'''5. Qe2''']] can defend the e-pawn, and though it looks unnatural, it is playable so long as White avoids the trap of playing 5...Bc5 6. e5?? (6...Ng4 and a crushing attack on f2 follows, e.g. 7. f3 Nf2 8. Rg1 Nd3+ 9. Qxd3 Qxd3 10. Bxd3 Bxg1{{Chess/not|---}}). 6. h3 h5 7. d3 instead, and Black's initiative fizzles out.

=== Move the pawn ===

[[/5. e5|'''5. e5!?''']], kicking the knight, is another interesting try for White. After 5...Ne4, White must play 6. d4 to disallow 6...Bc5. 6. d3??, to kick the knight again, allows 6...Bc5 as in the [[Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nf6/3. Nxe5/3...Nc6#History|Lowens vs Stafford 1950 game]], where Black threatens mate in 2 (...Bxf2+ Ke2 ...Bg4#); 7. dxe4 and Black wins White's queen, 7. Be3 and Black wins an exchange.

== Sample game ==
{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6|caption=The position after 4...dxc6. White has accepted the Stafford gambit.|frame=1|float=right}}
=== Sergey Erenburg v Eric Rosen (2024) ===

This miniature played at the FIDE World Blitz championship illustrates the sharpness of the gambit and the precariousness of inaccurate play by White.

'''1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6'''

White accepts the Stafford gambit and this position is reached. 

'''5. Nc3'''

White chooses to defend the e4 pawn with their knight. 5. d3 is a sound alternative. 

'''5...Bc5'''

Black develops their bishop and pressures f2. Black threatens ...Ng4, attacking f2 twice and threatening ...Bxf2+. If White does not react to this threat they lose their advantage and more, e.g. 6. d3? Ng4 7. Be3 Nxe3 8. fxe3 Bxe3 {{Chess/not|--}}. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2|caption=Position after 6. Be2. White attempts defence with both Nc3 and Be2. This allows Black to win back the pawn, but Black chooses to push for a larger advantage.|frame=1|float=right}}

'''6. Be2?'''

Be2 protects the g4 square, but it is a mistake in the 5. Nc3 line. Black can win back the pawn with 6...Qd4!, forming a battery against f2 and forking the e4 pawn. 7. O-O Nxe4 8. Nxe4 Qxe4 {{Chess/not}}.

Be2 works in the 5. d3 line because the e4 pawn is guarded with a pawn. 5. d3 Bc5 6. Be2 Qd4? 7. O-O Nxe4?? 8. dxe4 Qxe4 9. Nc3 {{Chess/not|+++}}. Black is down an knight and a pawn for two pawns. 

'''6...h5!?''' 

Black chooses not to play 6...Qd4 (despite what the commentators may have thought, Rosen ''was'' familiar with the idea but opted for this tricky line instead).&lt;ref name=&quot;rosen_youtube&quot;&gt;[https://www.youtube.com/watch?v=qJdrMhfk5XA  Stafford Gambit Crushes Grandmaster at the World Championship - Eric Rosen (Youtube)]&lt;/ref&gt;

6...h5 supports the g4 square so that Black may play ...Ng4. Black's real intention is to use trading their knight on g4 to open the h file: ...Ng4 Bxg4 hxg4!

'''7. h3?'''

White's move seems to add control over g4 and so prevent the knight from coming in. However, Black is interested in prying open the h file and this will help them to do so. 

An improvement would be 7. d3. Black can half-open the h file with 7...Ng4 8. Bxg4 hxg4 but this is not so dangerous if White avoids short-castling. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2 h5 7. h3 Qd4|caption=Position after 7...Qd4. White must deal with the threat of Qxf2.|frame=1|float=right}}

'''7...Qd4''' 

''Now'' Black threatens mate on f2, encouraging White to castle into the danger. 

'''8. O-O Ng4 9. hxg4?? hxg4'''

An improvement would be 8. Rf1 to prevent mate.

Black sacrifices the knight to pry open the h file. Even here White had a choice: 9. Bxg4 hxg4 and White needn't take back on g4. 10. d3 and if 10...gxh3 11. g3 h2+ 12. Kh1 and the king is quite safe. White is only slightly worse. 

'''10. g3'''

N.B. 10. d3?? Qe5 (mate in 4) 11. g3?? Qxg3#. 

'''10...Qe5'''

Threatens mate in 1 (f-pawn is pinned).

'''11. Kg2'''

Unpins f-pawn. 

'''11...Bxf2'''

11...Qh5? fails to Rh1. Bxf2 deflects the rook from the back rank. 

'''12. Rxf2'''

If 12. Kxf2 Rh2+ 13. Ke3 (Ke1?? Qxg3#) Qxg3+ 14. Bf3{{Chess/not|---}}

'''12...Qh5 13. Bf3''' 

13. Bf3 creates an escape route for the king via f1 and e2.

'''13...gxf3+ 14. Qxf3?? Qh1# 0-1'''

White blunders mate in one in time pressure. 14. Rxf3 Qh2+ 15. Kf1{{Chess/not|---}} allows the game to go on.

== Theory table ==

{{ChessMid}}
==References==
{{reflist}}
{{wikipedia|Petrov Defence}}
{{Chess Opening Theory/Footer}}</text>
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      <contributor>
        <username>JCrue</username>
        <id>2226064</id>
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|parent=[[../|Stafford gambit]]
}}
== 4...dxc6 ==

Black recaptures with the d-pawn. This opens the d-file for Black's queen and the diagonal for Black's light square bishop. The g4-square is now covered by Black's light square bishop. Black is prepared for a lighting-fast attack on White's position. They want to play ...Bc5 and pressure f2 with ...Ng4 or ...Nxe4, or by forming a battery with the bishop with ...Qd4. To prove White's advantage, they must keep their head and play solidly, stalling out Black's momentum. 

White's priorities are, in order:

# Find a way to protect their e-pawn to hold on to their material lead. 
# Find a way to prevent the knight from coming to g4. 

=== Defend with a pawn ===

The first option is [[/5. d3|'''5. d3''']] to defend e4. Though this appears to limit's White's light-square bishop, White plans to follow 5...Bc5 with 6. Be2 and control g4 that way, then build a strong centre with c3 and d4. 

[[/5. f3|'''5. f3''']] is an interesting try that derails the usual Stafford line. Ignoring common wisdom to &quot;never play f3&quot;, White defends e4 and g4 at the same time. Now 5...Bc5 runs into 6. c3 and 7. d4, and Black has a hard time making progress. The best move is instead 5...Be6. 

=== Defend while developing a piece ===

[[/5. Nc3|'''5. Nc3''']] is the most natural move, to develop a piece while defending e4, and staying flexible regarding the light-square bishop. This is a sharper, counterattacking alternative that generates tactics for both sides: playable but double-edged. 

White may hope to play d4 in one move, but will find it hard to do so as they no longer have a knight on f3 to support it, and White's knight on c3 prevents the c3 and d4 plan to shutdown Black's bishop. It is important in the 5. Nc3 line that White then controls g4 with 6. h3, and not try the combination 5. Nc3 and 6. Be2?? together. This is because White's bishop prevents Qf3 (see explanation in sample game). 

[[/5. Qe2|'''5. Qe2''']] can defend the e-pawn, and though it looks unnatural, it is playable so long as White avoids the trap of playing 5...Bc5 6. e5?? (6...Ng4 and a crushing attack on f2 follows, e.g. 7. f3 Nf2 8. Rg1 Nd3+ 9. Qxd3 Qxd3 10. Bxd3 Bxg1{{Chess/not|---}}). 6. h3 h5 7. d3 instead, and Black's initiative fizzles out.

=== Move the pawn ===

[[/5. e5|'''5. e5!?''']], kicking the knight, is another interesting try for White. After 5...Ne4, White must play 6. d4 to disallow 6...Bc5. 6. d3??, to kick the knight again, allows 6...Bc5 as in the [[Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nf6/3. Nxe5/3...Nc6#History|Lowens vs Stafford 1950 game]], where Black threatens mate in 2 (...Bxf2+ Ke2 ...Bg4#); 7. dxe4 and Black wins White's queen, 7. Be3 and Black wins an exchange.

== Sample game ==
{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6|caption=The position after 4...dxc6. White has accepted the Stafford gambit.|frame=1|float=right}}
=== Sergey Erenburg v Eric Rosen (2024) ===

This miniature played at the FIDE World Blitz championship illustrates the sharpness of the gambit and the precariousness of inaccurate play by White.&lt;ref&gt;[https://www.chessgames.com/perl/chessgame?gid=2820214 Erenburg v Rosen, 2024. Chessgames.com]&lt;/ref&gt;

'''1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6'''

White accepts the Stafford gambit and this position is reached. 

'''5. Nc3'''

White chooses to defend the e4 pawn with their knight. 5. d3 is a sound alternative. 

'''5...Bc5'''

Black develops their bishop and pressures f2. Black threatens ...Ng4, attacking f2 twice and threatening ...Bxf2+. If White does not react to this threat they lose their advantage and more, e.g. 6. d3? Ng4 7. Be3 Nxe3 8. fxe3 Bxe3 {{Chess/not|--}}. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2|caption=Position after 6. Be2. White attempts defence with both Nc3 and Be2. This allows Black to win back the pawn, but Black chooses to push for a larger advantage.|frame=1|float=right}}

'''6. Be2?'''

Be2 protects the g4 square, but it is a mistake in the 5. Nc3 line. Black can win back the pawn with 6...Qd4!, forming a battery against f2 and forking the e4 pawn. 7. O-O Nxe4 8. Nxe4 Qxe4 {{Chess/not}}.

Be2 works in the 5. d3 line because the e4 pawn is guarded with a pawn. 5. d3 Bc5 6. Be2 Qd4? 7. O-O Nxe4?? 8. dxe4 Qxe4 9. Nc3 {{Chess/not|+++}}. Black is down an knight and a pawn for two pawns. 

'''6...h5!?''' 

Black chooses not to play 6...Qd4 (despite what the commentators may have thought, Rosen ''was'' familiar with the idea but opted for this tricky line instead).&lt;ref name=&quot;rosen_youtube&quot;&gt;[https://www.youtube.com/watch?v=qJdrMhfk5XA  Stafford Gambit Crushes Grandmaster at the World Championship - Eric Rosen (Youtube)]&lt;/ref&gt;

6...h5 supports the g4 square so that Black may play ...Ng4. Black's real intention is to use trading their knight on g4 to open the h file: ...Ng4 Bxg4 hxg4!

'''7. h3?'''

White's move seems to add control over g4 and so prevent the knight from coming in. However, Black is interested in prying open the h file and this will help them to do so. 

An improvement would be 7. d3. Black can half-open the h file with 7...Ng4 8. Bxg4 hxg4 but this is not so dangerous if White avoids short-castling. 

{{Chess/board|moves=1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nc6 4. Nxc6 dxc6 5. Nc3 Bc5 6. Be2 h5 7. h3 Qd4|caption=Position after 7...Qd4. White must deal with the threat of Qxf2.|frame=1|float=right}}

'''7...Qd4''' 

''Now'' Black threatens mate on f2, encouraging White to castle into the danger. 

'''8. O-O Ng4 9. hxg4?? hxg4'''

An improvement would be 8. Rf1 to prevent mate.

Black sacrifices the knight to pry open the h file. Even here White had a choice: 9. Bxg4 hxg4 and White needn't take back on g4. 10. d3 and if 10...gxh3 11. g3 h2+ 12. Kh1 and the king is quite safe. White is only slightly worse. 

'''10. g3'''

N.B. 10. d3?? Qe5 (mate in 4) 11. g3?? Qxg3#. 

'''10...Qe5'''

Threatens mate in 1 (f-pawn is pinned).

'''11. Kg2'''

Unpins f-pawn. 

'''11...Bxf2'''

11...Qh5? fails to Rh1. Bxf2 deflects the rook from the back rank. 

'''12. Rxf2'''

If 12. Kxf2 Rh2+ 13. Ke3 (Ke1?? Qxg3#) Qxg3+ 14. Bf3{{Chess/not|---}}

'''12...Qh5 13. Bf3''' 

13. Bf3 creates an escape route for the king via f1 and e2.

'''13...gxf3+ 14. Qxf3?? Qh1# 0-1'''

White blunders mate in one in time pressure. 14. Rxf3 Qh2+ 15. Kf1{{Chess/not|---}} allows the game to go on.

== Theory table ==

{{ChessMid}}
==References==
{{reflist}}
{{wikipedia|Petrov Defence}}
{{Chess Opening Theory/Footer}}</text>
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  <page>
    <title>Geometry/Unified Angles</title>
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      <text bytes="7672" sha1="qft0vu7cyhyxv2irr0gfuun6b781tkg" xml:space="preserve">[[File:Angle vertex.svg|thumb|right|250px|Intersecting rays are the common feature of the three types of angle]]
Taking area of a region as a primitive notion, three species of angles are given a common basis. The magnitude of a slope, a hyperbolic or circular angle is determined by the area of an appropriate sector.
==Preview: From locus to group==
Given the base of a triangle, what is the locus of the vertex if the triangular area is to be kept constant? The answer, a line parallel to the base, is a feature of Euclidean geometry. This classically stated evocation of an image has been translated into group theory with its own terms. Furthermore, with analytic geometry, the image has expression with linear algebra as shear mapping. The collection of shear mappings forms a one-parameter group, in this case a transformation group that expresses a group action.

Another image is a rectangle {(0,0), (''x'',0), (0,''y''), (''x,y'')}, x&gt;0, y&gt;0, in the first quadrant of the Cartesian plane. What is the locus of (''x,y'') if the rectangle is to have constant area? The curve given by ''xy'' = constant is called a rectangular hyperbola. In this case, for ''c'' &gt; 0, the squeeze mapping (''x,y'') → ( ''c x, y''/''c'' ) permutes the hyperbola – the area of the rectangle after transformation is the same as the area before. The collection of squeeze mappings forms a group with elements corresponding to ''c''. Applied to the whole plane, a squeeze is a group action. Here is another one-parameter group.

==Introduction==
The hyperbolic angle is somewhat obscure but has been described in the wikibook [[Calculus/Hyperbolic angle|Calculus]]. This chapter of ''Geometry'' will clarify the connection to circular angle, the one measured from zero to 360 degrees since Alexandria. The analogy between circular and hyperbolic functions as determined by corresponding ''sectors'' was noted by [[w:Robert Baldwin Hayward]] in 1892.

The unification here requires differences of slope as a third angle species.  One can use the phrase &quot;angle of arc&quot; in the unification, as hyperbolic and circular arcs arise and line segments serve as &quot;arcs&quot; for the third angle species. The arcs also connote motion along the arc, such as rotation of a circular arc about its center, permuting the points of the extended arc. These motions can be described by 2x2 real matrices with determinant 1. Some reference to group theory is made: an additive group of angles corresponds under exponential isomorphism to a multiplicative group. Division in the group corresponds to subtraction of angles. Using standard positioning, the differences indicate directed angles. For instance, two oppositely directed angles are mapped by the exponential function to multiplicative inverses.

==Signed areas==
Traditionally circular arc has been measured as the ratio of its length to the radius, but here we use the area of sector of the arc when the radius squared is two, or r = &amp;radic; 2. Then the circumference becomes an arc measured to be &amp;pi; ''r''&lt;sup&gt;2&lt;/sup&gt; = 2&amp;pi;. A fractional sector has proportional area and gives the corresponding circular angle of arc. The notion that angle of arc should be measured with an area ratio was expressed by [[w:Alexander Macfarlane]] in his 1894 essay on the definitions of trigonometric functions (page 9): &quot;true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of the sector to the square of the radius.&quot;

Hyperbolic sectors corresponding to natural logarithm are constructed according to whether x is greater or less than one. A variable right triangle with area 1/2 is &lt;math&gt;V = \{(x, 1/x), \ (x,0), \ (0,0)\} .&lt;/math&gt; The isosceles case is &lt;math&gt;T = \{(1,1),\ (1,0),\ (0,0)\}.&lt;/math&gt; The natural logarithm is known as the area under ''y'' = 1/''x'' between one and ''x''. A positive hyperbolic angle is given by the area of 
&lt;math&gt;\int_1^x \frac{dt}{t} + T - V.&lt;/math&gt; A negative hyperbolic angle is given by the negative of the area &lt;math&gt;\int_x^1 \frac{dt}{t} + V - T.&lt;/math&gt; This convention is in accord with a negative natural logarithm for ''x'' in (0,1). Since ''T'' and ''V'' each have area 1/2, their difference is zero, so the hyperbolic angle is given by the natural logarithm.

The third angle species is easily described using slopes. A point (''x,y''), ''x''&gt;0, determines a slope ''m''=''y''/''x'', and indicates an angle between the x-axis and the ray from the origin to (''x,y''). A triangle of area ''m'' is formed by the x-axis, the slope ''m'' line, and ''x''= &amp;radic;2: ''A''=(1/2)(&amp;radic;2)(''m'' &amp;radic;2). For any two points on ''x''=&amp;radic;2 an angle at the origin has magnitude equal to the area formed by rays to the points.

For this angle, the arc is a line segment that can be taken as the base of a triangle. A classical theorem of Euclidean geometry is &quot;If the base and the area of a triangle be given, the locus of its vertex is a straight line parallel to the base.&quot; (see Robert Potts (1865) ''Euclid's Element of Geometry'', page 285.)

==Motions==
Each species of angle has a planar motion that moves it but keeps its magnitude constant. Since rotation leaves length invariant, area is also invariant, so circular angle is not modified in magnitude by rotation. The motions of the other species of angle are NOT length-preservers, but they are area-preservers.

In the case of hyperbolic angle, the motion squeezes a square into a rectangle of the same area. A seeming paradox arises with area and the hyperbola ''y'' = 1/''x'': take the harmonic series
&lt;math&gt;\{\sum_1^m \tfrac{1}{n} \}_{m=1}^\infty .&lt;/math&gt;  The terms in the sum become very small yet there is no upper bound on the sequence of partial sums. This divergence indicates that there is an infinite area between a hyperbola and its asymptotes.

Start with one '''wing''', the unit area &lt;math&gt;\int_1^e \tfrac{dx}{x} = \log e .&lt;/math&gt; In fact, there is a wing of hyperbolic angle between e&lt;sup&gt;n&lt;/sup&gt; and e&lt;sup&gt;n+1&lt;/sup&gt; for any ''n'', so the number of wings is infinite. A step from one wing to the next is made by the linear transformation that squeezes the unit square to a rectangle of length e and height 1/e. This feature of ''y''=1/''x'' was presented in 1647 by G. de Saint-Vincent as a characteristic of the quadrature of the hyperbola, and provided a geometric expression of natural logarithm, a more common designation of area also associated with hyperbolic angle, and here reinforced by wing measure. 

A shear mapping takes a rectangle to a parallelogram of the same area as the rectangle. This motion of the plane increases or decreases the slopes of lines through the origin by a constant amount.
The arc of this third species of angle is a segment on x= &amp;radic;2 that moves up or down with shearing, but the triangle with this segment as base, and apex at the origin, has constant area.

In the ring M(2,R) of [[Abstract Algebra/2x2 real matrices|2x2 real matrices]], the ones with determinant equal to 1 preserve area, and are elements of the special linear group SL(2,R), which forms a type of unit sphere in M(2,R). Three types of subgroups of SL(2,R) arise as exponential images of angles, which also express the planar motions characteristic of each angle species.

Though [[w:Euler's formula|Leonard Euler]] is associated with the correspondence for circular angle, the other angles have been absorbed into a more general study of general linear groups GL(n,''F'') over a field ''F'', and also of tangent vectors at the group identity, initiated by [[w:Lie theory|Sophus Lie]]. Indeed, the algebra of tangents at 1 is called ''Lie  algebra''.

{{BookCat}}</text>
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  <page>
    <title>Module:List</title>
    <ns>828</ns>
    <id>426613</id>
    <revision>
      <id>4631265</id>
      <parentid>4247961</parentid>
      <timestamp>2026-04-18T18:43:05Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>3441010</id>
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      <text bytes="5643" sha1="80u633hue68752ie9c34bo23bvxg416" xml:space="preserve">local libUtil = require('libraryUtil')
local checkType = libUtil.checkType
local mTableTools = require('Module:TableTools')

local p = {}

local listTypes = {
	['bulleted'] = true,
	['unbulleted'] = true,
	['horizontal'] = true,
	['ordered'] = true,
	['horizontal_ordered'] = true
}

function p.makeListData(listType, args)
	-- Constructs a data table to be passed to p.renderList.
	local data = {}

	-- Classes and TemplateStyles
	data.classes = {}
	data.templatestyles = ''
	if listType == 'horizontal' or listType == 'horizontal_ordered' then
		table.insert(data.classes, 'hlist')
		data.templatestyles = mw.getCurrentFrame():extensionTag{
			name = 'templatestyles', args = { src = 'Hlist/styles.css' }
		}
	elseif listType == 'unbulleted' then
		table.insert(data.classes, 'plainlist')
		data.templatestyles = mw.getCurrentFrame():extensionTag{
			name = 'templatestyles', args = { src = 'Plainlist/styles.css' }
		}
	end
	table.insert(data.classes, args.class)

	-- Main div style
	data.style = args.style

	-- Indent for horizontal lists
	if listType == 'horizontal' or listType == 'horizontal_ordered' then
		local indent = tonumber(args.indent)
		indent = indent and indent * 1.6 or 0
		if indent &gt; 0 then
			data.marginLeft = indent .. 'em'
		end
	end
	
	-- List style types for ordered lists
	-- This could be &quot;1, 2, 3&quot;, &quot;a, b, c&quot;, or a number of others. The list style
	-- type is either set by the &quot;type&quot; attribute or the &quot;list-style-type&quot; CSS
	-- property.
	if listType == 'ordered' or listType == 'horizontal_ordered' then 
		data.listStyleType = args.list_style_type or args['list-style-type']
		data.type = args['type']

		-- Detect invalid type attributes and attempt to convert them to
		-- list-style-type CSS properties.
		if data.type 
			and not data.listStyleType
			and not tostring(data.type):find('^%s*[1AaIi]%s*$')
		then
			data.listStyleType = data.type
			data.type = nil
		end
	end
	
	-- List tag type
	if listType == 'ordered' or listType == 'horizontal_ordered' then
		data.listTag = 'ol'
	else
		data.listTag = 'ul'
	end

	-- Start number for ordered lists
	data.start = args.start
	if listType == 'horizontal_ordered' then
		-- Apply fix to get start numbers working with horizontal ordered lists.
		local startNum = tonumber(data.start)
		if startNum then
			data.counterReset = 'listitem ' .. tostring(startNum - 1)
		end
	end

	-- List style
	 -- ul_style and ol_style are included for backwards compatibility. No
	 -- distinction is made for ordered or unordered lists.
	data.listStyle = args.list_style

	-- List items
	-- li_style is included for backwards compatibility. item_style was included
	-- to be easier to understand for non-coders.
	data.itemStyle = args.item_style or args.li_style
	data.items = {}
	for _, num in ipairs(mTableTools.numKeys(args)) do
		local item = {}
		item.content = args[num]
		item.style = args['item' .. tostring(num) .. '_style']
			or args['item_style' .. tostring(num)]
		item.value = args['item' .. tostring(num) .. '_value']
			or args['item_value' .. tostring(num)]
		table.insert(data.items, item)
	end
	
	return data
end

function p.renderList(data)
	-- Renders the list HTML.
	
	-- Return the blank string if there are no list items.
	if type(data.items) ~= 'table' or #data.items &lt; 1 then
		return ''
	end
	
	-- Render the main div tag.
	local root = mw.html.create('div')
	for _, class in ipairs(data.classes or {}) do
		root:addClass(class)
	end
	root:css{['margin-left'] = data.marginLeft}
	if data.style then
		root:cssText(data.style)
	end

	-- Render the list tag.
	local list = root:tag(data.listTag or 'ul')
	list
		:attr{start = data.start, type = data.type}
		:css{
			['counter-reset'] = data.counterReset,
			['list-style-type'] = data.listStyleType
		}
	if data.listStyle then
		list:cssText(data.listStyle)
	end

	-- Render the list items
	for _, t in ipairs(data.items or {}) do
		local item = list:tag('li')
		if data.itemStyle then
			item:cssText(data.itemStyle)
		end
		if t.style then
			item:cssText(t.style)
		end
		item
			:attr{value = t.value}
			:wikitext(t.content)
	end

	return data.templatestyles .. tostring(root)
end

function p.renderTrackingCategories(args)
	local isDeprecated = false -- Tracks deprecated parameters.
	for k, v in pairs(args) do
		k = tostring(k)
		if k:find('^item_style%d+$') or k:find('^item_value%d+$') then
			isDeprecated = true
			break
		end
	end
	local ret = ''
	if isDeprecated then
		ret = ret .. '[[Category:List templates with deprecated parameters]]'
	end
	return ret
end

function p.makeList(listType, args)
	if not listType or not listTypes[listType] then
		error(string.format(
			&quot;bad argument #1 to 'makeList' ('%s' is not a valid list type)&quot;,
			tostring(listType)
		), 2)
	end
	checkType('makeList', 2, args, 'table')
	local data = p.makeListData(listType, args)
	local list = p.renderList(data)
	local trackingCategories = p.renderTrackingCategories(args)
	return list .. trackingCategories
end

for listType in pairs(listTypes) do
	p[listType] = function (frame)
		local mArguments = require('Module:Arguments')
		local origArgs = mArguments.getArgs(frame, {
			frameOnly = ((frame and frame.args and frame.args.frameonly or '') ~= ''),
			valueFunc = function (key, value)
			if not value or not mw.ustring.find(value, '%S') then return nil end
			if mw.ustring.find(value, '^%s*[%*#;:]') then
				return value
			else
				return value:match('^%s*(.-)%s*$')
			end
			return nil
		end
		})
		-- Copy all the arguments to a new table, for faster indexing.
		local args = {}
		for k, v in pairs(origArgs) do
			args[k] = v
		end
		return p.makeList(listType, args)
	end
end

return p</text>
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    <title>Wikibooks:Sandbox</title>
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    <id>464822</id>
    <revision>
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      <parentid>4631032</parentid>
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      <contributor>
        <username>JackBot</username>
        <id>396820</id>
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      <minor />
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      <origin>4631215</origin>
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&lt;!-- Hello! Feel free to try your formatting and editing skills below this line. As this page is for editing experiments, this page will automatically be cleaned every 12 hours. --&gt;</text>
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    <title>User:JJPMaster (bot)/markAdmins-Data.json</title>
    <ns>2</ns>
    <id>471107</id>
    <revision>
      <id>4631242</id>
      <parentid>4630791</parentid>
      <timestamp>2026-04-18T15:49:14Z</timestamp>
      <contributor>
        <username>JJPMaster (bot)</username>
        <id>3488561</id>
      </contributor>
      <minor />
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  <page>
    <title>Nursing Home Social Services Reference</title>
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      <text bytes="17427" sha1="ldqgmtno5neelo1lb1jdt09mkq0sn7p" xml:space="preserve">= '''Everything You Wanted to Know About Social Services Work in Nursing Homes (but were afraid to ask!)''' =
= Table of Contents =

== Opening ==

* [[Nursing Home Social Services Reference/Frontspiece/Foreword ]] Comments on goals of the book

* [[Nursing Home Social Services Reference/Frontspiece/Prologue ]] Structure of book and background of authors 

* [[Nursing Home Social Services Reference/Frontspiece/Instructions for Authors ]]  Instructions for contributors

== Chapter 1 Introduction to Social Services ==
* [[Nursing Home Social Services Reference/Introduction to Social Services/Overview of the Career]]  Overview of Social Services as a Career

* [[Nursing Home Social Services Reference/Introduction to Social Services/Prerequisites]] Good things to have to get and do the job

* [[Nursing Home Social Services Reference/Introduction to Social Services/Areas of Competency]] Competencies required for the job

* [[Nursing Home Social Services Reference/Introduction to Social Services/Rewards of the Job]] Why do this job?

* [[Nursing Home Social Services Reference/Introduction to Social Services/Possible Pitfalls]] What to avoid (title)

* [[Nursing Home Social Services Reference/Introduction to Social Services/Typical Job Description]] Typical job description annotated

* [[Nursing Home Social Services Reference/Introduction to Social Services/Applicable Social Work Principles]] What you learned in School that might help

* [[Nursing Home Social Services Reference/Introduction to Social Services/Applicable Social Work Skills]] What skills you have developed that might help you including: Empathy Communication Organization Critical Thinking Active Listening Self-care Cultural  Patience Professional Advocacy Rapport
* [[Nursing Home Social Services Reference/Introduction to Social Services/Quality Assurance Performance Improvement]] The process for improving performance at your facility. What to know about the process, how to contribute, and how to track improvements in the SS Department 

== Chapter 2 Flow of Care ==

* [[Nursing Home Social Services Reference/Flow of Care/Pre-Admission]] What SS does prior to admission

* [[Nursing Home Social Services Reference/Flow of Care/Admission]] What SS does as part of the admission process

* [[Nursing Home Social Services Reference/Flow of Care/Intake ]]SS documentation at admission
* [[Nursing Home Social Services Reference/Flow of Care/Advanced Directives]] documentation of residents wishes for resuscitation and life sustaining treatment

* [[Nursing Home Social Services Reference/Flow of Care/Verifying Benefits and Finances]] SS part of helping res get benefits

* [[Nursing Home Social Services Reference/Flow of Care/Personal Needs Funds]] Medicaid offers some of a resident's SSA funds to be made available to the resident for personal needs. The amount varies by benefit mix and state. 
* [[Nursing Home Social Services Reference/Flow of Care/Initial Care Plans]] Care plan at admission based on medical record or MD orders

* [[Nursing Home Social Services Reference/Flow of Care/Behaviors and Behavior Plans]] If there are bx issues

* [[Nursing Home Social Services Reference/Flow of Care/Adjustment and Transitions]] What resident may need in coming to SNF

* [[Nursing Home Social Services Reference/Flow of Care/Roommates and Room Moves]] How room moves work or should work

* [[Nursing Home Social Services Reference/Flow of Care/Changes in Condition]] What if a resident declines and needs more care or improves and requires less care

* [[Nursing Home Social Services Reference/Flow of Care/Hospitalization]] What SS does to help a resident get to hospital

* [[Nursing Home Social Services Reference/Flow of Care/Changes in Benefits]] SS awareness of any change of benefits d/t hospitalization, lottery win, etc. 

* [[Nursing Home Social Services Reference/Flow of Care/Hospice]] SS awareness of coordination for hospice care

* [[Nursing Home Social Services Reference/Flow of Care/Discharges and Transfers]] How to discharge or transfer resident safely and successfully

== Chapter 3: Surveys and Complaints ==
* [[Nursing Home Social Services Reference/State Surveys/What and Why of State Surveys]] Awareness of goals of survey

* [[Nursing Home Social Services Reference/State Surveys/Highest Practicable Level of Well-being]] Goal of CMS for resident

* [[Nursing Home Social Services Reference/State Surveys/Social Services' Tags]] Which Tags will be SS responsibility
* [[Nursing Home Social Services Reference/State Surveys/Survey Readiness]] Being prepared for the day the surveyors arrive

* [[Nursing Home Social Services Reference/State Surveys/Plans of Correction (POC)]] How to respond to survey tags

* [[Nursing Home Social Services Reference/State Surveys/Documenting POC Compliance]] How to document compliance

== Chapter 4: Principles of Care ==
* [[Nursing Home Social Services Reference/Principles of Care/Dignity]] Offer residents dignity and respect

* [[Nursing Home Social Services Reference/Principles of Care/Highest Level of Independence]] Help residents maintain capabilities

* [[Nursing Home Social Services Reference/Principles of Care/Trauma Informed Care]] Offer care that helps residents avoid trauma triggers; reduce unwanted bxs and resident safety

* [[Nursing Home Social Services Reference/Principles of Care/Control Over Care]] Help residents exercise their rights over medical choices Proxies, Medical POAs, Guardians or conservators

* [[Nursing Home Social Services Reference/Principles of Care/Physical and Pharmacological Constraints]] Residents are to be free of these in the SNF; if these are needed residents need to be in a psychiatric facility

* [[Nursing Home Social Services Reference/Principles of Care/Equitable Access to Care and Privileges]] MCD, private and MCR residents should have access to appropriate services and care regardless of their insurance status

* [[Nursing Home Social Services Reference/Principles of Care/Advocacy for Residents]] SS workers are ethically required to advocate for residents who may need this

* [[Nursing Home Social Services Reference/Principles of Care/Confidentiality]] HIPAA requires that PHI be kept undisclosed unless there is written permission from resident (or guardian) for disclosure

* [[Nursing Home Social Services Reference/Principles of Care/On-Going Support]] SS workers cannot withdraw their care and support if it is needed by a resident on an on-ongoing basis

* [[Nursing Home Social Services Reference/Principles of Care/Managing Grief, Loss and Depression]] These mental health issues may require special help and often is a part of resident life as a residents experience functional decline and lose friends and family

* [[Nursing Home Social Services Reference/Principles of Care/Differences between Dementia, Delirium, Depression]] SS worker may have to distinguish these for proper support, appropriate care plans, workable assessments and useful consultation at PsychPharm meetings

== Chapter 5: Emergencies and Crises ==

* [[Nursing Home Social Services Reference/Emergencies and Crises/Behavioral Issues and Changes]]  D/t drugs, psychosis, delusions or poor emotional control, bx issues will need to be planned for and managed; SS is the nexus of this care

* [[Nursing Home Social Services Reference/Emergencies and Crises/Mental Health Crisis]] Like behavioral crises, episodes that leave resident in mental crisis must be dealt with (30 day bed hold, Medicare uplevel)

* [[Nursing Home Social Services Reference/Emergencies and Crises/Addiction and Substance Abuse]] This is a difficult topic requiring coordination with nursing, admin and SS

* [[Nursing Home Social Services Reference/Emergencies and Crises/Family Crises]] Resident family can often be a source of conflict and bx issues for their resident and others

* [[Nursing Home Social Services Reference/Emergencies and Crises/Death]] A resident death has an impact on other residents and the community; SS has a part in helping the dying resident and the community process this; disposing of deceased's belongings

* [[Nursing Home Social Services Reference/Emergencies and Crises/Elopement]] When a resident disappears or leaves without notice, there are steps to take to cover SNF's responsibility; SS is an important part of the search and rescue team

== Chapter 6: Contexts of Care ==
* [[Nursing Home Social Services Reference/Context of Care/Business of Nursing Homes]] Knowing how the SNF runs its business will help SS make choices for the department and resources for residents

* [[Nursing Home Social Services Reference/Context of Care/Resident Family Issues]] Issues with resident family that do not rise to the level of crises can still cause issues to be dealt with or managed

* [[Nursing Home Social Services Reference/Context of Care/Safe Discharges]] Residents who leave the facility for non-financial reasons have a right to expect a discharge to a safe and appropriate level of care

== Chapter 7 Psychopharmacology ==

* [[Nursing Home Social Services Reference/Psychopharmacology/Classes of Psychopharm Medications]] Drugs come in various categories and are not always used for s/s for which the drugs might appear to be designed

* [[Nursing Home Social Services Reference/Psychopharmacology/Gradual Dose Reductions]] GDRs are req'd by CMS and are complicated; SS may have to be aware or record changes in bx

* [[Nursing Home Social Services Reference/Psychopharmacology/Black Box Warnings]] Some Rxs are &quot;off label&quot; and need special monitoring by the team; nurses are key here 

* [[Nursing Home Social Services Reference/Psychopharmacology/Side Effects]] Every drug has side-effects; good to know the major ones and how to observe and plan for them

* [[Nursing Home Social Services Reference/Psychopharmacology/Polypharmacy and Drug Interactions]] Drugs can interact with each other in good and bad ways; as eyes and ears for the PCP, it is good to know some

* [[Nursing Home Social Services Reference/Psychopharmacology/Overmedication]] This is important to detect for the resident's well-being. Nurses might be first to see it; SS should know the signs

* [[Nursing Home Social Services Reference/Psychopharmacology/Abuse]] Cheeking meds and saving them is a tactic for abuse, selling or for a suicide plan; nurses will be the first line; but know what to look for and when to look

== Chapter 8 Pathologies with Psychiatric Signs and Symptoms ==
* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/UTI]] UTIs can cause delirium/hallucinations in older people and it is common among older people (including men).

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Delirium]] Delirium can be caused by fevers, meds and other conditions; it requires medical assessment and may appear like psychosis

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Parkinson's Disease]] causes visual hallucinations in some sufferers

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Hepatic Encephalopathy]] Some liver conditions can impact brain function and appear like delirium and / or dementia

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Brain Tumors and TBIs]] This kind of damage can affect emotions, thoughts, hallucinations, cause seizures and other issues

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Dementia with Behaviors and/or Psychosis]] Dementia can cause behavioral or psychotic episodes as part of the cognitive decline and brain damage that occurs

* [[Nursing Home Social Services Reference/Medical Psychiatric Pathologies/Huntington's Disease]] Huntington's affects physical strength and coordination but can also impact mental capacities causing hallucinations and delusions as well as distress.

== Chapter 9 Career Paths ==

* [[Nursing Home Social Services Reference/Career Paths/License]] A SS job qualifies as supervisable work toward an LCSW. It also provides valuable experience for the Nursing Home Administrator (NHA) license.

* [[Nursing Home Social Services Reference/Career Paths/Social Services Director]] SSD is a first step up from Social Services staff member or Assistant and is a path to a corporate SS director or possibly an Adminisrator (NHA)
* [[Nursing Home Social Services Reference/Career Paths/Corporate Social Services|Nursing Home Social Services Reference/Career Paths/Corporate Social Services Coordinator]] In facilities run by corporations, there is often a coordinator/director or corporate Vice President in charge of social services 

* [[Nursing Home Social Services Reference/Career Paths/Nursing Home Administrator]]  Operating a nursing home requires an executive with an understanding of SS and many more talents and skills. 

* [[Nursing Home Social Services Reference/Career Paths/Community Mental Health]]  CMH Agency's offer important skills for maintaining mental well-being for residents. 

* [[Nursing Home Social Services Reference/Career Paths/Medical Social Worker]]  With the experience of being a SS worker, transitioning to Medical Social Worker can be rewarding and use all the knowledge and skills learned in the nursing home. 

== Chapter 10 Self Care ==
*[[Nursing Home Social Services Reference/Self Care Intro]]  Overview and enumeration of important areas of self care
*[[Nursing Home Social Services Reference/Boundaries|Nursing Home Social Services Reference/Self Care/Boundaries in Self Care]] Discussion of boundaries: which boundaries, how to create them and enforce them consistently 
*[[Nursing Home Social Services Reference/Self Care/Support Systems]] Actively building communities of support 
*[[Nursing Home Social Services Reference/Self Care/Self-Awareness]] Being self aware to see issues of self care before it's too late 
*[[Nursing Home Social Services Reference/Self Care/Review and Reflection]] Making time to think about career, self-expression, power and resilience 
*[[Nursing Home Social Services Reference/Self Care/Rest and Relaxation]] Scheduling time for sufficient rest and recharging 
*[[Nursing Home Social Services Reference/Self Care/Self-Compassion]] Giving yourself unconditional self-regard 
*[[Nursing Home Social Services Reference/Self Care/Meningful Engagement|Nursing Home Social Services Reference/Self Care/Meaningful Engagement]] Engaging with communities, movements or activities such as advocacy that uplift your values and vision of the world. 

== Appendix A: Resources ==

=== Resident  and Resident-Family Advocacy Organizations ===

# LTCCC https://nursinghome411.org/about/
# Consumer Voice https://theconsumervoice.org/about/
# Nursing Home Abuse Center https://www.nursinghomeabusecenter.com/about/
# Justice in Aging https://justiceinaging.org/about/

===PsychPharm Aids===
* [[Nursing Home Social Services Reference/Appendix/Beer's List]] List of recommended maximum doses of psychoactive meds for demented res (guideline)  NIH Version of article: https://pmc.ncbi.nlm.nih.gov/articles/PMC3571677/  USC copy of pocket version: https://gwep.usc.edu/wp-content/uploads/2023/11/AGS-2023-BEERS-Pocket-PRINTABLE.pdf

===There's an App for That!===
* [[Nursing Home Social Services Reference/Appendix/PsychoTech]] Applications on iPads or other tablets and laptops can provide some supportive care for anxiety, insomnia and depression. This has the advantage of being accessible with the resident needs them and the disadvantage that familiarity with such technology is required. For older adults that already have these skills, this option is ideal. For those who might require training, there may be opportunities for Occupational Therapy interventions. Groups that assist with this could also be identified as an activity for residents sponsored by the Activities Department of the facility.  

===Certificate Programs===
* [[Nursing Home Social Services Reference/Appendix/Foundational Competencies in Older Adult Mental Health]]  This course is highly recommended as an overview of geriatric issues. It is on-line and offers CE or free non-CE participation. Non-CE participation still obtains the certificate. Colorado licensing does not require CEs. See Rush Medical School for more information or search for the above title in your favorite search engine.

* [[Nursing Home Social Services Reference/Appendix/Non-Violent_Crisis_Intervention]]  This course helps with intervening with behaviors that could lead to violence. 

=== Glossary ===
[[Nursing Home Social Services Reference/Appendix/Non-Violent_Crisis_Intervention|Nursing Home Social Services Reference/Appendix/Glossary]]  A list of terms that you may encounter and the meaning of these terms.

==Appendix B: References==

===Minimum Data Set (MDS)===
B. Fries and K. Murphy, ''Development of the Nursing Home Resident Assessment Instrument in the USA.'' Age and Aging, 1997 found at:
https://www.academia.edu/15976682/Development_of_the_Nursing_Home_Resident_Assessment_Instrument_in_the_USA?auto=download&amp;email_work_card=download-paper
[Retrieved June 30, 2025]

=== Survey Tags and Severity Matrix ===
LTCCC Guide Nursing Home Overview, Appendix 1,2 and 3 at https://nursinghome411.org/wp-content/uploads/2021/10/Guide-Appendices.pdf [Retrieved January 24, 2025]
__INDEX__

{{BookCat}}
[[Category:Social Work]]
[[Category:Nursing Homes]]</text>
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== Plain list ==
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    <title>User:Dom walden/Multivariate Analytic Combinatorics/Stratified Singular Varieties</title>
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      <text bytes="20093" sha1="qj3dx1kzijh619bi0m62wxbuq6pjkum" xml:space="preserve">== Introduction ==

We hinted in the [[User:Dom_walden/Multivariate_Analytic_Combinatorics/Cauchy-Hadamard_Theorem_and_Exponential_Bounds#Domain_of_convergence|previous chapter]] that singularities and the domain of convergence in the multivariate case are more complicated than in the univariate case. In this chapter, we present this in more detail.

First, we demonstrate the different types of what, in the multivariate case, are called '''singular varieties'''. Second, we show how to decompose the more complex singular varieties using '''Whitney stratification'''. Finally, we present '''critical points''' as the points of the singular varieties we are interested in for asymptotics.

== Singular varieties ==

For a function &lt;math display=&quot;inline&quot;&gt;F(z) = \frac{P(z)}{Q(z)}&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;z \in \C^d&lt;/math&gt;, we define the '''singular variety''' &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; the set of points in &lt;math display=&quot;inline&quot;&gt;\C^d&lt;/math&gt; such that &lt;math display=&quot;inline&quot;&gt;Q(z) = 0&lt;/math&gt;, i.e. &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ z \in \C^d : Q(z) = 0 \}&lt;/math&gt;.

As we saw in the single variable (&lt;math display=&quot;inline&quot;&gt;z \in \C&lt;/math&gt;), [[Analytic_Combinatorics/Meromorphic_Functions|meromorphic]] case, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; consisted of isolated points.

In the multivariate case, this can be much more complicated and can consist of one or more of the following:
* A single hyperplane
* Intersecting hyperplanes (called arrangements which may or may not be transverse)
* A cone (called cone points)

=== Complex hyperplanes ===

For complex vectors &lt;math display=&quot;inline&quot;&gt;z = (x_1 + iy_1, \cdots, x_d + iy_d)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;w = (u_1 + iv_1, \cdots, u_d + iv_d)&lt;/math&gt; the '''Hermitian scalar product''' is defined&lt;ref&gt;Shabat 1992, pp. 2.&lt;/ref&gt;

&lt;math display=&quot;block&quot;&gt;\langle z, w \rangle = \sum_{i=1}^d x_i u_i + \sum_{i=1}^d y_i v_i + i \sum_{i=1}^d (y_i u_i - x_i v_i).&lt;/math&gt;

A '''complex hyperplane''' means a set of points &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; in complex space such that for a fixed, non-zero vector &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; and constant complex number &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;&lt;ref&gt;Shabat 1992, pp. 2.&lt;/ref&gt;

&lt;math display=&quot;block&quot;&gt;\langle z, a \rangle = b.&lt;/math&gt;

This happens when the set of points &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; are perpendicular or '''orthogonal''' to &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;.

[image?]

=== Single hyperplane ===

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{1 - x - y} = \sum_{n \geq 0} \sum_{m \geq 0} \binom{n + m}{m} x^m y^n&lt;/math&gt;

there are singularities at any point in &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ (x, y) : 0 \leq x \leq 1, y = 1 - x \}&lt;/math&gt;. Below we see &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; for different values of &lt;math display=&quot;inline&quot;&gt;|x|, |y|&lt;/math&gt;.&lt;ref&gt;Mishna 2020, pp. 143.&lt;/ref&gt;

[[File:Singular variety of generating function for binomial coefficients.png|400px]]

Bear in mind that the above graph loses some information due to the axes being the modulus of the two inputs &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;. For example, &lt;math display=&quot;inline&quot;&gt;|x| = |y| = 1&lt;/math&gt; is a singularity when &lt;math display=&quot;inline&quot;&gt;x = (1/2, i \sqrt{3}/2)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y = (1/2, -i \sqrt{3}/2)&lt;/math&gt; (roughly).

=== Intersecting hyperplanes ===

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(3-2x-y)(3-x-2y)}&lt;/math&gt;

the singular variety is a union of two singular varieties, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-2x-y} = \{ (x, y) : y = 3 - 2x \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-x-2y} = \{ (x, y) : y = \frac{3 - x}{2} \}&lt;/math&gt;. We use the definition of the scalar product above to demonstrate that both are complex hyperplanes. For &lt;math display=&quot;inline&quot;&gt;x = x_0 + ix_1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y = y_0 + iy_1&lt;/math&gt; and some fixed vector &lt;math display=&quot;inline&quot;&gt;(u_0 + iu_1, v_0 + iv_1) \in \C^2&lt;/math&gt; the scalar product is

&lt;math display=&quot;block&quot;&gt;\langle (x, y), (u, v) \rangle = x_0 u_0 + y_0 v_0 + x_1 u_1 + y_1 v_1 + i((x_1 u_0 - x_0 u_1) + (y_1 v_0 - y_0 v_1)) = b&lt;/math&gt;

for constant complex number &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;.

Below we see &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt;, plotting separately the real parts of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; (left figure) and the imaginary parts of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; (right figure). The left figure demonstrates that &lt;math display=&quot;inline&quot;&gt;\langle (x_0, y_0), (u_0, v_0) \rangle = x_0 u_0 + y_0 v_0 = c_0&lt;/math&gt; for fixed &lt;math display=&quot;inline&quot;&gt;c_0&lt;/math&gt; and the right that &lt;math display=&quot;inline&quot;&gt;\langle (x_1, y_1), (u_1, v_1) \rangle = x_1 u_1 + y_1 v_1 = c_1&lt;/math&gt; for fixed &lt;math display=&quot;inline&quot;&gt;c_1&lt;/math&gt;. Notice that the two blue graphs and the two yellow graphs have the same slope. Therefore, &lt;math display=&quot;inline&quot;&gt;(u_1, v_1)&lt;/math&gt; is just a translation of &lt;math display=&quot;inline&quot;&gt;(u_0, v_0)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;(-u_1, -v_1)&lt;/math&gt; is the same but flipped 180 degrees and, as a result, &lt;math display=&quot;inline&quot;&gt;\langle (x_0, y_0), (-u_1, -v_1) \rangle = -x_0 u_1 - y_0 v_1 = c_2&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\langle (x_1, y_1), (u_0, v_0) \rangle = x_1 u_0 + y_1 v_0 = c_3&lt;/math&gt;. Therefore, we can find our &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;\langle (x, y), (u, v) \rangle = c_0 + c_1 + i(c_2 + c_3) = b.&lt;/math&gt;

[union_of_hyperplanes.py]

We look at the point where &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-2x-y}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-x-2y}&lt;/math&gt; intersect and draw both their slopes or '''tangent spaces''', as shown in the below image. Taking all possible linear combinations of the vectors which form the basis of these tangent spaces, we get a space called the '''span''' of the vectors. When the span of the tangent spaces is equal to &lt;math display=&quot;inline&quot;&gt;\C^d&lt;/math&gt; (in our case &lt;math display=&quot;inline&quot;&gt;\C^2&lt;/math&gt;), we call these types of arrangements '''transverse'''.

[image?]

We can demonstrate this separately for the real and imaginary parts to show that &lt;math display=&quot;inline&quot;&gt;((r_0 + ir_1, s_0 + is_1) (t_0 + it_1, q_0 + iq_1))&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;r_0, s_0&lt;/math&gt; is determined by the blue real, &lt;math display=&quot;inline&quot;&gt;t_0, q_0&lt;/math&gt; the yellow real, &lt;math display=&quot;inline&quot;&gt;r_1, s_1&lt;/math&gt; by the blue imaginary and &lt;math display=&quot;inline&quot;&gt;t_1, q_1&lt;/math&gt; by the yellow imaginary. We see that &lt;math display=&quot;inline&quot;&gt;((r_0, s_0), (t_0, q_0))&lt;/math&gt; spans the two-dimensional real space and &lt;math display=&quot;inline&quot;&gt;((r_1, s_1), (t_1, q_1))&lt;/math&gt; spans the two-dimensional &quot;imaginary&quot; space.

[union_of_hyperplanes_tangent_spaces.py]

Another example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(3 - 2x - y)(3 - x - 2y)(2 - x - y)}&lt;/math&gt;

all three singular varieties &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3 - 2x - y} = \{ (x, y) : y = 3 - 2x \}&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3 -x - 2y} = \{ (x, y) : y = \frac{3 - x}{2} \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{2 - x - y} = \{ (x, y) : y = 2 - x \}&lt;/math&gt; intersect at &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;. Their slopes are much like our first example in this section but with an extra one slope which is a combination of the other two slopes [example?]. Therefore, they are not transverse. Instead, we look at the set of points where the hyperplanes intersect, called the '''intersection lattice''', and the set of points where their tangent planes intersect (also called the intersection lattice). If these are equal, it is an '''arrangement'''.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 329.&lt;/ref&gt;

[union_of_hyperplanes_arrangement2.py]

However, note that the three hyperplanes are transverse when taken pairwise. We will show how to decompose such an arrangement in later chapters.

A final example of something that is not an arrangement, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(2 - x - y)(1 - xy)}&lt;/math&gt;

the singular varieties &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{2 - x - y} = \{ (x, y) : y = 2 - x \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{1 - xy} = \{ (x, y) : y = 1/x \}&lt;/math&gt; intersect at &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt; with identical slopes and, therefore, cannot span more than one dimension and, therefore, are not transverse. Nor are they arrangements as the intersection lattice of their hyperplanes is a single point (&lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;) but the intersection lattice of their tangent planes is the line &lt;math display=&quot;inline&quot;&gt;y = 2 - x&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 314.&lt;/ref&gt;

[union_of_hyperplanes_arrangement.py]

=== Cone point ===

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y, z) = \frac{1}{1 + xyz - (1/3)(x + y + z + xy + yz + xz)}&lt;/math&gt;

the singular varieties meet at a single point &lt;math display=&quot;inline&quot;&gt;(1, 1, 1)&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 228.&lt;/ref&gt;

== Whitney stratification ==

In the case of a single hyperplane we can skip this step. The '''Whitney stratification''' just contains one strata, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt;.

Otherwise, our aim is to decompose &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; into a union of (not necessarily connected) submanifolds such that:
# each submanifold is closed and smooth
# if &lt;math display=&quot;inline&quot;&gt;S_\alpha \subset \overline{S_\beta}&lt;/math&gt; then any sequences where &lt;math display=&quot;inline&quot;&gt;\{x_i\} \subset S_\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\{y_i\} \subset S_\alpha&lt;/math&gt; which both converge to &lt;math display=&quot;inline&quot;&gt;y \in S_\alpha&lt;/math&gt;, the lines &lt;math display=&quot;inline&quot;&gt;l_i = \overline{x_iy_i}&lt;/math&gt; converge to a line &lt;math display=&quot;inline&quot;&gt;l&lt;/math&gt; and the tangent planes &lt;math display=&quot;inline&quot;&gt;T_{x_i}(S_\beta)&lt;/math&gt; converge to a plane &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; then &lt;math display=&quot;inline&quot;&gt;T_y(S_\alpha) \subset T&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;l \subseteq T&lt;/math&gt; (these are the '''Whitney conditions'''.)&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 534.&lt;/ref&gt;

For example,

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(1 - x)(1 - y)(1 - z)}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; consists of three planes where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;. At the lines where they intersect they fail to be smooth. Therefore, we put the intersection lines in their own strata leaving us the strata:
* The &lt;math display=&quot;inline&quot;&gt;xy&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt;
* The &lt;math display=&quot;inline&quot;&gt;xz&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;
* The &lt;math display=&quot;inline&quot;&gt;yz&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (1, 1, z) \in \C^3 \}&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (1, y, 1) \in \C^3 \}&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (x, 1, 1) \in \C^3 \}&lt;/math&gt;

This does not meet the Whitney conditions at all points. For example, any sequence of points along the &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;-axis converging to a point on the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis will have tangents along the &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;-axis, but any point on the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis to which it converges will have tangent along the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis. Therefore, we need to separate the point &lt;math display=&quot;inline&quot;&gt;(1, 1, 1)&lt;/math&gt; into its own strata.&lt;ref&gt;Mishna 2020, pp. 179.&lt;/ref&gt;

== Critical points ==

Given a height function &lt;math display=&quot;inline&quot;&gt;h_r(z) = - \sum_{i=1}^d r_i \log(|z_i|)&lt;/math&gt; with derivative given by the Jacobian matrix &lt;math display=&quot;inline&quot;&gt;dh_r(z) = [-r_1/z_1 \cdots -r_d/z_d]&lt;/math&gt;.

For each strata, we want to find out where the derivative of our height function, when restricted to that strata, equals the matrix with entries all zero. The point at which this happens is called a '''critical point'''.

Suppose we have function &lt;math display=&quot;inline&quot;&gt;\frac{1}{1 - x - y}&lt;/math&gt; in the direction &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;, which has the singular variety &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ (x, y) : 0 \leq x \leq 1, y = 1 - x \}&lt;/math&gt;). The derivative of the height function restricted to &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;dh(x, 1 - x) = [-(1/x) - (1 / (x - 1)), 0]&lt;/math&gt;. This is zero at the critical point &lt;math display=&quot;inline&quot;&gt;(x, y) = (1/2, 1/2)&lt;/math&gt;.

[height1.py]

The above shows a graph of this. The critical point is highlighted in red. The value of the height function at the critical point, its '''critical value''', is roughly &lt;math display=&quot;inline&quot;&gt;1.4&lt;/math&gt;.

If we change the direction to &lt;math display=&quot;inline&quot;&gt;(1, 2)&lt;/math&gt;, the derivative changes to &lt;math display=&quot;inline&quot;&gt;dh(x, 1 - x) = [-(1/x) - (2 / (x - 1)), 0]&lt;/math&gt; and the critical point changes to &lt;math display=&quot;inline&quot;&gt;(x, y) = (1/3, 2/3)&lt;/math&gt; with critical value of roughly &lt;math display=&quot;inline&quot;&gt;1.9&lt;/math&gt;. This is graphed below.

[height3.py]

We can also see the height function as the dot product of two vectors &lt;math display=&quot;inline&quot;&gt;-(r_1, \cdots, r_d) \cdot (\log|z_1|, \cdots, \log|z_d|)&lt;/math&gt;.&lt;ref&gt;Melczer 2021, pp. 195.&lt;/ref&gt; This is minimised when both vectors are parallel.

This happens when &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x) = (f_{e^{x_1}}(e^{x_1}) e^{x_1} \cdots f_{e^{x_d}}(e^{x_d}) e^{x_d})&lt;/math&gt; is parallel to &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt;. We solve for &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and then take its exponent to get the critical point.

[jsklfjsld.py]

The above graph shows the case of &lt;math display=&quot;inline&quot;&gt;f(x, y) = \frac{1}{1 - x - y}&lt;/math&gt; for the direction &lt;math display=&quot;inline&quot;&gt;r = (1, 2)&lt;/math&gt;. The black arrows show the vector &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x)&lt;/math&gt;. The blue arrows show the vector &lt;math display=&quot;inline&quot;&gt;-(1, 2)&lt;/math&gt;. The point in the same direction at roughly &lt;math display=&quot;inline&quot;&gt;(-1.1, -0.4)&lt;/math&gt;.

Equivalently, we can use the fact that &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x)&lt;/math&gt; is identical to what it called the '''logarithmic gradient''' &lt;math display=&quot;inline&quot;&gt;\nabla_\log f(z) = (z_1 f_{z_1}(z_1) \cdots z_d f_{z_d}(z_d))&lt;/math&gt;, if we set &lt;math display=&quot;inline&quot;&gt;z = e^x&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 214-215.&lt;/ref&gt; Then, we can solve directly for &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; without having to take exponents. We use the logarithmic gradient when calculating critical points below.

[Need to explain Morse index?]

== Computing Whitney stratifications and critical points ==

[ https://melczer.ca/textbook/ ]

=== Computing with ideals ===

A polynomial ring K[z] is a sum of the form \sum_{i=0}^n c_i z^i where
c_i \in K. For example, 1 + 3z + 2z^2.

A polynomial ideal I is a subset of K[z] such that if a and b are in I
then a + b is also in I (closed under addition in I) and if a is in I
and c is any element in K[z] then ca is in I (closed under
multiplication in K[z]).

A set {g_1, ..., g_n} generates the ideal I if any member of I is a
linear combination of the members of the generating set.

A Grobner basis of I is a generating set with the property that any
non-zero member of I has leading term divisible by the leading term of
some member of the Grobner basis. A reduced Grobner basis is a GB such
that no monomial of g_i is divisible by the leading term of g_j for i
\neq j.

V(I) is the set of all roots in K of all elements of I.

=== GBs for Whitney stratification ===

=== GBs for critical points ===

Given a function &lt;math display=&quot;inline&quot;&gt;F(z) = \frac{P(z)}{Q(z)}&lt;/math&gt;, a direction &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt; and a Whitney stratified space &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \mathcal{F}_0 \subset \mathcal{F}_1 \cdots \subset \mathcal{F}_k = \empty&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;1 \leq m \leq k&lt;/math&gt;, we form the set &lt;math display=&quot;inline&quot;&gt;I_m = \{ f \in K[z] : f(z) = 0, z \in \mathcal{F}_m \}&lt;/math&gt; and calculate a prime decomposition &lt;math display=&quot;inline&quot;&gt;I_m = P_1 \cap \cdots \cap P_l&lt;/math&gt;. The zero set of each &lt;math display=&quot;inline&quot;&gt;P_j&lt;/math&gt; corresponds to a different hyperplane within the stratum and we calculate the critical point on each zero set &lt;math display=&quot;inline&quot;&gt;\mathcal{V}(P_j)&lt;/math&gt;.

If &lt;math display=&quot;inline&quot;&gt;P_j = \langle p_1, \cdots, p_s \rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_j&lt;/math&gt; has codimension &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, then a critical point happens when &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt; is &quot;parallel&quot; to the space spanned by &lt;math display=&quot;inline&quot;&gt;\nabla_\log p_i(z) = (z_1 \partial p_i / \partial z_1, \cdots, z_d \partial p_i / \partial z_d)&lt;/math&gt; for each &lt;math display=&quot;inline&quot;&gt;i&lt;/math&gt;. This has dimension &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;(?) and so we want the below matrix to have at most &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; linearly independent rows (i.e. have a rank of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  &amp; \nabla_\log p_1(z) &amp; \\
  &amp; \vdots &amp; \\
  &amp; \nabla_\log p_s(z) &amp; \\
  r_1 &amp; \cdots &amp; r_d
 \end{pmatrix}&lt;/math&gt;

Solving the system of equations defined by each &lt;math display=&quot;inline&quot;&gt;p_i(z)&lt;/math&gt; and the determinants of each &lt;math display=&quot;inline&quot;&gt;c + 1&lt;/math&gt; minor being zero and the non-vanishing of the generating set of &lt;math display=&quot;inline&quot;&gt;I_1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;z_1 \cdots z_d&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 244-245.&lt;/ref&gt;

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y, z) = \frac{1}{A(x, y, z) B(x, y, z)} = \frac{1}{(4 - x - 2y - z)(4 - 2x - y - z)}&lt;/math&gt;

To construct &lt;math display=&quot;inline&quot;&gt;I_0&lt;/math&gt;, we form an ideal generated by the basis &lt;math display=&quot;inline&quot;&gt;AB&lt;/math&gt; (i.e. &lt;math display=&quot;inline&quot;&gt;I_0 = \langle AB \rangle&lt;/math&gt;). Its prime decomposition is &lt;math display=&quot;inline&quot;&gt;I_0 = P_1 \cap P_2 = \langle A \rangle \cap \langle B \rangle&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;P_1&lt;/math&gt; our matrix is

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  x A_x &amp; y A_y &amp; z A_z \\
  r &amp; s &amp; t
 \end{pmatrix}&lt;/math&gt;

leading to the system of equations

&lt;math display=&quot;block&quot;&gt;\begin{align}
4 - x - 2y - z &amp;= 0 \\
tx - rz &amp;= 0 \\
2ty - sz &amp;= 0
\end{align}&lt;/math&gt;

giving the critical point &lt;math display=&quot;inline&quot;&gt;(4r, 2s, 4t)&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;2r = s&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;I_1 = P_1 = \langle A, B \rangle&lt;/math&gt; we have the matrix

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  x A_x &amp; y A_y &amp; z A_z \\
  x B_x &amp; y B_y &amp; z B_z \\
  r &amp; s &amp; t
 \end{pmatrix}&lt;/math&gt;

giving the system of equations

&lt;math display=&quot;block&quot;&gt;\begin{align}
4 - x - 2y - z &amp;= 0 \\
4 - 2x - y - z &amp;= 0 \\
ryz + sxz - 3txy &amp;= 0
\end{align}&lt;/math&gt;

which has the critical point &lt;math display=&quot;inline&quot;&gt;(4/3)(r + s, r + s, 3t)&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 246-248.&lt;/ref&gt;

[Section 10.4 is about how to recognise multiple points]

== Notes ==
{{Reflist}}

== References ==
* {{cite book | last=Melczer | first=Stephen | title=An Invitation to Analytic Combinatorics: From One to Several Variables | publisher=Springer Texts &amp; Monographs in Symbolic Computation | year=2021 | url=https://melczer.ca/files/Melczer-SubmittedManuscript.pdf }}
* {{cite book | last=Mishna | first=Marni | title=Analytic Combinatorics: A Multidimensional Approach | publisher=Taylor &amp; Francis Group, LLC | year=2020 }}
* {{cite book | last1=Pemantle | first1=Robin | last2=Wilson | first2=Mark C. | last3=Melczer | first3=Stephen | title=Analytic Combinatorics in Several Variables | publisher=Cambridge University Press | year=2024 | edition=2nd | url=https://acsvproject.com/PemantleWilsonMelczer23.pdf }}
* {{cite book | last=Shabat | first=B. V. | title=Introduction to Complex Analysis. Part II: Functions of Several Variables | publisher=American Mathematical Society, Providence, Rhode Island | year=1992 }}</text>
      <sha1>qj3dx1kzijh619bi0m62wxbuq6pjkum</sha1>
    </revision>
    <revision>
      <id>4631294</id>
      <parentid>4631293</parentid>
      <timestamp>2026-04-19T08:02:59Z</timestamp>
      <contributor>
        <username>Dom walden</username>
        <id>3209423</id>
      </contributor>
      <comment>/* Cone point */</comment>
      <origin>4631294</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20502" sha1="o0355eyeu0e1qlwra4y1vzqcsy58io0" xml:space="preserve">== Introduction ==

We hinted in the [[User:Dom_walden/Multivariate_Analytic_Combinatorics/Cauchy-Hadamard_Theorem_and_Exponential_Bounds#Domain_of_convergence|previous chapter]] that singularities and the domain of convergence in the multivariate case are more complicated than in the univariate case. In this chapter, we present this in more detail.

First, we demonstrate the different types of what, in the multivariate case, are called '''singular varieties'''. Second, we show how to decompose the more complex singular varieties using '''Whitney stratification'''. Finally, we present '''critical points''' as the points of the singular varieties we are interested in for asymptotics.

== Singular varieties ==

For a function &lt;math display=&quot;inline&quot;&gt;F(z) = \frac{P(z)}{Q(z)}&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;z \in \C^d&lt;/math&gt;, we define the '''singular variety''' &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; the set of points in &lt;math display=&quot;inline&quot;&gt;\C^d&lt;/math&gt; such that &lt;math display=&quot;inline&quot;&gt;Q(z) = 0&lt;/math&gt;, i.e. &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ z \in \C^d : Q(z) = 0 \}&lt;/math&gt;.

As we saw in the single variable (&lt;math display=&quot;inline&quot;&gt;z \in \C&lt;/math&gt;), [[Analytic_Combinatorics/Meromorphic_Functions|meromorphic]] case, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; consisted of isolated points.

In the multivariate case, this can be much more complicated and can consist of one or more of the following:
* A single hyperplane
* Intersecting hyperplanes (called arrangements which may or may not be transverse)
* A cone (called cone points)

=== Complex hyperplanes ===

For complex vectors &lt;math display=&quot;inline&quot;&gt;z = (x_1 + iy_1, \cdots, x_d + iy_d)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;w = (u_1 + iv_1, \cdots, u_d + iv_d)&lt;/math&gt; the '''Hermitian scalar product''' is defined&lt;ref&gt;Shabat 1992, pp. 2.&lt;/ref&gt;

&lt;math display=&quot;block&quot;&gt;\langle z, w \rangle = \sum_{i=1}^d x_i u_i + \sum_{i=1}^d y_i v_i + i \sum_{i=1}^d (y_i u_i - x_i v_i).&lt;/math&gt;

A '''complex hyperplane''' means a set of points &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; in complex space such that for a fixed, non-zero vector &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; and constant complex number &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;&lt;ref&gt;Shabat 1992, pp. 2.&lt;/ref&gt;

&lt;math display=&quot;block&quot;&gt;\langle z, a \rangle = b.&lt;/math&gt;

This happens when the set of points &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; are perpendicular or '''orthogonal''' to &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;.

[image?]

=== Single hyperplane ===

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{1 - x - y} = \sum_{n \geq 0} \sum_{m \geq 0} \binom{n + m}{m} x^m y^n&lt;/math&gt;

there are singularities at any point in &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ (x, y) : 0 \leq x \leq 1, y = 1 - x \}&lt;/math&gt;. Below we see &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; for different values of &lt;math display=&quot;inline&quot;&gt;|x|, |y|&lt;/math&gt;.&lt;ref&gt;Mishna 2020, pp. 143.&lt;/ref&gt;

[[File:Singular variety of generating function for binomial coefficients.png|400px]]

Bear in mind that the above graph loses some information due to the axes being the modulus of the two inputs &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;. For example, &lt;math display=&quot;inline&quot;&gt;|x| = |y| = 1&lt;/math&gt; is a singularity when &lt;math display=&quot;inline&quot;&gt;x = (1/2, i \sqrt{3}/2)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y = (1/2, -i \sqrt{3}/2)&lt;/math&gt; (roughly).

=== Intersecting hyperplanes ===

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(3-2x-y)(3-x-2y)}&lt;/math&gt;

the singular variety is a union of two singular varieties, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-2x-y} = \{ (x, y) : y = 3 - 2x \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-x-2y} = \{ (x, y) : y = \frac{3 - x}{2} \}&lt;/math&gt;. We use the definition of the scalar product above to demonstrate that both are complex hyperplanes. For &lt;math display=&quot;inline&quot;&gt;x = x_0 + ix_1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y = y_0 + iy_1&lt;/math&gt; and some fixed vector &lt;math display=&quot;inline&quot;&gt;(u_0 + iu_1, v_0 + iv_1) \in \C^2&lt;/math&gt; the scalar product is

&lt;math display=&quot;block&quot;&gt;\langle (x, y), (u, v) \rangle = x_0 u_0 + y_0 v_0 + x_1 u_1 + y_1 v_1 + i((x_1 u_0 - x_0 u_1) + (y_1 v_0 - y_0 v_1)) = b&lt;/math&gt;

for constant complex number &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;.

Below we see &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt;, plotting separately the real parts of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; (left figure) and the imaginary parts of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt; (right figure). The left figure demonstrates that &lt;math display=&quot;inline&quot;&gt;\langle (x_0, y_0), (u_0, v_0) \rangle = x_0 u_0 + y_0 v_0 = c_0&lt;/math&gt; for fixed &lt;math display=&quot;inline&quot;&gt;c_0&lt;/math&gt; and the right that &lt;math display=&quot;inline&quot;&gt;\langle (x_1, y_1), (u_1, v_1) \rangle = x_1 u_1 + y_1 v_1 = c_1&lt;/math&gt; for fixed &lt;math display=&quot;inline&quot;&gt;c_1&lt;/math&gt;. Notice that the two blue graphs and the two yellow graphs have the same slope. Therefore, &lt;math display=&quot;inline&quot;&gt;(u_1, v_1)&lt;/math&gt; is just a translation of &lt;math display=&quot;inline&quot;&gt;(u_0, v_0)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;(-u_1, -v_1)&lt;/math&gt; is the same but flipped 180 degrees and, as a result, &lt;math display=&quot;inline&quot;&gt;\langle (x_0, y_0), (-u_1, -v_1) \rangle = -x_0 u_1 - y_0 v_1 = c_2&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\langle (x_1, y_1), (u_0, v_0) \rangle = x_1 u_0 + y_1 v_0 = c_3&lt;/math&gt;. Therefore, we can find our &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;

&lt;math display=&quot;block&quot;&gt;\langle (x, y), (u, v) \rangle = c_0 + c_1 + i(c_2 + c_3) = b.&lt;/math&gt;

[union_of_hyperplanes.py]

We look at the point where &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-2x-y}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3-x-2y}&lt;/math&gt; intersect and draw both their slopes or '''tangent spaces''', as shown in the below image. Taking all possible linear combinations of the vectors which form the basis of these tangent spaces, we get a space called the '''span''' of the vectors. When the span of the tangent spaces is equal to &lt;math display=&quot;inline&quot;&gt;\C^d&lt;/math&gt; (in our case &lt;math display=&quot;inline&quot;&gt;\C^2&lt;/math&gt;), we call these types of arrangements '''transverse'''.

[image?]

We can demonstrate this separately for the real and imaginary parts to show that &lt;math display=&quot;inline&quot;&gt;((r_0 + ir_1, s_0 + is_1) (t_0 + it_1, q_0 + iq_1))&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;r_0, s_0&lt;/math&gt; is determined by the blue real, &lt;math display=&quot;inline&quot;&gt;t_0, q_0&lt;/math&gt; the yellow real, &lt;math display=&quot;inline&quot;&gt;r_1, s_1&lt;/math&gt; by the blue imaginary and &lt;math display=&quot;inline&quot;&gt;t_1, q_1&lt;/math&gt; by the yellow imaginary. We see that &lt;math display=&quot;inline&quot;&gt;((r_0, s_0), (t_0, q_0))&lt;/math&gt; spans the two-dimensional real space and &lt;math display=&quot;inline&quot;&gt;((r_1, s_1), (t_1, q_1))&lt;/math&gt; spans the two-dimensional &quot;imaginary&quot; space.

[union_of_hyperplanes_tangent_spaces.py]

Another example, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(3 - 2x - y)(3 - x - 2y)(2 - x - y)}&lt;/math&gt;

all three singular varieties &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3 - 2x - y} = \{ (x, y) : y = 3 - 2x \}&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{3 -x - 2y} = \{ (x, y) : y = \frac{3 - x}{2} \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{2 - x - y} = \{ (x, y) : y = 2 - x \}&lt;/math&gt; intersect at &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;. Their slopes are much like our first example in this section but with an extra one slope which is a combination of the other two slopes [example?]. Therefore, they are not transverse. Instead, we look at the set of points where the hyperplanes intersect, called the '''intersection lattice''', and the set of points where their tangent planes intersect (also called the intersection lattice). If these are equal, it is an '''arrangement'''.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 329.&lt;/ref&gt;

[union_of_hyperplanes_arrangement2.py]

However, note that the three hyperplanes are transverse when taken pairwise. We will show how to decompose such an arrangement in later chapters.

A final example of something that is not an arrangement, for

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(2 - x - y)(1 - xy)}&lt;/math&gt;

the singular varieties &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{2 - x - y} = \{ (x, y) : y = 2 - x \}&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\mathcal{V}_{1 - xy} = \{ (x, y) : y = 1/x \}&lt;/math&gt; intersect at &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt; with identical slopes and, therefore, cannot span more than one dimension and, therefore, are not transverse. Nor are they arrangements as the intersection lattice of their hyperplanes is a single point (&lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;) but the intersection lattice of their tangent planes is the line &lt;math display=&quot;inline&quot;&gt;y = 2 - x&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 314.&lt;/ref&gt;

[union_of_hyperplanes_arrangement.py]

=== Cone point ===

The singular variety may look like a cone.

For example,

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{x^2 + y^2}&lt;/math&gt;

has singularities in the case that &lt;math display=&quot;inline&quot;&gt;y = ix&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;y = -ix&lt;/math&gt;. When plotted, it looks like two cones whose points meet at the origin. This is not the same as a union of hyperplanes as this is a single variety [right?]

[cone_point.py]

Another example, for

&lt;math display=&quot;block&quot;&gt;F(x, y, z) = \frac{1}{1 + xyz - (1/3)(x + y + z + xy + yz + xz)}&lt;/math&gt;

the singular varieties meet at a single point &lt;math display=&quot;inline&quot;&gt;(1, 1, 1)&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 228.&lt;/ref&gt;

== Whitney stratification ==

In the case of a single hyperplane we can skip this step. The '''Whitney stratification''' just contains one strata, &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt;.

Otherwise, our aim is to decompose &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; into a union of (not necessarily connected) submanifolds such that:
# each submanifold is closed and smooth
# if &lt;math display=&quot;inline&quot;&gt;S_\alpha \subset \overline{S_\beta}&lt;/math&gt; then any sequences where &lt;math display=&quot;inline&quot;&gt;\{x_i\} \subset S_\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\{y_i\} \subset S_\alpha&lt;/math&gt; which both converge to &lt;math display=&quot;inline&quot;&gt;y \in S_\alpha&lt;/math&gt;, the lines &lt;math display=&quot;inline&quot;&gt;l_i = \overline{x_iy_i}&lt;/math&gt; converge to a line &lt;math display=&quot;inline&quot;&gt;l&lt;/math&gt; and the tangent planes &lt;math display=&quot;inline&quot;&gt;T_{x_i}(S_\beta)&lt;/math&gt; converge to a plane &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; then &lt;math display=&quot;inline&quot;&gt;T_y(S_\alpha) \subset T&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;l \subseteq T&lt;/math&gt; (these are the '''Whitney conditions'''.)&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 534.&lt;/ref&gt;

For example,

&lt;math display=&quot;block&quot;&gt;F(x, y) = \frac{1}{(1 - x)(1 - y)(1 - z)}&lt;/math&gt;

&lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; consists of three planes where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;. At the lines where they intersect they fail to be smooth. Therefore, we put the intersection lines in their own strata leaving us the strata:
* The &lt;math display=&quot;inline&quot;&gt;xy&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt;
* The &lt;math display=&quot;inline&quot;&gt;xz&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;
* The &lt;math display=&quot;inline&quot;&gt;yz&lt;/math&gt; plane at &lt;math display=&quot;inline&quot;&gt;x = 1&lt;/math&gt; with the lines removed where &lt;math display=&quot;inline&quot;&gt;y = 1&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;z = 1&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (1, 1, z) \in \C^3 \}&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (1, y, 1) \in \C^3 \}&lt;/math&gt;
* The line &lt;math display=&quot;inline&quot;&gt;\{ (x, 1, 1) \in \C^3 \}&lt;/math&gt;

This does not meet the Whitney conditions at all points. For example, any sequence of points along the &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;-axis converging to a point on the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis will have tangents along the &lt;math display=&quot;inline&quot;&gt;y&lt;/math&gt;-axis, but any point on the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis to which it converges will have tangent along the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-axis. Therefore, we need to separate the point &lt;math display=&quot;inline&quot;&gt;(1, 1, 1)&lt;/math&gt; into its own strata.&lt;ref&gt;Mishna 2020, pp. 179.&lt;/ref&gt;

== Critical points ==

Given a height function &lt;math display=&quot;inline&quot;&gt;h_r(z) = - \sum_{i=1}^d r_i \log(|z_i|)&lt;/math&gt; with derivative given by the Jacobian matrix &lt;math display=&quot;inline&quot;&gt;dh_r(z) = [-r_1/z_1 \cdots -r_d/z_d]&lt;/math&gt;.

For each strata, we want to find out where the derivative of our height function, when restricted to that strata, equals the matrix with entries all zero. The point at which this happens is called a '''critical point'''.

Suppose we have function &lt;math display=&quot;inline&quot;&gt;\frac{1}{1 - x - y}&lt;/math&gt; in the direction &lt;math display=&quot;inline&quot;&gt;(1, 1)&lt;/math&gt;, which has the singular variety &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \{ (x, y) : 0 \leq x \leq 1, y = 1 - x \}&lt;/math&gt;). The derivative of the height function restricted to &lt;math display=&quot;inline&quot;&gt;\mathcal{V}&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;dh(x, 1 - x) = [-(1/x) - (1 / (x - 1)), 0]&lt;/math&gt;. This is zero at the critical point &lt;math display=&quot;inline&quot;&gt;(x, y) = (1/2, 1/2)&lt;/math&gt;.

[height1.py]

The above shows a graph of this. The critical point is highlighted in red. The value of the height function at the critical point, its '''critical value''', is roughly &lt;math display=&quot;inline&quot;&gt;1.4&lt;/math&gt;.

If we change the direction to &lt;math display=&quot;inline&quot;&gt;(1, 2)&lt;/math&gt;, the derivative changes to &lt;math display=&quot;inline&quot;&gt;dh(x, 1 - x) = [-(1/x) - (2 / (x - 1)), 0]&lt;/math&gt; and the critical point changes to &lt;math display=&quot;inline&quot;&gt;(x, y) = (1/3, 2/3)&lt;/math&gt; with critical value of roughly &lt;math display=&quot;inline&quot;&gt;1.9&lt;/math&gt;. This is graphed below.

[height3.py]

We can also see the height function as the dot product of two vectors &lt;math display=&quot;inline&quot;&gt;-(r_1, \cdots, r_d) \cdot (\log|z_1|, \cdots, \log|z_d|)&lt;/math&gt;.&lt;ref&gt;Melczer 2021, pp. 195.&lt;/ref&gt; This is minimised when both vectors are parallel.

This happens when &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x) = (f_{e^{x_1}}(e^{x_1}) e^{x_1} \cdots f_{e^{x_d}}(e^{x_d}) e^{x_d})&lt;/math&gt; is parallel to &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt;. We solve for &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and then take its exponent to get the critical point.

[jsklfjsld.py]

The above graph shows the case of &lt;math display=&quot;inline&quot;&gt;f(x, y) = \frac{1}{1 - x - y}&lt;/math&gt; for the direction &lt;math display=&quot;inline&quot;&gt;r = (1, 2)&lt;/math&gt;. The black arrows show the vector &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x)&lt;/math&gt;. The blue arrows show the vector &lt;math display=&quot;inline&quot;&gt;-(1, 2)&lt;/math&gt;. The point in the same direction at roughly &lt;math display=&quot;inline&quot;&gt;(-1.1, -0.4)&lt;/math&gt;.

Equivalently, we can use the fact that &lt;math display=&quot;inline&quot;&gt;\nabla f(e^x)&lt;/math&gt; is identical to what it called the '''logarithmic gradient''' &lt;math display=&quot;inline&quot;&gt;\nabla_\log f(z) = (z_1 f_{z_1}(z_1) \cdots z_d f_{z_d}(z_d))&lt;/math&gt;, if we set &lt;math display=&quot;inline&quot;&gt;z = e^x&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 214-215.&lt;/ref&gt; Then, we can solve directly for &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt; without having to take exponents. We use the logarithmic gradient when calculating critical points below.

[Need to explain Morse index?]

== Computing Whitney stratifications and critical points ==

[ https://melczer.ca/textbook/ ]

=== Computing with ideals ===

A polynomial ring K[z] is a sum of the form \sum_{i=0}^n c_i z^i where
c_i \in K. For example, 1 + 3z + 2z^2.

A polynomial ideal I is a subset of K[z] such that if a and b are in I
then a + b is also in I (closed under addition in I) and if a is in I
and c is any element in K[z] then ca is in I (closed under
multiplication in K[z]).

A set {g_1, ..., g_n} generates the ideal I if any member of I is a
linear combination of the members of the generating set.

A Grobner basis of I is a generating set with the property that any
non-zero member of I has leading term divisible by the leading term of
some member of the Grobner basis. A reduced Grobner basis is a GB such
that no monomial of g_i is divisible by the leading term of g_j for i
\neq j.

V(I) is the set of all roots in K of all elements of I.

=== GBs for Whitney stratification ===

=== GBs for critical points ===

Given a function &lt;math display=&quot;inline&quot;&gt;F(z) = \frac{P(z)}{Q(z)}&lt;/math&gt;, a direction &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt; and a Whitney stratified space &lt;math display=&quot;inline&quot;&gt;\mathcal{V} = \mathcal{F}_0 \subset \mathcal{F}_1 \cdots \subset \mathcal{F}_k = \empty&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;1 \leq m \leq k&lt;/math&gt;, we form the set &lt;math display=&quot;inline&quot;&gt;I_m = \{ f \in K[z] : f(z) = 0, z \in \mathcal{F}_m \}&lt;/math&gt; and calculate a prime decomposition &lt;math display=&quot;inline&quot;&gt;I_m = P_1 \cap \cdots \cap P_l&lt;/math&gt;. The zero set of each &lt;math display=&quot;inline&quot;&gt;P_j&lt;/math&gt; corresponds to a different hyperplane within the stratum and we calculate the critical point on each zero set &lt;math display=&quot;inline&quot;&gt;\mathcal{V}(P_j)&lt;/math&gt;.

If &lt;math display=&quot;inline&quot;&gt;P_j = \langle p_1, \cdots, p_s \rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_j&lt;/math&gt; has codimension &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, then a critical point happens when &lt;math display=&quot;inline&quot;&gt;r&lt;/math&gt; is &quot;parallel&quot; to the space spanned by &lt;math display=&quot;inline&quot;&gt;\nabla_\log p_i(z) = (z_1 \partial p_i / \partial z_1, \cdots, z_d \partial p_i / \partial z_d)&lt;/math&gt; for each &lt;math display=&quot;inline&quot;&gt;i&lt;/math&gt;. This has dimension &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;(?) and so we want the below matrix to have at most &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; linearly independent rows (i.e. have a rank of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  &amp; \nabla_\log p_1(z) &amp; \\
  &amp; \vdots &amp; \\
  &amp; \nabla_\log p_s(z) &amp; \\
  r_1 &amp; \cdots &amp; r_d
 \end{pmatrix}&lt;/math&gt;

Solving the system of equations defined by each &lt;math display=&quot;inline&quot;&gt;p_i(z)&lt;/math&gt; and the determinants of each &lt;math display=&quot;inline&quot;&gt;c + 1&lt;/math&gt; minor being zero and the non-vanishing of the generating set of &lt;math display=&quot;inline&quot;&gt;I_1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;z_1 \cdots z_d&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 244-245.&lt;/ref&gt;

For example, for

&lt;math display=&quot;block&quot;&gt;F(x, y, z) = \frac{1}{A(x, y, z) B(x, y, z)} = \frac{1}{(4 - x - 2y - z)(4 - 2x - y - z)}&lt;/math&gt;

To construct &lt;math display=&quot;inline&quot;&gt;I_0&lt;/math&gt;, we form an ideal generated by the basis &lt;math display=&quot;inline&quot;&gt;AB&lt;/math&gt; (i.e. &lt;math display=&quot;inline&quot;&gt;I_0 = \langle AB \rangle&lt;/math&gt;). Its prime decomposition is &lt;math display=&quot;inline&quot;&gt;I_0 = P_1 \cap P_2 = \langle A \rangle \cap \langle B \rangle&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;P_1&lt;/math&gt; our matrix is

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  x A_x &amp; y A_y &amp; z A_z \\
  r &amp; s &amp; t
 \end{pmatrix}&lt;/math&gt;

leading to the system of equations

&lt;math display=&quot;block&quot;&gt;\begin{align}
4 - x - 2y - z &amp;= 0 \\
tx - rz &amp;= 0 \\
2ty - sz &amp;= 0
\end{align}&lt;/math&gt;

giving the critical point &lt;math display=&quot;inline&quot;&gt;(4r, 2s, 4t)&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;2r = s&lt;/math&gt;.

For &lt;math display=&quot;inline&quot;&gt;I_1 = P_1 = \langle A, B \rangle&lt;/math&gt; we have the matrix

&lt;math display=&quot;block&quot;&gt;\begin{pmatrix}
  x A_x &amp; y A_y &amp; z A_z \\
  x B_x &amp; y B_y &amp; z B_z \\
  r &amp; s &amp; t
 \end{pmatrix}&lt;/math&gt;

giving the system of equations

&lt;math display=&quot;block&quot;&gt;\begin{align}
4 - x - 2y - z &amp;= 0 \\
4 - 2x - y - z &amp;= 0 \\
ryz + sxz - 3txy &amp;= 0
\end{align}&lt;/math&gt;

which has the critical point &lt;math display=&quot;inline&quot;&gt;(4/3)(r + s, r + s, 3t)&lt;/math&gt;.&lt;ref&gt;Pemantle, Wilson and Melczer 2024, pp. 246-248.&lt;/ref&gt;

[Section 10.4 is about how to recognise multiple points]

== Notes ==
{{Reflist}}

== References ==
* {{cite book | last=Melczer | first=Stephen | title=An Invitation to Analytic Combinatorics: From One to Several Variables | publisher=Springer Texts &amp; Monographs in Symbolic Computation | year=2021 | url=https://melczer.ca/files/Melczer-SubmittedManuscript.pdf }}
* {{cite book | last=Mishna | first=Marni | title=Analytic Combinatorics: A Multidimensional Approach | publisher=Taylor &amp; Francis Group, LLC | year=2020 }}
* {{cite book | last1=Pemantle | first1=Robin | last2=Wilson | first2=Mark C. | last3=Melczer | first3=Stephen | title=Analytic Combinatorics in Several Variables | publisher=Cambridge University Press | year=2024 | edition=2nd | url=https://acsvproject.com/PemantleWilsonMelczer23.pdf }}
* {{cite book | last=Shabat | first=B. V. | title=Introduction to Complex Analysis. Part II: Functions of Several Variables | publisher=American Mathematical Society, Providence, Rhode Island | year=1992 }}</text>
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{{tmbox|type=notice|text=This is a sandbox page, a place for experimenting with Wikibooks.}}

This is a history of Chrysler factories that are being or have been used to produce cars, vans, SUVs, trucks, and automobile components. 

For '''Chrysler brand''' only: Plant code in 1955-1957 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number.

For '''cars''': Plant code in 1958 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number. Plant code was the number in the 4th position of the 10-digit serial number for 1959-1965. Plant code was the number in the 7th position of the 13-digit serial number for 1966-1967. Plant code was the letter in the 7th position of the 13-digit serial number for 1968-1980.

Canadian-built cars had the plant code as the number in the 5th position of the 11-digit serial number for 1965.

For '''trucks''': The first digit of the 7-digit sequence number (4th position overall of the 10-digit serial number) indicated which plant built the truck for 1967-1968. Plant code was the number in the 4th position of the 10-digit serial number for 1969. Plant code was the letter in the 7th position of the 13-digit serial number for 1970-1980.

Canadian-built models used a different system for VINs until 1968, when Chrysler of Canada adopted the same system as the US. Plant code for trucks was the number in the 5th position of the 10-digit serial number for 1961-1967.

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

For '''AMC passenger cars from 1958 through mid-1966''': Plant code is indicated by the 2nd letter of the serial number. If there is no 2nd letter, then it was made in Kenosha, WI. If the 2nd letter is a T, then it was made in Brampton, ON, Canada. If the 2nd letter is a K, then it was a knock-down export car. 

For '''AMC passenger cars from mid-1966 through 1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. 0-6 is Kenosha, WI and 7-8 is Brampton, ON, Canada.

For '''Canadian-built Jeeps built by AMC Canada for 1979-1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. It was made in Canada if it's either an 8 (1979) or a 7 (1980). All other Jeeps from 1980 and earlier were made in Toledo. 

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

==Current factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| D (1968-),&lt;br&gt; 4 (1966-1967)
| [[w:Belvidere Assembly|Belvidere Assembly]] 
| [[w:Belvidere, Illinois|Belvidere, Illinois]]
| [[w:United States|United States]]
| 1965
| Feb. 2023
| 
| Located at 3000 West Chrysler Drive. '''Belvidere Satellite Stamping Plant''' adjoins the main assembly plant. Began production on July 7, 1965. The first vehicle produced was a 1966 Plymouth Fury four-door. In 1972, the Chrysler Town and Country station wagon was added to the Belvidere plant. In 1977, the plant was converted to build front-wheel drive subcompacts. Production of the Dodge Omni and Plymouth Horizon began on December 5, 1977. All L-body derivatives were made at Belvidere through 1987. In 1987, Belvidere was converted to build Chrysler's midsize, fwd C-body sedans. Belvidere then switched back to building small cars and began production of the Neon on November 10, 1993. Belvidere built the last Plymouth, a silver 2001 Neon LX on June 28, 2001. Neon production ended in September 2005. Dodge Caliber began production in January 2006, followed by Jeep Compass in June 2006 and Jeep Patriot in December 2006. Caliber ended production on December 19, 2011. Dodge Dart began production on April 30, 2012, and ended on October 4, 2016. Compass and Patriot production ended on December 23, 2016. Jeep Cherokee production began on June 1, 2017 and ended on February 28, 2023. Belvidere was then idled. &lt;br&gt; Past models: [[w:Plymouth Fury|Plymouth Fury]] (1966-1974), [[w:Plymouth Gran Fury#1975–1977|Plymouth Gran Fury]] (1975-1977), [[w:Dodge Monaco|Dodge Monaco]] (1966-1976), [[w:Dodge Royal Monaco#1977 (Royal Monaco)|Dodge Royal Monaco]] (1977), [[w:Dodge Polara|Dodge Polara]] (1966-1973), [[w:Chrysler Newport|Chrysler Newport]] (1977), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1973-1977), [[w:Dodge Omni|Dodge Omni]] (1978-1987), [[w:Plymouth Horizon|Plymouth Horizon]] (1978-1987), [[w:Dodge Omni 024|Dodge Omni 024]] (1979-1980), [[w:Plymouth Horizon TC3|Plymouth Horizon TC3]] (1979-1980), [[w:Dodge Omni 024|Dodge 024]] (1981-1982), [[w:Plymouth Horizon TC3|Plymouth TC3]] (1981-1982), [[w:Dodge Charger (1981)|Dodge Charger]] (1983-1987), [[w:Plymouth Turismo|Plymouth Turismo]] (1983-1987), [[w:Dodge Rampage|Dodge Rampage]] (1982-1984),  [[w:Plymouth Scamp|Plymouth Scamp]] (1983), [[w:Dodge Dynasty|Dodge Dynasty]] (1988-1993), [[w:Chrysler Dynasty|Chrysler Dynasty]] (Canada: 1988-1993), [[w:Chrysler New Yorker#1988–1993|Chrysler New Yorker]] (1988-1993), [[w:Chrysler New Yorker Fifth Avenue#1990–1993: New Yorker Fifth Avenue|Chrysler New Yorker Fifth Avenue]] (1990-1993), [[w:Chrysler Imperial#1990–1993|Chrysler Imperial]] (1990-1993), [[w:Dodge Neon|Dodge Neon]] (1995-2005), [[w:Plymouth Neon|Plymouth Neon]] (1995-2001), Chrysler Neon (Canada: 2000-02),&lt;br&gt; Dodge SX 2.0 (Canada: 2003-05),&lt;br&gt; [[w:Dodge Caliber|Dodge Caliber]] (2007-2012), [[w:Jeep Compass#First generation (MK49; 2006)|Jeep Compass]] (2007-2017), [[w:Jeep Patriot|Jeep Patriot]] (2007-2017), [[w:Dodge Dart (PF)|Dodge Dart]] (2013-2016), [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2017-2023)
|-
| H (1989-),&lt;br&gt; A (1988)
| [[w:Brampton Assembly|Brampton Assembly]] (Formerly Bramalea Assembly)
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1987
| Dec. 2023
| 
| Located at 2000 Williams Parkway East. Factory was built by AMC and Renault. The plant was acquired by Chrysler as part of its takeover of AMC. Began production on September 28, 1987. Plant was originally known as Bramalea Assembly. Plant was renamed Brampton Assembly in 1992 after Chrysler closed and sold the old AMC plant on Kennedy Road in Brampton. The attached '''Brampton Satellite Stamping Plant''' was added in December 1991 and was built for the launch of the Chrysler LH platform. On December 17, 1991, Eagle Premier and Dodge Monaco production ended. Production of the Chrysler LH platform cars began in June 1992. Production switched to the rear-wheel drive Chrysler LX platform cars in January 2004. The Chrysler 300 was also built for export to mainland Europe as the Lancia Thema from 2011-2014. Production ended on December 22, 2023 and the plant was idled. Last vehicle off the line was a Pitch-Black 2023 Dodge Challenger Demon 170. 7,147,888 vehicles were produced through 2023.&lt;br&gt; Past models: [[w:Eagle Premier|Eagle Premier]] (1988-1992), [[w:Dodge Monaco#Fifth generation (1990–1992)|Dodge Monaco]] (1990-1992),&lt;br&gt; [[w:Dodge Intrepid|Dodge Intrepid]] (1993-2004),&lt;br&gt; [[w:Chrysler Intrepid|Chrysler Intrepid]] (Canada: 1993-2004),&lt;br&gt; [[w:Chrysler Concorde|Chrysler Concorde]] (1993-2004),&lt;br&gt; [[w:Eagle Vision|Eagle Vision]] (1993-1997),&lt;br&gt; [[w:Chrysler New Yorker#1994–1996|Chrysler New Yorker]] (1994-1996),&lt;br&gt; [[w:Chrysler LHS|Chrysler LHS]] (1994-1997, 1999-2001),&lt;br&gt; [[w:Chrysler 300M|Chrysler 300M]] (1999-2004),&lt;br&gt; [[w:Chrysler 300|Chrysler 300]] (2005-2023),&lt;br&gt; [[w:Dodge Magnum#Chrysler LX platform (2005–2008)|Dodge Magnum]] (2005-2008),&lt;br&gt; [[w:Dodge Charger (2006)|Dodge Charger]] (2006-2023),&lt;br&gt; [[w:Dodge Challenger (2008)|Dodge Challenger]] (2008-2023),&lt;br&gt; [[w:Lancia Thema#Second generation (2011–2014)|Lancia Thema]] (For export: 2011-2014)
|-
| C
| [[w:Jefferson North Assembly|Detroit Assembly Complex – Jefferson]] / Jefferson North Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]] (1993-), [[w:Dodge Durango#WD|Dodge Durango]] (2011-)
| Located at 2101 Conner Street. Jefferson North replaced the previous Jefferson Assembly plant that closed in 1990 and was demolished in 1991. Jefferson North is across the street from the old Jefferson Assembly plant, on the north side of Jefferson Ave. Jefferson North was built on the site of Chrysler's old Kercheval Avenue Body Plant, which, in 1955, had been connected to the old Jefferson Assembly plant by a bridge crossing over Jefferson Ave. The Jefferson North plant site also absorbed what had been the axle plant and service parts buildings of the old [[w:Hudson Motor Car Company|Hudson]] plant, which were located at Connor St. and Vernor Hwy. The main Hudson plant is now the parking lot on the corner of Jefferson Ave. and Connor St. After 2017, the Jefferson North plant complex also absorbed the site of the former Budd Co. body- &amp; parts-making plant at Connor St. and Charlevoix Avenue, which had previously belonged to the Liberty Motor Car Co. The Budd plant extended north on Connor from Charlevoix most of the way to Mack Ave. Since the nearby Mack Ave. Assembly Plant began operations in 2021, the 2 plants have operated as the Detroit Assembly Complex. Jefferson North began production on January 14, 1992 with the original Grand Cherokee (ZJ). The 2nd gen. Grand Cherokee began production on July 17, 1998. The 3rd gen. Grand Cherokee began production on July 26, 2004 followed by the Jeep Commander on July 18, 2005. The 4th gen. Grand Cherokee began production on May 10, 2010 followed by the Dodge Durango on December 14, 2010. The 5th gen. Grand Cherokee began production in May 2022. The 4xe plug-in hybrid version of the Grand Cherokee began production at Jefferson North in March 2023. On Aug. 13, 2013, Jefferson North built its 5 millionth vehicle, a silver 2014 Grand Cherokee Overland. On May 25, 2016, Jefferson North built its 6 millionth vehicle, a Granite Crystal (silvery gray) 2016 Grand Cherokee 75th anniversary Edition.&lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010),&lt;br&gt; [[w:Jeep Grand Cherokee (WK2)#Grand Cherokee WK (2022)|Jeep Grand Cherokee WK]] (2022)&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2023-2025)
|-
| 8
| [[w:Detroit Assembly Complex – Mack|Detroit Assembly Complex – Mack]] / Mack Ave. Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2021
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2021-), [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]] (2022-)
| Located at 4000 St. Jean Avenue. The Mack Avenue Assembly Plant is built on the site of the former Mack Ave. Engine Plants I &amp; II, the New Mack Assembly Plant, &amp; the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The two plants that comprised the former Mack Avenue Engine Complex were converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North plants have operated as the Detroit Assembly Complex since 2021. Production began in March 2021 with the 3-row Grand Cherokee L. The 2-row Grand Cherokee followed in the fall of 2021. The 4xe plug-in hybrid version of the Grand Cherokee began production at Mack Ave. in August 2022. &lt;br&gt; Past models: [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2022-2025)
|-
|
| [[w:Dundee Engine Plant|Dundee Engine Plant]] (Formerly [[W:Global Engine Manufacturing Alliance|GEMA]]) 
| [[w:Dundee, Michigan|Dundee, Michigan]]
| [[w:United States|United States]]
| 2005
| 
| [[w:Prince engine#1.6-litre turbocharged (PSA)|1.6L turbo PSA/BMW Prince EP6CDTX Hybrid I4]],&lt;br&gt; [[w:FCA Global Medium Engine|2.0L turbo Hurricane4 EVO I4 (GME-T4 EVO)]],&lt;br&gt; Engine components
| Located at 5800 North Ann Arbor Road. Plant was originally part of the the Global Engine Manufacturing Alliance, a 3-way engine manufacturing joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. The North Plant launched in October 2005, followed by the South Plant in November 2006. The plant began with production of the I4 World Gasoline Engine, which was developed by the Global Engine Alliance, a 3-way engine development joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. Originally, the plant was envisioned as supplying engines to Mitsubishi and Hyundai as well as Chrysler however the plant only ever supplied engines to Chrysler. Mitsubishi and Hyundai each set up engine production at their own engine plants. On August 31, 2009, Chrysler bought Mitsubishi’s and Hyundai’s stakes in the group and now wholly owns both the Global Engine Manufacturing Alliance and its primary engine-building plant in Dundee, Michigan. In January 2012, the plant was renamed Dundee Engine Plant. Production of the Fiat 1.4-liter FIRE I4 engine began in November 2010. The Tigershark engine, an evolution of the World engine, began production in 2012 for the 2.0L and in May 2013 for the 2.4L. Tigershark engine production ended on March 16, 2023. In November 2019, the Pentastar V6 began production at Dundee, moving from the Mack Ave. Engine Plant in Detroit, which was to be converted into a vehicle assembly plant. The Pentastar V6 ended production at Dundee on August 18, 2023. In 2025, Dundee began making a North American-spec version of the European-developed Prince engine for the 2026 Jeep Cherokee Hybrid. &lt;br&gt; Past engines: [[w:World Gasoline Engine|1.8L/2.0L/2.4L/2.4L Turbo I4 World Gasoline Engine]], [[w:World Gasoline Engine#Tigershark|2.0L/2.4L I4 Tigershark Engine]], [[w:FIRE engine|Fiat 1.4L/1.4L Turbo FIRE MultiAir I4 engine]], [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]] 
|-
| 
| Etobicoke Casting Plant
| [[w:Etobicoke|Etobicoke District]], [[w:Toronto|Toronto]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1964
| 
| Aluminum die castings, Engine and Transmission Components
| Located at 15 Brown's Line. Etobicoke used to be a separate city but became part of Toronto in 1998. Factory was originally built in 1942 by the Canadian government and operated by Alcan Aluminum, which used it to make molds for military aircraft parts during World War II. It produced precision aircraft parts and other high quality aluminum castings. The aluminum foundry was purchased by Chrysler in April 1964 from Alcan Aluminum. The plant was expanded in 1965 and 1998. Etobicoke Casting is making oil pans for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee.
|-
|
| [[w:Indiana Transmission#Indiana Transmission I|Indiana Transmission Plant I]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1998
| 
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, Gear machining and final assembly of electric drive modules
| Located at 3660 North U.S. Highway 931. RFE transmission production ended in January 2025. More than 8 million RFE transmissions were produced at Indiana Transmission Plant I. Production of the nine-speed transmission began in May 2013. The 9-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. &lt;br&gt; Past transmissions:&lt;br&gt; [[w:Chrysler RFE transmission|Chrysler RFE 4-/5-/6-speed auto. trans.]]
|-
|
| [[w:Kokomo Casting|Kokomo Casting Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1965
| 
| Aluminum parts for automotive components, transmission and transaxle cases; engine block castings
| Located at 1001 East Boulevard. Kokomo Casting is the world’s largest die-cast facility. Plant was expanded in 1969, 1986, 1995 and 1997. Over 18 million four-speed transmission cases were made at Kokomo Casting from 1988 through July 2014. Kokomo Casting started making nine-speed transmission cases in 2013. Kokomo Casting is making engine blocks for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee. Kokomo Casting is making gearbox covers for the electric drive modules made at the Indiana Transmission Plant. 
|-
|
| [[w:Kokomo Engine Plant|Kokomo Engine Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2022 (as Kokomo Engine)
| 
| [[w:FCA Global Medium Engine|2.0L turbo GME-T4 I4]]
| Located at 3360 North U.S. Highway 931. Plant was previously known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Starting in 2020, the plant was converted to engine production. Engine production began in late February 2022. 
|-
|
| [[w:Kokomo Transmission|Kokomo Transmission Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1956
| 
| [[w:ZF 8HP transmission|850RE]] 8-speed auto. transmission,&lt;br&gt; [[w:ZF 8HP transmission|880RE]] 8-speed auto. transmission,&lt;br&gt; Machining of engine block castings and transmission components,&lt;br&gt; Machined components for the &lt;br&gt; 9-speed auto. transmission
| Located at 2401 South Reed Road. On October 9, 2020, Kokomo Transmission built its last 41TE 4-speed auto. transmission, assembling more than 17 million 4-speed auto. transmissions since production began in 1988. Kokomo Transmission began building 6-speed auto. transmissions in 2006. Production of the eight-speed automatic transmission began in September 2012. The 8-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. On August 8, 2023, Kokomo Transmission built its 6 millionth 8-speed transmission. Kokomo Transmission is machining gearbox covers for the electric drive modules made at the Indiana Transmission Plant. &lt;br&gt; Past transmissions: &lt;br&gt;[[w:TorqueFlite|TorqueFlite 3-/4-speed auto. trans.]],&lt;br&gt; [[w:Ultradrive|Ultradrive (TE/AE/LE/RLE/TES/TEA)&lt;br&gt; 4-/6-speed auto. trans.]],&lt;br&gt; [[w:ZF 8HP transmission|845RE]] 8-speed auto. transmission,&lt;br&gt; SI-EVT trans. (eFlite) for Pacifica Hybrid
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo North Engine Plant]] 
| [[w:Ramos Arizpe|Ramos Arizpe]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1981
| 
| [[w:Chrysler Hemi engine#Third generation: 2003–present|5.7L/6.4L/6.2L supercharged Hemi V8]], [[w:Stellantis Hurricane engine|Chrysler 3.0L twin-turbo Hurricane GME-T6 I6]]
|  Production began on May 8, 1981. Hemi V8 production began in June 2002 with the 5.7L. Production of the supercharged 6.2L Hellcat Hemi V8 began in the third quarter of 2014. Tigershark I4 production began in the first quarter of 2014. &lt;br&gt; Past engines: [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L, 2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|2.0L, 2.4L, 2.4L Turbo I4 &quot;Neon engine&quot;]],&lt;br&gt; [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4 engine]], [[w:Chrysler Hemi engine#6.1|6.1L Hemi V8]]
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo South Engine Plant]] 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]]
| Plant opened on October 29, 2010. The plant has produced over 6 million Pentastar V6 engines. &lt;br&gt; Past engines: Chrysler 3.6L Pentastar V6 engine (EH3) for Pacifica Plug-in Hybrid
|-
| 
| Saltillo Stamping Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1997
| 
| Stampings and assemblies including Body panels
| Part of the Saltillo Truck Assembly Complex.
|-
| G
| Saltillo Truck Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1995
| 
| [[w:Ram Heavy Duty (fifth generation)|Ram HD pickup &amp; chassis cab]] (2019-)
|  Production began in 1995. As of 2025, also known as Saltillo Truck Heavy Duty Plant. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1995-2012),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-2023),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram HD pickup &amp; chassis cab]] (2013-2018), [[w:Sterling Bullet|Sterling Truck Bullet]] (2008-2009)
|-
| 4
| Saltillo Truck Extension Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2025
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2025-)
| Also known as Saltillo Truck Light Duty Plant. 2 new buildings were constructed for the new light duty plant. Production began in May 2025. Initially, production was for export but production for the domestic Mexican market began in February 2026. The plant also includes a seat assembly line, the first Stellantis plant in North America to integrate this process into its own production chain.
|-
| E
| Saltillo Van Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2013
| 
| [[w:Ram ProMaster|Ram ProMaster]] (2014-),&lt;br&gt; [[w:Ram ProMaster#E-Ducato and Ram ProMaster EV (2024)|Ram ProMaster EV]] (2024-)
| Production started in July 2013. &lt;br&gt; Past models: [[w:Fiat Ducato|Fiat Ducato]] (Export to Brazil/Argentina: 2018-2022)
|-
| N
| [[w:Sterling Heights Assembly|Sterling Heights Assembly Plant]] 
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1984
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2019-)
| Located at 38111 Van Dyke Ave. The plant was originally built by the US Navy as a jet engine plant in 1953. It was called Naval Industrial Reserve Aircraft Plant and was owned by the US Navy. When the jet engine project was cancelled, the plant was transferred to the US Army, which contracted with Chrysler to build missiles at the plant, which was now known as the Michigan Ordinance Missile Plant. Chrysler began production of the PGM-11 Redstone missile at the Sterling Heights plant on September 27, 1954. The final Redstone was built in 1961. Chrysler also built the PGM-19 Jupiter missile at Sterling Heights from 1958-December 1960. Chrysler also built the first stage of the Saturn I rocket at the Sterling Heights plant. Chrysler vacated the Michigan Army Missile Plant at the end of 1969. Meanwhile, LTV Corp. (previously Ling-Temco-Vought) built the MGM-52 Lance missile at the Sterling Heights plant. The Army turned the plant over to the state of Michigan, which was then sold it to Volkswagen in 1980. VW converted the plant to automotive production and intended to make the Jetta there but VW's US sales declined and VW never ended up building anything there. VW then sold the plant to Chrysler in 1983. Chrysler LeBaron GTS and Dodge Lancer production began in September 1984 and ended on April 7, 1989. Shadow and Sundance production began on August 25, 1986 and ended on March 9, 1994. The Dodge Daytona was also moved from St. Louis to Sterling Heights in 1991 and was produced there through February 26, 1993. Chrysler then built a succession of midsize cars at Sterling Heights from June 1994. During Chrysler's bankruptcy in 2009, Sterling Heights Assembly was initially left behind in &quot;old Chrysler&quot; and was supposed to close by December 2010 but during 2010, &quot;new Chrysler&quot; changed its mind and bought the plant from &quot;old Chrysler&quot; for $20 million. The 2011 Chrysler 200 and Dodge Avenger sedans began production on December 6, 2010 followed by the Chrysler 200 Convertible in February 2011. The Chrysler 200 Convertible was also built for export to Europe as the Lancia Flavia from March 2012. The 2nd gen. Chrysler 200 sedan began production on March 14, 2014 and ended on December 2, 2016. The plant was then idled for a lengthy retooling to build body-on-frame pickups and began production of the new generation DT-series Ram 1500 in March 2018. &lt;br&gt; Past models: [[w:Dodge Lancer#1985–1989: Lancer|Dodge Lancer]] (1985-1989), [[w:Chrysler LeBaron#1985–1989 LeBaron GTS|Chrysler LeBaron GTS (H-body)]] (1985-1989), [[w:Plymouth Sundance|Plymouth Sundance]] (1987-1994), [[w:Dodge Shadow|Dodge Shadow]] (1987-1994), [[w:Dodge Daytona|Dodge Daytona]] (1992-1993), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1992-1993), [[w:Chrysler Cirrus|Chrysler Cirrus]] (1995-2000), [[w:Dodge Stratus|Dodge Stratus]] sedan (1995-2006), [[w:Plymouth Breeze|Plymouth Breeze]] (1996-2000), [[w:Chrysler Sebring|Chrysler Sebring]] sedan (2001-2010), [[w:Chrysler Sebring|Chrysler Sebring]]  convertible (2001-2006, 2008-2010), [[w:Dodge Avenger#Dodge Avenger sedan (2008–2014)|Dodge Avenger]] sedan (2008-2014), [[w:Chrysler 200|Chrysler 200]] sedan (2011-2017), [[w:Chrysler 200|Chrysler 200]] convertible (2011-2014), [[w:Chrysler 200#Lancia Flavia|Lancia Flavia]] (For export: 2012-2014)  
|-
| 
| Sterling Stamping Plant
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1965
| 
| Stampings and assemblies including hoods, roofs, liftgates, side apertures, fenders, and floorpans
| Located at 35777 Van Dyke Ave. This plant is a separate facility but is located next door to the Sterling Heights Assembly Plant. Sterling Stamping is the largest stamping plant in the world. Sterling Stamping supplies stampings to many Chrysler assembly plants, not only Sterling Heights Assembly. The first stampings were produced in January 1965. 
|-
| W
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo North Assembly Plant]] 
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]] (2018-)
| Located at 4400 Chrysler Drive. Toledo North first built the Jeep Liberty, which began in April 2001. Dodge Nitro production began in August 2006 and ended on December 16, 2011. The 2nd gen. Jeep Liberty began production in July 2007. Jeep Liberty ended production on August 16, 2012. Toledo North then closed for retooling to build the all-new 2014 Cherokee. The Jeep Cherokee began production at Toledo North on June 24, 2013 and ended on April 6, 2017. The Cherokee was then moved to Belvidere Assembly. Toledo North was then retooled to build the Jeep Wrangler, which began on November 15th, 2017. The 4xe plug-in hybrid version of the Wrangler began production in December 2020. &lt;br&gt; Past models: [[w:Jeep Liberty|Jeep Liberty]] (2002-2012), [[w:Dodge Nitro|Dodge Nitro]] (2007-2011),&lt;br&gt; [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2014-2017),&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler 4xe (JL)]] (2021-2025)
|-
| L
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo Supplier Park Plant]] (South plant)
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Gladiator (JT)|Jeep Gladiator]] (2020-)
| Located along Stickney Ave. The Toledo Supplier Park Plant is built on the site of the former Stickney Ave. plant that became part of Chrysler in 1987 as part of the acquisition of AMC. That plant was acquired by Kaiser Jeep in 1964 from Autolite and was built in 1942. The Toledo Supplier Park Plant includes body and chassis operations in partnership with Kuka and Hyundai Mobis, respectively. The paint shop was originally run in partnership with Magna but Chrysler took over the paint operation in the first quarter of 2011. The Toledo Supplier Park Plant began Wrangler production in August 2006. Wrangler production ended on April 27, 2018. Gladiator pickup production began in March 2019. &lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Wrangler (JK)|Jeep Wrangler (JK)]] (2007-2017),&lt;br&gt; [[w:Jeep Wrangler (JK)#2018 model year update|Jeep Wrangler JK]] (2018)
|-
| 
| [[w:Toledo Machining|Toledo Machining Plant]] 
| [[w:Perrysburg, Ohio|Perrysburg, Ohio]]
| [[w:United States|United States]]
| 1966
| 
| Steering Columns,&lt;br&gt; Torque Converters for 8-spd. rwd and 9-spd. fwd auto. transmissions
| Located at 8000 Chrysler Drive. Production began in 1966 and the plant was expanded in 1969. &lt;br&gt; Past products: Power Electronics module for Wrangler 4xe PHEV
|-
| T,&lt;br&gt; V (1985 only)
| [[w:Toluca Car Assembly|Toluca Car Assembly]] 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1968
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (2017-), [[w:Jeep Wagoneer S|Jeep Wagoneer S EV]] (2024-),&lt;br&gt; [[w:Jeep Cherokee (KM)|Jeep Cherokee (KM)]] (2026-), [[w:Jeep Recon|Jeep Recon EV]] (2026-)
| Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Began production on December 9, 1968. An adjacent supplier park was opened in 2007. Journey production began in early 2008. PT Cruiser production ended on July 9, 2010. Fiat 500 production began in December 2010. Journey production ended in December 2020. Jeep Compass production began on January 16, 2017. &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Dart (rwd), Dodge Dart K, Dodge Dart E, Chrysler Valiant Volaré (rwd), Chrysler Valiant Volaré K, Chrysler Valiant Volaré E, [[w:Dodge Magnum#First generation|Dodge Magnum]] [rwd] (1981-1982), [[w:Dodge Magnum#Second generation|Dodge Magnum 400/Magnum]] [fwd] (1983-1988), [[w:Chrysler Phantom|Chrysler Phantom]], [[w:Chrysler Shadow|Chrysler Shadow]], [[w:Chrysler Spirit|Chrysler Spirit]] &lt;br&gt; Export to US: [[w:Dodge Aries|Dodge Aries]] (1984-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1984-1989), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe]] (1987-1989, 1992), [[w:Dodge Shadow|Dodge Shadow]] (1988, 1991-1994), [[w:Plymouth Sundance|Plymouth Sundance]] (1988, 1992-1994), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron sedan]] (1990-1994), [[w:Dodge Spirit|Dodge Spirit]] (1991-1995), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1991-1995), [[w:Dodge Neon#First generation (1994)|Dodge Neon]] (1995-1999), [[w:Plymouth Neon#First generation (1994)|Plymouth Neon]] (1995-1999), [[w:Chrysler Sebring#Convertible (1996–2000)|Chrysler Sebring Convertible]] (1996-2000), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2001-10), [[w:Dodge Journey|Dodge Journey]] (2009-2020),&lt;br&gt; [[w:Fiat 500 (2007)|Fiat 500]] (2012-2019), [[w:Fiat 500 (2007)#Fiat 500e (2013)|Fiat 500e]] (2013-2019)&lt;br&gt; Export to Brazil/Europe/Australia:&lt;br&gt; [[w:Fiat Freemont|Fiat Freemont]] (2011-2016)
|-
| 
| Toluca Stamping Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1994
| 
| Stampings and assemblies including Body panels
| Part of the Toluca Assembly Complex.
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine South Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler Pentastar V6 engine]]
| Located at 2300 Van Horn Road. Production began in March 2010 with the 3.6L Pentastar V6. In 2022, Trenton South was upgraded with a flexible engine line that could build both the Classic and Upgrade versions of the 3.6L Pentastar V6. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production.
|-
| 
| Warren Stamping Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1949
| 
| Stampings and assemblies including roofs, tailgates, side apertures, fenders, and floorpans
| Located at 22800 Mound Road. This plant is a separate facility but is located next door to the Warren Truck Assembly Plant. The plant was expanded in 1952, 1964, 1965, and 1986. Warren Stamping supplies stampings to many Chrysler assembly plants, not only Warren Truck Assembly.
|-
| S (1970-), 1 (1967-1969),&lt;br&gt; 2 (For A-series)
| Warren Truck Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1938
| 
| [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer]] (2022-), [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer L]] (2023-)
| Located at 21500 Mound Road. Formerly known as the Dodge City Truck Plant. Also referred to as Warren Truck #1 in the 1970's when there were 2 other truck plants nearby. Began production in October 1938. The Mitsubishi Raider began production in September 2005 and ended on June 11, 2009. Dodge Dakota production ended on August 23, 2011. Full-size Jeep SUV production began in 2021. Ram 1500 Classic production ended in October 2024, ending production of Dodge/Ram trucks at Warren after 86 years. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge T-, V-, W-Series|Dodge T-, V-, W-Series]] (1939-1947), Dodge military trucks, [[w:Dodge B series#Pickup truck|Dodge B series]] (1948-1953), [[w:Dodge C series|Dodge C series]] (1954-1960),&lt;br&gt; [[w:Dodge Town Panel and Town Wagon|Dodge Town Panel/Town Wagon]] (1954-66), [[w:Dodge D series|Dodge D/W series]] (1961-1980), [[w:Dodge Ram|Dodge Ram pickup]] (1981-2012), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-24), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1977-1978, 1981-1985), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1977-1978, 1981), [[w:Dodge M-series chassis|Dodge M-series chassis]] (1968-1979),&lt;br&gt; [[w:Dodge A100|Dodge A100/A108]] (1964-1970),&lt;br&gt; [[w:Dodge Dakota|Dodge Dakota]] (1987-2011),&lt;br&gt; [[w:Mitsubishi Raider|Mitsubishi Raider]] (2006-2009),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer]] (2022-2025),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer L]] (2023-2025),&lt;br&gt; [[w:Fargo Trucks|Fargo Trucks]] (For export)
|-
| R (1968-),&lt;br&gt; 9 (1959-1967)
| [[w:Windsor Assembly|Windsor Assembly]] 
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1929
| 
| [[w:Chrysler Pacifica (minivan)|Chrysler Pacifica (minivan)]] (2017-), [[w:Chrysler Voyager#Sixth generation (2020–present)|Chrysler Voyager]] (2020-2026), Chrysler Grand Caravan (Canada: 2021-),&lt;br&gt; [[w:Dodge Charger (2024)|Dodge Charger]] (2024-)
| Located at 2199 Chrysler Centre. Today's Windsor Assembly Plant was originally Windsor Plant 3 or Windsor Car Assembly Plant. Before the 1965 US-Canada Auto Pact, Windsor Assembly made most of the cars sold by Chrysler Canada, including some unique-to-Canada variations. On June 10, 1983, Windsor ended rwd car production and was converted to build fwd minivans. Windsor began building minivans on October 7, 1983. For the first 2 generations of Chrysler minivans, Windsor only built the SWB models. For the 3rd generation, Windsor built both SWB and LWB models. For the 4th generation, Windsor only built LWB models. The minivan-based Pacifica SUV began production in January 2003 and  ended production on November 23, 2007. Production of the VW Routan began in August 2008. Production of the Ram C/V began on August 31, 2011, and ended in early 2015. Pacifica minivan production began on February 29, 2016 followed by the PHEV version on December 1, 2016. Town &amp; Country production ended on March 21, 2016 while Dodge Grand Caravan production ended on August 21, 2020. Production of the electric Charger Daytona began in December 2024 followed by the gas-powered Charger Sixpack in December 2025. &lt;br&gt; Past models: [[w:Chrysler Imperial|Chrysler Imperial]] (1929-1937) [https://www.web.imperialclub.info/registry/vin_decode.htm#1931-54], [[w:Dodge Kingsway|Dodge Kingsway]] (Canada: 1940-41, 1951-52), [[w:Dodge Regent|Dodge Regent]] (Canada: 1951-1959), [[w:Dodge Crusader|Dodge Crusader]] (Canada: 1951-1958), [[w:Dodge Mayfair|Dodge Mayfair]] (Canada: 1953-1959), [[w:Dodge Viscount|Dodge Viscount]] (Canada: 1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1961), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:Chrysler Windsor|Chrysler Windsor]] (1957-1966), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1963), [[w:Chrysler 300 non-letter series|Chrysler Saratoga 300]] (1964-1965), [[w:Chrysler Newport#1961–1964|Chrysler Newport]] (1961-1963), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1963-64, 1966), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Plymouth Fury|Plymouth Fury]] (1959-1970), [[w:Dodge Polara|Dodge Polara]] (1960, 1964-1969), Dodge 220 (Canada: 1963), [[w:Dodge 330|Dodge 330]] (1963-1965), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Monaco|Dodge Monaco]] (1967-1968), [[w:Plymouth Valiant#Canada (1960–1966)|Valiant]] (Canada: 1960-1966), [[w:Plymouth Barracuda#First generation (1964–1966)|Valiant Barracuda]] (Canada: 1964-1965), [[w:Plymouth Valiant|Plymouth Valiant]] (1970-1974), [[w:Plymouth Duster|Plymouth Duster]] (1970), [[w:Dodge Dart|Dodge Dart (compact)]] (1966, 1970-1975), [[w:Plymouth Satellite#Third generation (1971–1974)|Plymouth Satellite]] (1972-1974),  [[w:Plymouth GTX|Plymouth GTX]] (1971), [[w:Plymouth Road Runner#Second generation (1971–1974)|Plymouth Road Runner]] (1971-1974), [[w:Dodge Charger (1966)#Fourth generation|Dodge Charger]] (1975-1978), [[w:Dodge Magnum#US and Canada (1978–1979)|Dodge Magnum]] (1978-1979), [[w:Chrysler Cordoba|Chrysler Cordoba]] (1975-1983), [[w:Dodge Mirada|Dodge Mirada]] (1980-1983), [[w:Imperial (automobile)#Sixth generation (1981–1983)|Imperial]] (1981-1983), [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979), [[w:Dodge Diplomat|Dodge Diplomat]] (1981-1983), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1982-83), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle]] (Canada: 1981-1982), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1983), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron]] (1981), [[w:Chrysler New Yorker#1982|Chrysler New Yorker]] (1982), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler New Yorker Fifth Avenue]] (1983), [[w:Plymouth Voyager|Plymouth Voyager]] (1984-'00), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1984-2000), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2020), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (2001-2016), [[w:Chrysler Voyager|Chrysler Voyager]] (2000), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler minivans (S)#Cargo van|Dodge Mini Ram Van]] (1984-1988), [[w:Chrysler minivans (RT)#2011 revision|Ram C/V]] (2012-2015), [[w:Volkswagen Routan|Volkswagen Routan]] (2009-14), [[w:Chrysler Voyager#Lancia Voyager|Lancia Voyager]] (For export: 2012-2015), [[w:Chrysler Pacifica (crossover)|Chrysler Pacifica (SUV)]] (2004-2008)
|-
| 
| CpK Interior Products, Inc. - Belleville Operations
| [[w:Belleville, Ontario|Belleville]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 134 River Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. - Guelph Operations
| [[w:Guelph|Guelph]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 500 Laird Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. -&lt;br&gt; Port Hope Operations
| [[w:Port Hope, Ontario|Port Hope]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 128 Peter St. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 4
| Arab American Vehicles Company (AAV)
| [[w:Cairo|Cairo]]
| [[w:Egypt|Egypt]]
| 1978 (production began) &lt;br&gt; 1987 (became part of Chrysler)
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2025-), [[w:Citroën C4#Third generation (C41; 2020)|Citroën C4X]] (2025-)
| Arab American Vehicles Company is a joint venture between the [[w:Arab Organization for Industrialization|Arab Organization for Industrialization]], which holds 51%, and Stellantis, which holds the other 49%. AAV was originally established in 1977 as a joint venture  with [[w:American Motors Corporation|AMC]] to produce Jeeps. Production began in 1978. It became part of Chrysler when Chrysler bought AMC in 1987. It continued to be a part of subsequent corporate entities DaimlerChrysler, Chrysler LLC, Chrysler Group LLC, [[w:Fiat Chrysler Automobiles|FCA]], and [[w:Stellantis|Stellantis]]. Production of the Jeep J8, a light military vehicle based on the JK Wrangler, began on November 13, 2008. Jeep Grand Cherokee L production began in September 2024. Citroen C4X production began in April 2025. AAV has also produced vehicles for other automakers including Toyota. Toyota Fortuner SUV production began in April 2012. &lt;br&gt; Past models: Jeep CJ6, Jeep Wagoneer (SJ), Jeep AM720 military vehicle, [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]], [[w:Jeep Wrangler (TJ)|Jeep TJL]], [[w:Jeep Wrangler (JK)#Military Jeep J8 (2007–present)|Jeep J8]] (2008-2019),  [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]], [[w:Jeep Liberty (KJ)|Jeep Cherokee (KJ)]], [[w:Jeep Liberty (KK)|Jeep Cherokee (KK)]], [[w:Jeep Grand Cherokee (WK2)|Jeep Grand Cherokee (WK2)]]
|}

==Current non-Chrysler FCA/Stellantis Factories Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| Y (700),&lt;br&gt; 9 (ProMaster Rapid),&lt;br&gt; 3 (Vision)
| Betim Plant
| [[w:Betim|Betim]], [[w:Minas Gerais|Minas Gerais]]
| [[w:Brazil|Brazil]]
| 1973
| 
| [[w:RAM 700|RAM 700]] (2015-),&lt;br&gt; [[w:ProMaster Rapid|Ram V700 Rapid]] (Peru)&lt;br&gt; Related models:&lt;br&gt; [[w:Fiat Strada|Fiat Strada]] ('99-),&lt;br&gt; [[w:Fiat Fiorino#Latin America (2013–present)|Fiat Fiorino]] ('14-)
| Fiat plant. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Dodge Vision|Dodge Vision]] (Mexico: 2015-2018),&lt;br&gt; [[w:ProMaster Rapid|Ram ProMaster Rapid]] (Mexico: '18-'24)
|-
| U
| Cordoba Plant
| [[w:Córdoba, Argentina|Córdoba]], [[w:Córdoba Province, Argentina|Córdoba Province]]
| [[w:Argentina|Argentina]]
| 1995
| 
| [[w:Peugeot Landtrek|Ram Dakota]] (2026-)
| Fiat plant.
|-
| F
| [[w:FCA India Automobiles|FCA India Automobiles Private Limited]]
| [[w:Ranjangaon|Ranjangaon]], [[w:Pune district|Pune district]], [[w:Maharashtra|Maharashtra]]
| [[w:India|India]]
| 1997
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]],&lt;br&gt; [[w:Jeep Meridian|Jeep Meridian/Commander]],&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]]
| Originally established as a 50/50 joint venture between Fiat and [[w:Tata Motors|Tata Motors]] called Fiat India Automobiles Private Limited. On June 1, 2017, Jeep Compass production began in India, the first Jeep built in India under its own brand. The JL-series Wrangler began to be assembled in India in 2021. The Jeep Meridian began production in May 2022. The Meridian is also exported to Japan as the Commander. The Jeep Grand Cherokee began assembly in India in November 2022. 
|-
| K
| Goiana Plant
| [[w:Goiana|Goiana]], [[w:Pernambuco|Pernambuco]]
| [[w:Brazil|Brazil]]
| 2015
|
| [[w:Jeep Renegade|Jeep Renegade]], [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Commander (2022)|Jeep Commander]], [[w:Ram Rampage|Ram Rampage]]&lt;br&gt; Related models: [[w:Fiat Toro|Fiat Toro]]
| [[w:Fiat Chrysler Automobiles|FCA]] plant. Production at Goiana began with the Jeep Renegade in February 2015. &lt;br&gt; Past Chrysler Group models: [[w:Ram 1000|Ram 1000]]
|-
| P
| Melfi Plant (formerly SATA = Società Automobilistica Tecnologie Avanzate [Advanced Technologies Automotive Company])
| [[w:Melfi|Melfi]], [[w:Province of Potenza|Province of Potenza]]
| [[w:Italy|Italy]]
| 1993
|
| [[w:Jeep Compass#Third generation (J4U; 2025)|Jeep Compass (J4U)]]&lt;br&gt; (Europe: '26-)
| Fiat plant. Jeep production began at Melfi in 2014 with the Renegade. Renegade production at Melfi ended on October 17, 2025. Production of the 3rd gen. Compass began on October 29, 2025. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Jeep Renegade|Jeep Renegade]] (US/Can.: '15-'23, Europe: '15-'25),&lt;br&gt; [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (Europe: '20-'25) &lt;br&gt; Related models:&lt;br&gt; [[w:Fiat 500X|Fiat 500X]] (US/Can.: '16-'23,&lt;br&gt; Europe: '15-'24)
|-
| B
| Porto Real Plant
| [[w:Porto Real|Porto Real]], [[w:Rio de Janeiro (state)|Rio de Janeiro state]]
| [[w:Brazil|Brazil]]
| 2000
|
| [[w:Jeep Avenger|Jeep Avenger]]
| PSA plant. The Jeep Avenger is the first Jeep to be made in a former PSA plant. 
|-
| U (2027-), 6 (2015-2022)
| [[w:Tofaş|Tofaş]]
| [[w:Bursa|Bursa]]
| [[w:Turkey|Turkey]]
| 1971
|
| [[w:Citroën Jumpy#Ram ProMaster City|Ram ProMaster City]] (2027-)
| Originally a Fiat joint venture, it is now 37.8% owned by Stellantis, 37.8% owned by [[w:Koç Holding|Koç Holding]], and 24.3% publicly traded on the Istanbul Stock Exchange. &lt;br&gt; Past Chrysler Group models: &lt;br&gt; [[w:Fiat Doblò#Ram ProMaster City|Ram ProMaster City]] (2015-2022), [[w:Chrysler Neon#Third generation (2016)|Dodge Neon]]&lt;br&gt; (Mexico &amp; Middle East: 2017-2020), [[w:Fiat Fiorino#Europe (2007–2024)|Ram V700 City]] (Chile: 2018-2023)
|-
| J,&lt;br&gt; 5 (Ypsilon)
| Tychy Plant
| [[w:Tychy|Tychy]], [[w:Silesian Voivodeship|Silesian Voivodeship]]
| [[w:Poland|Poland]]
| 1975
|
| [[w:Jeep Avenger|Jeep Avenger]]
| Fiat plant. Production began in September 1975 when the plant was owned by Polish automaker [[w:Fabryka Samochodów Małolitrażowych|FSM]], which built Fiat-based models. Fiat took over FSM in 1992. Fiat subsequently became [[w:Fiat Chrysler Automobiles|FCA]] and then Stellantis. Jeep Avenger production began on January 31, 2023. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Chrysler Ypsilon|Chrysler Ypsilon]] (UK/Ireland/Japan)
|}

==Current partner factories making Chrysler Group vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| GAC Hangzhou plant
| [[w:Hangzhou|Hangzhou]], [[w:Zhejiang|Zhejiang]]
| [[w:China|China]]
| 2021 (production began for Chrysler)
|
| [[w:Trumpchi GS5#Dodge Journey|Dodge Journey]] (Mexico: 2022-)
| [[w:GAC Group|GAC]] plant. 
|-
| 3
| GAC Yichang plant
| [[w:Yichang|Yichang]], [[w:Hubei|Hubei]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Dodge Attitude#Fourth generation (2025)|Dodge Attitude]] (Mexico: 2025-)
| [[w:GAC Group|GAC]] plant. 
|-
| 5
| Shenzhen Baoneng Motor Co., Ltd.
| [[w:Shenzhen|Shenzhen]], [[w:Guangdong|Guangdong]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Peugeot Landtrek|Ram 1200]] (Mexico: 2025-)
| [[w:Baoneng Group#Automotive business|Shenzhen Baoneng Motor Co., Ltd.]] plant. 
|}

==Former factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Closed
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
|
| Ajax Trim plant
| [[w:Ajax, Ontario|Ajax]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1953, 1964 (became part of Chrysler)
| 2003
| Automotive Soft Trim Components, Seat Cushion and Seatback Covers, Foam-in-place Covers
| Built in 1953 by Canadian Automotive Trim. Purchased by Chrysler in 1964. Closed by December 2003.
|-
| J (1989-1992),&lt;br&gt; B (1981-1988),&lt;br&gt; 7-8 (1966-1980),&lt;br&gt; T (1958-1966) 
| [[w:Brampton Assembly (AMC)|AMC Brampton Assembly]]
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]] 
| 1961,&lt;br&gt; 1987 (became part of Chrysler)
| 1992
| [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1987-92), [[w:Jeep CJ#CJ-5|Jeep CJ-5]] (1979-1980),&lt;br&gt; [[w:Jeep CJ#CJ-7|Jeep CJ-7]] (1979-1980),&lt;br&gt; [[w:AMC Eagle|AMC Eagle]] (1981-1987), [[w:AMC Eagle|Eagle Wagon]] (1988), [[w:AMC Concord|AMC Concord]] (1978, 1981-1983), [[w:AMC Spirit|AMC Spirit]] (1983), [[w:AMC Hornet|AMC Hornet]] (1970-77), [[w:AMC Gremlin|AMC Gremlin]] (1970-1978),&lt;br&gt; [[w:AMC Rebel|AMC Rebel]] (1968-1970), [[w:Rambler Rebel#Fifth generation|Rambler Rebel]] (1967), [[w:Rambler Classic|Rambler Classic]] (1961-1966),&lt;br&gt; [[w:AMC Ambassador|AMC Ambassador]] (1966-1968), [[w:AMC Ambassador|Rambler Ambassador]] ('63-'65), [[w:Rambler American|Rambler American]] (1962-68), [[w:Rambler American|AMC Rambler]] (1969)
| [[w:American Motors Corporation|AMC]] plant. Became part of Chrysler in the 1987 buyout of AMC. Located at at the corner of Kennedy Road South and Steeles Avenue East. Production began on January 26, 1961 with the Rambler Classic. This was the last plant to produce AMC vehicles. Eagle Wagon (formerly AMC Eagle) ended production on December 11, 1987. Production ended in April 1992 and Jeep Wrangler production was moved to Toledo, OH. Buildings on the west side of the plant were demolished in 2005 and buildings on the east side were demolished in 2007. A Lowe's and a Wal-Mart now occupy some of the former plant site.
|-
| V
| [[w:Conner Avenue Assembly|Conner Avenue Assembly]] 
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1996
| 2017
| [[w:Dodge Viper|Dodge Viper]] &lt;br&gt; GTS: '96, &lt;br&gt; All models: &lt;br&gt; '97-'06, '08-'10, '13-'17, [[w:Plymouth Prowler|Plymouth Prowler]]&lt;br&gt; ('97, '99-'01),&lt;br&gt; [[w:Chrysler Prowler|Chrysler Prowler]] ('01-'02),&lt;br&gt; [[w:Viper engine|8.0L/8.3L/8.4L aluminum &lt;br&gt; Viper V10 engine]] &lt;br&gt;(5/01-2017)
| Actually located at 20000 Connor Street. The plant was originally built in 1966 to make spark plugs by Champion. After Champion was bought by Cooper Industries in 1990, the plant was closed. It remained empty until Chrysler bought it in 1995. Viper production was moved to the Connor Avenue plant from the New Mack plant beginning with the GTS coupe for 1996, followed by the RT/10 roadster for 1997. Prowler production began in May 1997 and ended on February 15, 2002. Production of the Viper's aluminum V10 engine was moved to Connor Avenue, where it was built alongside the Viper itself, in May 2001 from the Mound Road Engine Plant, which closed in 2002. Viper production ended on July 2, 2010 and the plant was dormant until production restarted in December 2012. Production ended on Aug. 31, 2017. In 2018, the plant was renamed Connor Center, a meeting and display space that will showcase Chrysler’s concept and historic vehicle collection. 
|-
|E
| [[w:Diamond-Star Motors|Diamond-Star Motors/Mitsubishi Motors Manufacturing America/Mitsubishi Motors North America Manufacturing Division]]
| [[w:Normal, Illinois|Normal, Illinois]]
| [[w:United States|United States]]
| 1988
| 2005 (end of Chrysler production)/&lt;br&gt; 2015 (end of Mitsubishi production)
| [[w:Plymouth Laser|Plymouth Laser]] (1990-1994),&lt;br&gt; [[w:Eagle Talon|Eagle Talon]] (1990-1998),&lt;br&gt; [[w:Eagle Summit#First generation (1989–1992)|Eagle Summit 4-d]] (1991-1992),&lt;br&gt; [[w:Dodge Avenger#Dodge Avenger Coupe (1995–2000)|Dodge Avenger]] (1995-2000),&lt;br&gt; [[w:Dodge Stratus#Stratus coupe (2001–2005)|Dodge Stratus Coupe]]&lt;br&gt; (2001-2005),&lt;br&gt; [[w:Chrysler Sebring|Chrysler Sebring Coupe]]&lt;br&gt; (1995-2005),&lt;br&gt;  [[w:Mitsubishi Eclipse|Mitsubishi Eclipse]] (1990-2012),&lt;br&gt; [[w:Mitsubishi Mirage#Third generation (1987)|Mirage sedan]] (1991-1992),&lt;br&gt; [[w:Mitsubishi Galant|Galant]] (1994-2012),&lt;br&gt; [[w:Mitsubishi Endeavor|Endeavor]] (2004-2011),&lt;br&gt; [[w:Mitsubishi ASX#First generation (GA; 2010)|Outlander Sport/RVR]] (2013-15)
| Originally established as Diamond-Star Motors, a 50/50 joint venture between Chrysler and Mitsubishi which assembled both Chrysler and Mitsubishi vehicles. Production began in September 1988. In October 1991, Chrysler sold its 50% stake in the plant to Mitsubishi but production for Chrysler continued under contract. On May 24, 1993, production of the Mitsubishi Galant began at the Diamond-Star plant. In July 1995, the plant was renamed Mitsubishi Motors Manufacturing America. In January 2002, the plant was renamed Mitsubishi Motors North America Manufacturing Division. In January 2003, production of the Mitsubishi Endeavor began, the first SUV built at the Mitsubishi plant. In February 2005, production of vehicles for Chrysler ended. All further production was only of Mitsubishi-branded vehicles. In mid-2012, the plant began producing the Mitsubishi Outlander Sport. The Outlander Sport is sold as the RVR in Canada. On November 30, 2015, vehicle production ended. The Mitsubishi plant produced 3,283,549 vehicles. The plant continued to produce replacement parts until May 2016, when the plant closed completely. In June 2016, the plant was sold to liquidation firm Maynards Industries. In January 2017, EV startup [[w:Rivian|Rivian Automotive]] bought the former Mitsubishi plant. Rivian began production at the Normal, IL plant in September 2021 with the [[w:Rivian R1T|R1T]] electric pickup.
|-
| U
| [[w:Eurostar Automobilwerk|Eurostar]] - Chrysler Graz Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1991
| 2002
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (1992-2002), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2002)
| Originally, a 50/50 joint venture between Chrysler and Steyr-Daimler-Puch founded in 1990. The Eurostar plant is next to the Steyr Fahrzeugtechnik plant solely owned by Steyr-Daimler-Puch. Production of the Chrysler Voyager and Grand Voyager minivans began in October 1991. This was the 2nd generation Chrysler minivan. The 3rd generation began production on September 25, 1995. In 1996, production of right-hand drive minivans began. The 4th generation began production in January 2001. Production of the PT Cruiser began in July 2001 on the same line as the Voyager minivans. In 1999, DaimlerChrysler bought the 50% stake in Eurostar held by Steyr-Daimler-Puch Fahrzeugtechnik, now majority owned by Magna International, making Eurostar a subsidiary of DaimlerChrysler. DaimlerChrysler sold 100% of Eurostar to Magna Steyr in July 2002. PT Cruiser production in Austria ended and was consolidated in Toluca, Mexico. Production of the Chrysler Voyager and Grand Voyager minivans moved next door to the main Magna Steyr plant for 2003. Magna International had acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. 
|-
| 3 (1959),&lt;br&gt; E (1958)
| Evansville Assembly
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1919, 1928 (became part of Chrysler)
| 1959 
| Graham Brothers trucks (through 1932),&lt;br&gt; Dodge Brothers trucks &lt;br&gt; (through 1932),&lt;br&gt; Plymouth cars (1936-1958),&lt;br&gt; Dodge cars (1937-1938),&lt;br&gt; [[w:Plymouth Savoy|Plymouth Savoy]] (1959), [[w:Plymouth Belvedere|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1959)
| Located at 1625 N. Garvin St. Built in 1919 by Graham Brothers Truck Company to build trucks. In 1925, Dodge Brothers bought a controlling 51% stake in Graham Brothers and then bought the rest in 1926, completely merging the 2 companies. This gave Dodge Brothers plants in Evansville, IN and Stockton, CA. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. At that point, trucks with a Dodge Brothers nameplate were rated as a half-ton; larger rated trucks were sold under the Graham Brothers name. On January 1, 1929, the Graham Brothers brand was eliminated, and all trucks produced became Dodge trucks. In 1932, Chrysler closed the Evansville plant due to the Great Depression. In 1935, as the economy improved, Chrysler reopened the Evansville plant and renovated and expanded it. It began building Plymouth cars for 1936. Dodge cars were also built for 1937 and 1938. During World War II, the plant became the Evansville Ordinance Plant, which produced more than 3.26 billion ammunition  cartridges - about 96% of all the .45 automatic ammunition produced for all the armed forces. The Evansville Ordinance Plant also rebuilt 1,600 Sherman tanks and 4,000 military trucks. After the war ended, Plymouth car production resumed. However, in the early 1950s, during the Korean War, the Evansville plant retooled and dedicated about a third of its space and manpower to building 60-foot aluminum hulls for Grumman  UF-1 Albatross air-sea rescue planes for the Navy and Coast Guard. Evansville built its 1 millionth Plymouth in March 1953. The plant was closed in 1959 and was replaced by the larger, more modern St. Louis plant in Fenton, MO, which had access to more railroad lines for shipping than Evansville did. 
|-
| 
| Evansville Stamping
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1935, 1953 (became part of Chrysler)
| 1959 
| Body panel stampings
| Opened by Briggs Manufacturing Company when Chrysler reopened Evansville Assembly in 1935 to build Plymouth cars. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Made body panels for the nearby Evansville Assembly plant. Closed when Evansville Assembly closed in 1959.
|-
| B (1968-1980),&lt;br&gt; 2 (1960-1967),&lt;br&gt; 2 (1959)
| [[w:Dodge Main|Hamtramck Assembly (Dodge Main)]]
| [[w:Hamtramck, Michigan|Hamtramck, Michigan]]
| [[w:United States|United States]]
| 1911,&lt;br&gt; 1928 (became part of Chrysler)
| 1980 
| Dodge (1914-1958),&lt;br&gt; [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1959),&lt;br&gt; [[w:Dodge Royal#Third generation (1957–1959)|Dodge Royal]] (1959),&lt;br&gt; [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959),&lt;br&gt; [[w:Dodge Matador|Dodge Matador]] (1960),&lt;br&gt; [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962),&lt;br&gt; [[w:Dodge Polara|Dodge Polara]] (1960-1964),&lt;br&gt; [[w:Dodge 330|Dodge 330]] (1963-1964),&lt;br&gt; [[w:Dodge 440|Dodge 440]] (1963-1964),&lt;br&gt; [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1975), [[w:Plymouth Duster|Plymouth Duster]] (1970-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1969, 1972-1975), [[w:Dodge_Dart#1971|Dodge Dart Demon]] (1971-1972),&lt;br&gt; [[w:Dodge Charger (1966)|Dodge Charger]] (1967-1969), [[w:Dodge Charger Daytona#First generation (1969)|Dodge Charger Daytona]] ('69), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-74), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970-1974), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volaré|Plymouth Volaré]] (1976-1980),&lt;br&gt; Engines
| Located at 7900 Joseph Campau Ave. This plant predated Dodge being part of Chrysler Corp. On November 4, 1914, the first Dodge Brothers passenger car was produced at the Hamtramck plant. Prior to that, Dodge Brothers made components for other automakers, primarily Ford. The Hamtramck plant was fully vertically integrated, capable of building almost every part needed to build a complete car. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. Even after the Chrysler takeover, Hamtramck remained Dodge's home plant. From the early 1950s, various operations were automated or moved to other plants and Hamtramck became more of an assembly plant by the early 1960s. Closed January 4, 1980. Last vehicle built was a Silver Metallic 1980 Dodge Aspen R/T 2-door. 13,943,221 vehicles were produced at the plant. Demolished in 1981. Replaced by the General Motors Detroit/Hamtramck Assembly Plant (Factory Zero), which opened in 1985. 
|-
| 
| [[w:Highland Park Chrysler Plant|Highland Park Plant]]
| [[w:Highland Park, Michigan|Highland Park, Michigan]]
| [[w:United States|United States]]
| 1909,&lt;br&gt; 1925 (became part of Chrysler)
| 1960s (end of manufacturing)
| Maxwell cars (1910-1925), Chrysler Series 50 (1926-27), Chrysler Series 52 (1928), Plymouth Model Q (1929), Chrysler Series 60 (1927), Chrysler Series 62 (1928), DeSoto Series K (1929-1930), DeSoto Series CK (1930), DeSoto Series CF (1930), Fargo Trucks (1928-1930) &lt;br&gt; Parts including fluid coupling and torque converter for [[w:Fluid Drive|Fluid Drive]]
| Located at at 12000 Chrysler Service Drive (formerly Chrysler Drive). Originally, this was [[w:Maxwell Motor Company|Maxwell Motor Company]]'s main plant. However, before Maxwell Motor Co.'s formation, parts of the site was used by several car and truck manufacturers: Grabowsky Power Wagon Company used 1 building, Brush Runabout Co. used another building, and Gray Motor Co. owned another building. Gray only used the western 1/3 of the building so they leased the middle third to Alden- Sampson Truck Co. and the eastern third to Maxwell-Briscoe Motor Co. All these companies, along with several others, combined under the [[w:United States Motor Company|United States Motor Company]] in 1910. In 1913, United States Motor Company collapsed and Maxwell was the only surviving part. The U.S. Motor Co. assets were purchased by Walter Flanders, who reorganized the company as the Maxwell Motor Co. Maxwell hired Walter P. Chrysler to turn the company around in 1921 after its finances deteriorated in the post-World War I recession in 1920. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including its Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924 at the Jefferson Ave. plant. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925 at Highland Park. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 built at Highland Park and was then reworked again into the first Plymouth in 1928, also built at Highland Park. Plymouth production was moved to the new Lynch Road Assembly plant in 1929 and DeSoto production also moved to the new Lynch Road Assembly plant in 1931. Highland Park still built parts but it no longer built vehicles. Highland Park was used more for design,  engineering, and management and served as Chrysler Corporation's headquarters through 1996. During the 1990's, Chrysler moved to its current headquarters at the Chrysler Technology Center in Auburn Hills. Much of the site has been demolished though Chrysler still has a small presence at the site with the FCA Detroit Office Warehouse. Other parts of the site are now occupied by several automotive suppliers including Magna, Valeo, Mobis, Avancez, and Yanfeng.
|-
|
| [[w:Indiana Transmission#Indiana Transmission Plant II|Indiana Transmission Plant II]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2003 (as Indiana Transmission Plant II)
| 2019 (as Indiana Transmission Plant II)
| [[w:W5A580|Mercedes W5A580 (A580) &lt;br&gt; 5-speed auto. trans.]], Transmission components
| Located at 3360 North U.S. Highway 931. Plant was originally known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Production began in November 2003. 5-speed auto. trans. production ended in August 2018 while production of components for the 8-speed auto. transmission ended in the fall of 2019. Starting in 2020, the plant was converted to engine production and is now known as Kokomo Engine Plant. Engine production began in late February 2022. 
|-
|
| Indianapolis Electrical Plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1953
| 1988
| Transmission plant: [[w:Chrysler PowerFlite transmission|Chrysler PowerFlite 2-speed auto. transmission]] &lt;br&gt; Electrical Plant: Alternators, distributors, starters, power steering units, voltage regulators, windshield wiper motors, and other electrical parts for cars
| Located at 2900 Shadeland Ave. Began making transmissions in 1953. In January 1959, Chrysler housed its new Electrical Division at the Shadeland Avenue plant, replacing transmission production. Became part of Chrysler's Acustar components subsidiary in 1987. Production ended on November 30, 1988 but shipping and other activities continued until March 1989 when the factory closed. Certain portions have been demolished and improvements were made to the remainder, which is now the Shadeland Business Center. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Naomi Street plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1946 (became part of Chrysler)
| 1970's
| Engine blocks
| Located at 1535 Naomi Street. Purchased from American Foundry Company in 1946. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Kept operating even after the Tibbs Ave. plant was launched. This location is now Wilco Gutter Supply. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Tibbs Avenue plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1950 
| 2005
| Engine heads and blocks and other components
| Located at 1100 S. Tibbs Avenue. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Production ended on September 30, 2005 and the facility closed. Demolished in 2006. 
|-
| C (1968-1990),&lt;br&gt; 3 (1960-1967),&lt;br&gt; 1 (1959)
| [[w:Detroit Assembly Complex – Jefferson#Jefferson Avenue Assembly|Jefferson Avenue Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1908,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| [[w:Chrysler Six|Chrysler Series 70 (B-70)]] (1924-1925), [[w:Chrysler Six|Chrysler Series 70 (G-70)]] (1926-1927), [[w:Chrysler Six|Chrysler Series 72]] (1928), [[w:Chrysler Royal|Chrysler Royal]] (1933, 1937-1950), DeSoto SD (1933), Chrysler Airflow (1934-1937), Chrysler Airstream (1935-36), [[w:Chrysler (brand)|Chrysler brand]] (1937-1958), Desoto Airflow (1934-1936), DeSoto Airstream (1935-1936), [[w:Chrysler Imperial|Chrysler Imperial]] (1926-1942, 1946-1954), [[w:Imperial (automobile)|Imperial]] (1955-1958, 1962-1975), [[w:Chrysler 300 letter series|Chrysler 300 letter series]] (1959-1965), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-1978), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1961), [[w:Chrysler Newport|Chrysler Newport]] (1961-1978), [[w:Chrysler 300 non-letter series|Chrysler 300 Sport Series]] (1962-1971), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1968-1972), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:DeSoto Fireflite|DeSoto Fireflite]] (1959-1960), [[w:DeSoto Adventurer|DeSoto Adventurer]] (1959-1960), [[w:DeSoto (automobile)#1961|DeSoto]] (1961), [[w:Dodge Matador|Dodge Matador]] (1960), [[w:Dodge Polara|Dodge Polara]] (1965-1966), [[w:Dodge D series|Dodge D/W series]] (1979-1980), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1979-1980), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1979-1980), [[w:Dodge Aries|Dodge Aries]] (1981-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1981-1989), [[w:Dodge 400|Dodge 400]] 4-d (1982-1983), [[w:Chrysler LeBaron|Chrysler LeBaron]] 4-d (1982-1984), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker (E-body)]] (1983-1987), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker Turbo (E-body)]] (1988), [[w:Dodge 600|Dodge 600]] 4-d (1983-1988), [[w:Chrysler E-Class|Chrysler E-Class]] (1983-1984), [[w:Plymouth Caravelle|Plymouth Caravelle]] (US: 1985-1988), [[w:Plymouth Caravelle|Plymouth Caravelle]] 4-d (Canada: 1983-1988), [[w:Dodge Omni|Dodge Omni]] (1989-1990), [[w:Plymouth Horizon|Plymouth Horizon]] (1989-1990),&lt;br&gt; Engines
| Located at 12200 East Jefferson Ave. Plant was originally opened by [[w:Chalmers Automobile|Chalmers Motor Co.]]. After falling on hard times, Chalmers agreed in 1917 to build cars for [[w:Maxwell Motor Company|Maxwell Motor Co.]] at the Jefferson Ave. plant in Detroit. In exchange, Chalmers cars would be sold through Maxwell dealers. After having its own financial problems, Maxwell stopped producing cars at the Chalmers plant in 1921. Maxwell hired Walter P. Chrysler to turn the company around in 1921. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including the Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 and was then reworked again into the first Plymouth in 1928. The Jefferson Ave. plant was the home plant of the Chrysler brand through 1978. It was also the home plant for the spin-off Imperial brand except for 1959-1961, when Imperial had its own exclusive plant on Warren Ave. in Dearborn. By the time it ended production on February 2, 1990, Jefferson Ave. Assembly had built 8,310,107 vehicles. Demolished in 1991. Replaced by the Jefferson North plant built across Jefferson Ave. from the old plant, where the Kercheval Body Plant used to be. The Jefferson North plant opened in January 1992.
|-
| W (1987-1989 Chrysler M-body) &lt;br&gt;&lt;br&gt; [K for 1981-1983 AMC and 1983-1987 Renault],&lt;br&gt; 0-6 (1966-1980 AMC)
| Kenosha I Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]]&lt;br&gt; (1987-1989),&lt;br&gt; [[w:Dodge Diplomat#Second generation (1980)|Dodge Diplomat]] (1987-1989), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1987-89), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1987-1989)
| This was the Kenosha Main Plant of [[w:American Motors Corporation|AMC]]. The Kenosha plant was the oldest still operating automobile factory in the world when it ended vehicle production in December 1988. It first built automobiles in 1902 for the Thomas B. Jeffery Company under the Rambler brand. The factory was purchased in 1900 from the Sterling Bicycle Co., which built it in 1895. In 1914, the Thomas B. Jeffery Company rebranded its vehicles under the Jeffery brand. In 1916, the Thomas B. Jeffery Company was bought by Charles Nash and renamed Nash Motors. Kenosha produced Nash vehicles from 1917-1957. Kenosha also produced Nash's entry-level Lafayette brand from 1934-1936. After Nash merged with Hudson to form American Motors in 1954, Kenosha also produced Hudson vehicles from 1955-1957. Kenosha then produced vehicles under the Rambler brand for AMC from 1958-1968 and under the AMC brand from 1966-1983. Kenosha also produced the Alliance for AMC shareholder Renault for 1983-1987 along with the Encore for 1984-1986 and the GTA for 1987. Chrysler signed a deal with AMC in September 1986 to utilize surplus capacity at AMC's Kenosha plant to build Chrysler's trio of rwd M-body sedans beginning in February 1987. Chrysler did not have any capacity left in its own plants to continue building the M-body sedans. The St. Louis North plant that had been building the M-body sedans had been converted to build the extended length minivans. This deal led to Chrysler's acquisition of AMC, announced in March 1987. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 and the M-bodies were discontinued. The site continued building engines until 2010. The plant has since been demolished.
|-
| Y
| Kenosha II Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Dodge Omni|Dodge Omni]] (1988-1989), [[w:Plymouth Horizon|Plymouth Horizon]] (1988-1989)
| This was the Kenosha Lakefront Plant of [[w:American Motors Corporation|AMC]], located on the shore of Lake Michigan. This property was originally a Simmons mattress manufacturing plant from 1870 to 1960. AMC bought it in 1960 to manufacture and paint auto bodies. Became part of Chrysler in the 1987 buyout of AMC. In September 1987, production of the Dodge Omni and Plymouth Horizon began. Production was moved to Kenosha from Chrysler's Belvidere, IL plant, which was being converted to build Chrysler's C-body sedans (Dynasty/New Yorker). Closed in December 1988. Omni &amp; Horizon production then moved to the Jefferson Ave. plant in Detroit. Demolished in 1990. In 1994, the City of Kenosha purchased the property for $1. The property was subsequently cleaned up and redeveloped into the HarborPark area, which includes a park and open space, a public museum, residential housing, and a marina. 
|-
| 
| [[w:Kenosha Engine|Kenosha Engine Plant]]
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2010
| [[w:AMC straight-4 engine|AMC straight-4 engine]],&lt;br&gt; [[w:AMC straight-6 engine|AMC straight-6 engine]],&lt;br&gt; [[w:AMC V8 engine|AMC V8 engine]],&lt;br&gt; [[w:Chrysler LH engine|Chrysler 2.7L DOHC V6]], [[w:Chrysler SOHC V6 engine#3.5|Chrysler 3.5L SOHC V6]]
| Located at 5555 30th Avenue. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 at the adjacent assembly plant but the site continued building engines until 2010. After the Chrysler buyout, the plant kept building the AMC 5.9L V8 for the SJ Jeep Grand Wagoneer through 1991, the AMC 2.5L I4 through 2002 for Jeeps and the Dodge Dakota, the AMC 4.2L I6 through 1990 for the Jeep Wrangler, and the AMC 4.0L I6 through 2006 for various Jeeps ('06 Wrangler was the last to use the 4.0L). Chrysler started building its own 2.7L V6 at Kenosha in 1997 and its own 3.5L V6 in 2003. Engine production ended in October 2010 and the plant closed. Demolished in 2012-2013.
|-
|
| Kercheval Body Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1920,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| Automobile Bodies
| Located at 12265 E Jefferson Ave., across Jefferson Ave. from Chrysler's Jefferson Ave. Assembly Plant, which had originally been the Chalmers plant. The assembly plant was on the south side of Jefferson Ave. while the body plant was on the north side. The plant was called Kercheval because the north side of the plant was bordered by Kercheval Ave. The plant was built in 1920 by Wadsworth Manufacturing Co. to replace a previous plant on the same site that burned down in 1919. That fire had also damaged the Chalmers plant across the street. In November 1920, Wadsworth Manufacturing was sold to American Motor Body Co., a division of the American Can Co. On July 1, 1923, American Motor Body Co. became American Motor Body Corp., run by Charles M. Schwab. On September 4, 1925, Chrysler Corp. bought the Detroit plant of the American Motor Body Corp. to quickly increase its manufacturing capacity. In 1955, Kercheval Body Plant was connected to the Jefferson Assembly plant by a bridge crossing over Jefferson Ave. Previously, bodies made at Kercheval had to be transported by truck across Jefferson Ave. to the assembly plant. The plant closed in Feb. 1990, at the same time the Jefferson Ave. Assembly Plant closed. The Kercheval plant was demolished and Chrysler built the new Jefferson North Assembly Plant on the site of the former Kercheval Body Plant, on the north side of Jefferson Ave. 
|-
|
| Kokomo - Home Ave. plant
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1937
| 1969
| Manual Transmissions (1937-1955), Aluminum die casting (1955-1969)
| Located at 1105 S. Home Ave. Chrysler bought this plant in 1937. This was Chrysler's first plant in Kokomo. It had previously belonged to the [[w:Haynes Automobile Company|Haynes Automobile Co.]], which went out of business in 1925. The plant had been dormant since then. 5,124,211 manual transmissions were built here from 1937-1955. Transmission production then shifted to a new plant about a mile southeast on South Reed Road. The Home Ave. plant then became an aluminum die casting plant until 1969, when that operation shifted to the new Kokomo Casting plant on East Boulevard. Chrysler subsequently sold this plant. The facility was last used by Warren's Auto Parts, an auto salvage yard, which closed in 2020 after nearly 50 years. It is currently empty though still standing as of 2025. 
|-
| M
| [[w:Lago Alberto Assembly|Lago Alberto Assembly]]
| [[w:Nuevo Polanco|Nuevo Polanco district]], [[w:Miguel Hidalgo, Mexico City|Miguel Hidalgo borough]], [[w:Mexico City|Mexico City]]
| [[w:Mexico|Mexico]]
| 1938
| 2002
| &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Savoy, Dodge Dart, Dodge 330, Dodge 440, Chrysler Valiant (1963-1969), Valiant Barracuda (1965-1969), Dodge Coronet, [[w:Dodge Ram|Dodge Ram pickup]] (1981-02), [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-96), [[w:Dodge Ramcharger#Third generation (1999–2001)|Dodge Ramcharger]] (1999-01) &lt;br&gt; Export to US:&lt;br&gt; [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-93), [[w:Dodge Ram#First generation (1981; D/W)|Dodge Ram pickup]] (1990-93), [[w:Dodge Ram#Second generation (1994; BR/BE)|Dodge Ram pickup]] (1994-02)
| Located at 320 Lago Alberto Street. Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Lago Alberto began exporting to the US with the 1986 Dodge Ramcharger, sourced exclusively from Mexico. The Lago Alberto plant was closed in 2002 and Mexican pickup production was consolidated in the newer, more modern Saltillo plant. 
|-
| E (1968-1971),&lt;br&gt; 5 (1960-1967),&lt;br&gt; 4 (1959),&lt;br&gt; L (1958),&lt;br&gt; L (1955-1957 Chrysler brand)
| [[w:Los Angeles (Maywood) Assembly|Los Angeles (Maywood) Assembly]]
| [[w:Commerce, California|City of Commerce, California]]
| [[w:United States|United States]]
| 1932
| 1971
| Plymouth (1946-1958), Dodge (1946-1953, 1955-1958), DeSoto Deluxe/Custom (1948-1952), DeSoto Powermaster Six (1953-1954), DeSoto Firedome (1952-57), DeSoto Fireflite (1955-1957), DeSoto Firesweep (1957-1958), Chrysler Windsor (1948-1958), Chrysler Royal (1949-1950), Chrysler Saratoga (1951-1952, 1957-1958), Chrysler New Yorker (1953-1958), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1960), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-60), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1957-1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Polara|Dodge Polara]] (1960-1964), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1958-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1959-1961), [[w:Plymouth Savoy|Plymouth Savoy]] (1958-1959, 1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1971), [[w:Plymouth Duster|Plymouth Duster]] (1970-1971), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1971), [[w:Dodge Dart#1971|Dodge Dart Demon]] (1971), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1966, 1970), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1971), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1971), [[w:Dodge Coronet|Dodge Coronet]] (1965-1971), [[w:Dodge Charger (1966)|Dodge Charger]] (1971), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971)
| Located at 5800 South Eastern Avenue and Slauson Avenue in Maywood, now part of City of Commerce. Across the street from the [[w:Maywood Assembly|Ford Maywood Assembly plant (Los Angeles Assembly plant No. 1)]]. 
|-
| A (1968-1981),&lt;br&gt; 1 (1960-1967),&lt;br&gt; 6 (1959)
| [[w:Lynch Road Assembly|Lynch Road Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1929
| 1981
| Plymouth (1929-1958), DeSoto (1931-1932), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-64), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (59-61), [[w:Plymouth Fury|Plymouth Fury]] (1959-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (59-61), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (62-70), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1970, 1973-1974), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1978), [[w:Plymouth GTX|Plymouth GTX]] (1967-1970), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1970, 1973, 1975), [[w:Plymouth Superbird|Plymouth Road Runner Superbird]] (1970), [[w:Dodge Coronet|Dodge Coronet]] (1965-1976), [[w:Dodge Monaco#Fourth generation (1977–1978)|Dodge Monaco]] (1977-1978), [[w:Dodge Charger (1966)#1966|Dodge Charger]] (1966), [[w:Dodge Charger (1966)#Third generation|Dodge Charger]] (1971-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971),  [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979-81), [[w:Chrysler New Yorker#1979–1981|Chrysler New Yorker]] (79-81), [[w:Dodge St. Regis|Dodge St. Regis]] (1979-81), [[w:Plymouth Gran Fury#1980–1981|Plymouth Gran Fury]] (80-81),&lt;br&gt; Engines
| Located at 6334 Lynch Road. This was originally Plymouth's home plant. At the time it opened in 1929, Lynch Road was the largest single story auto plant in the world. DeSoto production was transferred from Highland Park to Lynch Road in 1931. In June 1932, DeSoto production was moved to the Jefferson Ave. plant when the DeSoto brand moved up in the brand hierarchy to between Dodge and Chrysler. Previously, DeSoto was between Plymouth and Dodge. During World War II, Lynch Road made tank transmissions, truck parts, and uranium enrichment diffusers for the Oak Ridge Gaseous Diffusion Plant in Oak Ridge, TN to produce enriched uranium for the atomic bomb. For 1965, Lynch Road began to focus on production of Plymouth and Dodge intermediate models. For 1979, Lynch Road was switched to build Chrysler's R-body full-size cars for all 3 Chrysler car brands. Production ended on April 3, 1981 and the factory closed. Last car produced was a white Plymouth Gran Fury police car. 
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant I]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1998
| 2019
| [[w:Chrysler PowerTech engine#4.7|4.7L PowerTech SOHC V8]],&lt;br&gt; [[w:Chrysler Pentastar engine|3.0L/3.2L/3.6L Pentastar V6]]
| Located at 4000 St. Jean Avenue. Mack Ave. Engine Plant I was built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in 1998. 4.7L V8 engine production ended in April 2013. Pentastar V6 engine production ended in 2019. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant II]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2000
| 2012
| [[w:Chrysler PowerTech engine#3.7 EKG|3.7L PowerTech SOHC 90° V6]]
| Located at 4500 St. Jean Avenue. Mack Ave. Engine Plant II was built next to the Mack Ave. Engine Plant I, which had been built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in November 2000. Production ended in September 2012. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
|
| Mack Ave. Stamping Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 1979
| Stampings
| Chrysler acquired the Mack Ave. Stamping Plant from Briggs Manufacturing Company in 1953. The plant was originally built in 1916 by the Michigan Stamping Company, which was taken over by Briggs Manufacturing in 1923. The plant was closed in 1979 and the site was basically abandoned. The city of Detroit bought the plant site in 1982 but was unable to find a purchaser or afford environmental remediation for the site and returned it to Chrysler. In 1990, Chrysler began cleanup and demolition of the old plant and built a new factory on the site, which became the New Mack Assembly Plant. The site later became the Mack Ave. Engine Complex and later, the Mack Ave. Assembly Plant, which is now part of the Detroit Assembly Complex.
|-
| 
| McGraw Stamping/McGraw Glass Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 19?
| 2003
| Oil pans, valve covers, and other small stampings,&lt;br&gt; Automotive Glass (1960-)
| Located at 9400 McGraw Ave. Was around the corner and behind the Wyoming Ave. DeSoto/Export plant. Originally, a stamping plant. In 1960, switched to making automotive glass. Used Safeguard brand. Glass was DOT# 21. Demolished. 
|-
|
| [[w:Mound Road Engine|Mound Road Engine Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 2002
| [[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]],&lt;br&gt; [[w:Chrysler LA engine#239 V6|3.9L 90° V6 engine]],&lt;br&gt; [[w:Chrysler LA engine#Magnum 8.0 L V10|8.0L iron Magnum V10 engine]],&lt;br&gt; [[w:Viper engine|8.0L aluminum Viper V10 engine]] (1992-5/01)
| Located at 20300 Mound Road. One of the plants Chrysler acquired from Briggs Manufacturing Company in 1953. Chrysler used the plant to produce aircraft parts from 1953-1954 and then transferred the plant to Plymouth in 1954 to build its new A-series V8 engine for 1956 model year cars. Converted into an engine plant and enlarged by 71,000 sq. ft., it began building V8 engines for Plymouth in July 1955. Dodge later used the A engine from 1959 in the US in cars and trucks and Chrysler used the A engine from 1960 in the US. The plant was closed in 2002 and demolished in 2003. The land was then paved over and is now used as a storage lot for vehicles produced at the nearby Warren Truck Assembly Plant. Warren Truck Assembly is just to the north of where Mound Road Engine was. Mt. Elliott Tool and Die was located immediately to the south of the Mound Road Engine Plant on Outer Drive East.
|-
|
| [[w:Mount Elliott Tool and Die|Mount Elliott Tool and Die]]/Outer Drive Manufacturing Technology Center/Outer Drive Stamping
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1938, 1953 (became part of Chrysler)
| 2018
| Stamping Dies, Checking Fixtures, Stamping Fixtures 
| Located at 3675 Outer Drive East. Built in 1938 by Briggs Manufacturing Company. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Chrysler renamed it Outer Drive Stamping. Stamping operations ended in 1983 and operations from the closed Vernor Tool &amp; Die plant were moved here. The plant was then renamed Outer Drive Manufacturing Technology Center. The plant now did tool and die work as well as pilot plant operations and engineering for new stamping technologies. Once the Chrysler Technology Center in Auburn Hills was built, Pilot Operations and Advanced Stamping Manufacturing Engineering moved there and the plant was renamed Mount Elliott Tool and Die. Operations at the plant ended in 2018 and the plant was sold to German automotive supplier Laepple Automotive in 2024. Laepple Automotive is producing stamped body parts at the plant. 
|-
| V
| [[w:Detroit Assembly Complex – Mack|New Mack Assembly Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 1995
| [[w:Dodge Viper|Dodge Viper RT/10]] (1992-96)
| Located at 4000 St. Jean Avenue. The New Mack Assembly Plant is built on the site of the former Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. Viper production began in May 1992 at the New Mack Assembly Plant. After ending production in 1995, New Mack Assembly was converted into the Mack Ave. Engine Plant I. A new addition was then built to create Mack Ave. Engine Plant II. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021. 
|-
| F (1968-2009),&lt;br&gt; 6 (1960-1967),&lt;br&gt; 5 (1959),&lt;br&gt; N (1958)
| [[w:Newark Assembly|Newark Assembly]]
| [[w:Newark, Delaware|Newark, Delaware]]
| [[w:United States|United States]]
| 1951,&lt;br&gt; 1957 (Automotive prod.)
| 2008
| Plymouth (1957-1958), Dodge (1958), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1958-1959, 1961), [[w:Plymouth Fury|Plymouth Fury]] (1959-1974), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge Polara|Dodge Polara]] (1960-1966), [[w:Dodge Monaco|Dodge Monaco]] (1965-1966, 1968), [[w:Chrysler Newport|Chrysler Newport]] (1965-1970), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1965-70), [[w:Chrysler Town &amp; Country (1941–1988)|Chrysler Town &amp; Country]] (1966-1967), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1964, 1974-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1964, 1974-1975), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volare|Plymouth Volare]] (1976-1980), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1979-1980), [[w:Plymouth Reliant|Plymouth Reliant]] sedan &amp; wagon (1981-1988), [[w:Dodge Aries|Dodge Aries]] sedan &amp; wagon (1981-1988), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (sedan: 1984-1988, wagon: 1982-1988), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1989-1995), [[w:Dodge Spirit|Dodge Spirit]] (1989-1995), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron Sedan (A-body)]] (1990, 1993-1994), [[w:Chrysler Saratoga#1989–1995|Chrysler Saratoga]] (For export: 1990-1992), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe (J-body)]] (1992-1993), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron convertible (J-body)]] (1992-1995), [[w:Dodge Intrepid#First generation (1993–1997)|Dodge Intrepid]] (1994-1996), [[w:Chrysler Intrepid#First generation (1993–1997)|Chrysler Intrepid]] (Canada: 1994-1995), [[w:Chrysler Concorde#First generation (1993–1997)|Chrysler Concorde]] (1995-1996), [[w:Dodge Durango#First generation (DN; 1998)|Dodge Durango (DN)]] (1998-2003), [[w:Dodge Durango#Second generation (HB; 2004)|Dodge Durango (HB)]] (2004-2009), [[w:Chrysler Aspen|Chrysler Aspen]] (2007-2009) 
| Chrysler began construction of the Delaware Tank Plant in January 1951 to build [[w:M48 Patton|M48 Patton]] tanks. Production began in April 1952. Production ended in May 1961 and the Tank Plant was closed in October 1961. Low rate initial production of the [[w:M60 tank|M60 tank]] was also done at the Newark plant in 1959 before production was moved to the [[w:Detroit Arsenal (Warren, Michigan)|Detroit Arsenal Tank Plant]] in Warren, MI in 1960. Conversion to automotive production began in 1956. Production of Plymouth and Dodge cars began on April 30, 1957. Production ended on December 19, 2008. Sold to the University of Delaware on October 24, 2009. Most of the plant was demolished in 2010-2011 except for the Administration Building near the front of the complex. The site is now the Science, Technology, and Advanced Research (STAR) campus. The old Chrysler Administration Building has been redesigned and is now being used by the College of Health Sciences. 
|-
| K
| [[w:Pillette Road Truck Assembly|Pillette Road Truck Assembly]]
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1976
| 2003
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1976-1980),&lt;br&gt; [[w:Dodge Ram Van|Dodge Ram Van]] (1981-2003), [[w:Dodge Ram Wagon|Dodge Ram Wagon]] (1981-'02), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1976-1983)
| Located at 2935 Pillette Road. Was originally Windsor Plant 6. The Pillette Road plant was located less than a mile away from Chryslers' main plant complex in Windsor. Production began in January 1976. Production ended on June 12, 2003. 2,309,399 units were built. Demolished in 2004. Part of the site is now the Grand Central Business Park. Another part was a logistics center serving Chrysler's Windsor Assembly Plant and operated by Syncreon. The Syncreon Automotive Windsor site closed in October 2022. The parts sorting and sequencing work done there was now going to be done in-house at Chrysler's Windsor Assembly Plant. 
|-
| 
| [[w:Los Angeles (Maywood) Assembly#San Leandro Assembly|San Leandro Assembly]]
| [[w:San Leandro, California|San Leandro, California]]
| [[w:United States|United States]]
| 1948
| 1954
| Plymouth (1949-1954),&lt;br&gt; Dodge (1949-1954)
| 
|-
| B (1996-2009),&lt;br&gt; G (1968-1991),&lt;br&gt; 7 (1960-1967),&lt;br&gt; 8 (1959)
| [[w:Saint Louis Assembly|St. Louis I Assembly]] - South plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1959
| 2008
| [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1960), [[w:Plymouth Fury|Plymouth Fury]] (1960-1964), [[w:Plymouth Savoy|Plymouth Savoy]] (1960-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1961), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1965, 1976), [[w:Plymouth Duster|Plymouth Duster]] (1973-1976), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1965, 1973-1976), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1965),&lt;br&gt; [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1974), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-75), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1976), [[w:Dodge Coronet|Dodge Coronet]] (1965-1973, 75), [[w:Dodge Charger (1966)|Dodge Charger]] (1968-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971), [[w:Dodge Diplomat|Dodge Diplomat]] (1977-1981), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1977-1981), [[w:Plymouth Caravelle|Plymouth Caravelle (M-body)]] (Canada: 1978-1981), [[w:Plymouth Reliant|Plymouth Reliant]] 2-d (1982-1986), [[w:Dodge Aries|Dodge Aries]] 2-d (1982-1986), [[w:Dodge 400|Dodge 400]] 2-d &amp; convertible (1982-1983), [[w:Dodge 600|Dodge 600]] 2-d &amp; convertible (1984-1986), [[w:Plymouth Caravelle|Plymouth Caravelle]] 2-d (Canada: 1983-1986), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (2-d &amp; convertible: 1982-1986), [[w:Chrysler Executive|Chrysler Executive]] (1983-1986), [[w:Dodge Daytona|Dodge Daytona]] (1984-1991), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1984-1991), [[w:Dodge Daytona#Chrysler Laser|Chrysler Laser]] (1984-1986), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe/convertible (J-body)]] (1987-1991), [[w:Plymouth Voyager|Plymouth Voyager]] (1996-2000), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1996-2007), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2009), [[w:Chrysler Voyager|Chrysler Voyager]] (2001-2003), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1996-2001, 2004-2007)
| Located at 1001 N. Hwy Dr. St. Louis South was idled in 1991. The Dodge Daytona was moved to Sterling Heights and the J-body Chrysler LeBaron was moved to Newark, DE. St. Louis South was reopened in 1995 to build minivans, which were moved from the St. Louis North plant. For the 3rd generation, St. Louis South built both SWB and LWB models. For the 4th generation, St. Louis South built all SWB models but also built some LWB models. Closed on October 31, 2008. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1996-2009),&lt;br&gt; X (1973-1995),&lt;br&gt; U (1970-1972),&lt;br&gt; 7 (1967-1969)
| [[w:Saint Louis Assembly|St. Louis II Assembly]] - North plant / Missouri Truck Assembly Plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1966
| 2009
| [[w:Dodge D series|Dodge D/W series]] (1967-1973), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1974-1977), [[w:Plymouth Trail Duster|Plymouth Trail Duster]] (1974-1976), [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1980), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1975-1976, 1980),&lt;br&gt; [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1984-1987), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1984-1987), [[w:Dodge Diplomat|Dodge Diplomat]] (1984-1987), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]] ('84-'87), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1987-1995), [[w:Dodge Caravan|Dodge Grand Caravan]] (1987-1995), [[w:Chrysler minivans (S)#Cargo van|Dodge Extended Mini Ram Van]] (1987-1988), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1990-1995),&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1996-'09)
| Originally opened to build trucks as the Missouri Truck Assembly Plant. In 1980, the plant was idled. Plant was reopened in 1983 to build the rwd, M-body sedans, which were moved from Windsor, ON, Canada so that Windsor could be converted to build minivans. Plant was renamed St. Louis II Assembly. During 1987, the M-body sedans were moved to AMC's plant in Kenosha, WI so that St. Louis North could be converted to build the new LWB minivans. For the first 2 generations of Chrysler minivans, St. Louis North only built the LWB models. For 1996, minivan production moved to St. Louis South and the North plant was converted to build full-size pickups. Closed on July 10, 2009. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1970-1978),&lt;br&gt; 6 (1968-1969),&lt;br&gt; 9 (1961-1967)
| Tecumseh Road Truck Assembly / Windsor Truck Assembly Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1916,&lt;br&gt; 1925 (became part of Chrysler)
| 1978
| Maxwell (1924-1925),&lt;br&gt; Chrysler (1924-1929),&lt;br&gt; Dodge Trucks (1931-1960), &lt;br&gt; Dodge D-Series Trucks: &lt;br&gt; D100 (1961), D200 (1964), W200 (1965), W100 (1966), D100/W100 (1967), D200 (1970), D500 (1968),&lt;br&gt; D500/D600/D700/D800 &lt;br&gt; medium-duty trucks (1970-1972), D500/D600/W600/D700/D800 medium-duty trucks (1974-1977),&lt;br&gt; D100 (1978),&lt;br&gt; Fargo Trucks (1936-1972)
| Located at 300 Tecumseh Road East. Was originally Windsor Plant 1. Became part of Chrysler upon its founding in 1925. Was previously a Maxwell-Chalmers plant and was originally Maxwell's Canadian plant from 1916. Switched to building trucks in 1931 after Plant 3 opened in 1929. Closed in 1978. Became the Imperial Quality Assurance Centre from 1980-1983, doing extra quality control on the 1981-1983 Imperial built at the Windsor Assembly Plant (the car plant - Plant 3) about 1.5 miles east of Plant 1. The Imperial Quality Assurance Centre closed in 1983 when the Imperial was discontinued. Subsequently demolished. Is now the Plaza 300 shopping mall.
|-
|
| Tipton Transmission Plant
| [[w:Tipton, Indiana|Tipton, Indiana]]
| [[w:United States|United States]]
| 2014
| 2023
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, SI-EVT transmission
| Located at 5880 W. State Road 28. Originally, the plant was supposed to be a [[w:Getrag|Getrag]] plant focused on supplying Chrysler with dual-clutch transmissions. Chrysler withdrew from the deal in 2008 after a dispute over financing and sued Getrag. The 80% completed facility then sat dormant until Chrysler Group purchased the facility in February 2013 and completed construction. Production began in April 2014 with the ZF-designed 9-speed auto. transmission, built under license from ZF. Production ended in June 2023 and 9-speed production was consolidated into Indiana Transmission Plant I in Kokomo, IN. The SI-EVT transmission for the Pacifica Hybrid was moved to Kokomo Transmission Plant. Sold to IRH Manufacturing LLC in September 2024 to make solar cells.
|-
| L (1989-2001),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Parkway|Toledo Assembly #1 Plant]] - Jeep Parkway plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2001 (ended final assembly), 2006 (ended body assembly)
| [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]] (1984-01), [[w:Jeep Cherokee (XJ)#Wagoneer|Jeep Wagoneer (XJ)]] (1984-90), [[w:Jeep Comanche|Jeep Comanche]] (1986-1992),&lt;br&gt; Bodies for vehicles made at Stickney Ave. plant &lt;br&gt; 
Models only made before Chrysler takeover: Willys Aero (1952-1955), Kaiser Manhattan (1954-1955), Jeep CJ (1946-1986), Jeep DJ, Jeep Jeepster (1948-1950), Jeep Jeepster Commando (1967-1971), Jeep Commando (1972-1973), Willys Jeep Station Wagon (1946-1964), Willys Jeep Truck (1947-1965), Jeep Gladiator (SJ) (1963-1971), Jeep J-series pickup, Jeep Wagoneer [SJ] (1963-1981), Jeep Cherokee [SJ] (1974-1981), Jeep Forward Control [FC] (1957-1965), Jeep FJ Fleetvan (1961-1975)
| Located at 1000 Jeep Parkway. The John North Willys-owned Overland Automobile Co. purchased the plant in 1909 from Pope-Toledo, another early automaker. Overland Automobile Co. became Willys-Overland in 1912. Began building Jeeps in the 1940s. This was the original Jeep assembly plant. Willys-Overland was bought by Kaiser in 1953. Kaiser then sold its Willow Run plant in Ypsilanti, MI to GM and moved its production to the Willys plant in Toledo, OH. Kaiser and Willys production in the US ended in 1955 and Toledo focused on Jeep production going forward. Kaiser Jeep was sold to AMC in 1970. Became part of Chrysler in the 1987 buyout of AMC. Final assembly ended in 2001 when the XJ Cherokee ended production but painted body production continued until June 30, 2006, when the TJ Wrangler ended production. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant a few miles away.  The Administration Building, used from 1915 through 1974, was imploded on April 14, 1979. A third of the plant was demolished in 2002 after final assembly ended including the Jeep Museum. The remainder was demolished in 2006-2007 after body production ended. One of the three large brick smokestacks was preserved and was dedicated in 2013 to the plant's history and workforce. A bronze plaque was mounted next to the smokestack, which still says &quot;Overland&quot; on it. Over 11 million vehicles were produced at the site, including military Jeeps during World War II. The site was sold to the Toledo-Lucas County Port Authority in 2010. The site has been redeveloped into the Overland Industrial Park. Dana Inc. and Detroit Manufacturing Systems are among the tenants in the Overland Industrial Park and those plants supply the current Jeep plants elsewhere in Toledo. All-Phase Electric Supply Co. is another tenant. 
|-
| P (1989-2006),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Stickney|Toledo Assembly #2 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2006 
| [[w:Jeep Wagoneer (SJ)#1984: SJ and XJ|Jeep Grand Wagoneer (SJ)]]&lt;br&gt; (1984-1991),&lt;br&gt; [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1993-95), [[w:Jeep Wrangler (TJ)|Jeep Wrangler (TJ)]] (1997-06)
Models only made before Chrysler takeover:&lt;br&gt; Jeep Wagoneer [SJ] (1981-1983), Jeep Cherokee [SJ] (1981-1983)
| Located at 4000 Stickney Ave. Originally opened in 1942 by the Electric Auto-Lite Co., a maker of spark plugs. Sold to Kaiser-Jeep in 1964, which used it as a machining and engine plant until 1981, when AMC converted it for vehicle production. AMC had taken over Kaiser Jeep in 1970. AMC built the SJ Wagoneer and Cherokee at the Stickney Ave. plant. Body assembly was done at an SJ- or later,  Wrangler-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant. Became part of Chrysler in the 1987 buyout of AMC. Production ended in 2006 with the end of the TJ Wrangler. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant built on the site of the old Stickney Ave. plant. 
|-
| W (1994-1996)
| [[w:Toledo Complex#Stickney|Toledo Assembly #3 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1993
| 1996 
| [[w:Dodge Dakota#First generation (1987–1996)|Dodge Dakota]] (1994-1996)
| Body assembly was done at a Dakota-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant.
|-
|
| Toluca Engine Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2002
| [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Closed in 2002
|-
|
| Toluca Transmission Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2001
| Automatic Transmissions for fwd cars
| Closed in 2001
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine North Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 1952
| 2022
| [[w:Chrysler B engine|Chrysler B V8 engine]],&lt;br&gt; [[w:Chrysler B engine#RB engines|Chrysler RB V8 engine]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]],&lt;br&gt; [[w:Volkswagen EA827 engine#1.7|VW 1.7L EA827 I4 engine]] (adding Chrysler parts to already built VW engines made in W. Germany),&lt;br&gt; [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L/2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|1.8L, 2.0L I4 &quot;Neon engine&quot;]], [[w:Chrysler 3.3 &amp; 3.8 engines|3.3L/3.8L OHV V6]], [[w:Chrysler SOHC V6 engine|3.5L/3.2L/4.0L SOHC V6]], [[w:Chrysler Pentastar engine|3.2L/3.6L Pentastar V6 engine]], [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4]],&lt;br&gt; Engine components,&lt;br&gt; Air raid sirens
| Located at 2000 Van Horn Road. Trenton North began production in fall 1952 and was expanded in 1964, 1967, 1969, 1976, and 1977. At first, Trenton North began by making water pumps and air raid sirens but engines quickly followed. Trenton Engine North was Chrysler’s first dedicated engine factory in the US, separate from the assembly plants. On September 29, 1978, V8 production ended. Trenton North added Chrysler parts such as the intake and exhaust manifolds, water pump, ignition system and other major parts to already built VW 1.7L EA827 I4 engines imported from Salzgitter, W. Germany for use in the Omni/Horizon. Production of the 3.5L V6 moved to Kenosha Engine in 2003. Trenton North was idled in May 2011 when the 3.8 V6 ended production following the end of 3.3 &amp; 4.0 V6 engine production in 2010. Chrysler then announced in June 2011 it would use a fifth of the plant to make components for the Pentastar V6 being made at Trenton South. In January 2012, Trenton North began producing the 3.6L Pentastar V6 engine. Chrysler then installed a flexible production line that could build both the Pentastar V6 and the Tigershark I4. In May 2013, Trenton began producing the 3.2L Pentastar V6. Tigershark I4 production began late in 3rd quarter 2013. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production. Trenton North has been repurposed for warehousing and other non-manufacturing opportunities. 
|-
|
| [[w:Twinsburg Stamping|Twinsburg Stamping]]
| [[w:Twinsburg, Ohio|Twinsburg, Ohio]]
| [[w:United States|United States]]
| 1957
| 2010
| Stampings and assemblies
| Located at 2000 East Aurora Road. Opened in August 1957. Closed July 31, 2010. Sold in 2011. Demolished in 2012-2013. Now the Cornerstone Business Park.
|-
|
| Vernor Tool &amp; Die plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1983
| Tooling &amp; Dies
| Located at 12026 E Vernor Highway. Operations moved to Mount Elliott Tool and Die. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
|
| Vernor Trim plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1970's
| Trim
| Located at 12025 E Vernor Highway. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
| 4 (1960-1961),&lt;br&gt; 7 (1959)
| Warren Avenue Plant
| [[w:Dearborn, Michigan|Dearborn, Michigan]]
| [[w:United States|United States]]
| 1927,&lt;br&gt; 1950 (opened as part of Chrysler)
| 1960s
| DeSoto bodies (1950-1958), DeSoto engines &lt;br&gt; (1951-1958),&lt;br&gt; [[w:Imperial (automobile)#Second generation (1957–1966)|Imperial]] (1959-1961)
| Located at 8505 West Warren Avenue. This was previously the factory of Paige and Graham-Paige. Chrysler leased half the plant in 1941 to use for military production. Chrysler produced aircraft components for the [[w:Martin B-26 Marauder|B-26 Marauder]] (nose and center fuselage sections) and the [[w:Boeing B-29 Superfortress|B-29 Superfortress]] (the pressurized nose section, wing leading edges, and engine cowlings). The B-29 had a nose so large that trenches had to be dug in the floor and some of the bracing for the plant’s roof girders had to be removed to accommodate the aircraft. Chrysler bought the plant in 1947. Began building bodies for DeSoto in August 1950. Engine production began in 1951. Production of the Imperial brand moved here from the Jefferson Ave. plant in Detroit for 1959 in an attempt to give the Imperial brand its own, exclusive factory however sales weren't high enough to support its own plant so Imperial production moved back to the Jefferson Ave. plant in Detroit for 1962. Production of small parts followed for a few years as did export operations. The plant was later sold. Most of the plant has seen been demolished but the front building facing on Warren Ave. is still there and is now used as Corporate HQ by Shatila Food Products. The DeSoto logo, featuring a stylized image of Hernando de Soto, can still be seen at the top of the building above the front door. 
|-
| T (1970-), 2 (1966-1969),&lt;br&gt; 
| Warren Truck #2 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1966
| 1975
| Dodge heavy-duty trucks
| Located at 6600 East 9 Mile Road, at the corner of Sherwood Ave and East 9 Mile Road. Closed in 1975 when Dodge exited the heavy-duty truck market. Site now belongs to Sundance Beverage Co., the parent of Everfresh Juice Co.
|-
| V (1971-1979)
| Warren Truck (Compact) #3 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1970
| 1979
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1979),&lt;br&gt;[[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1974-1979)
| Located on Hoover Road between 8 Mile Road and 9 Mile Road. 
|-
| 
| Windsor Engine Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1938
| 1980
| [[w:Chrysler flathead engine#Straight-6|Chrysler flathead inline 6]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], &lt;br&gt;[[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Was originally Windsor Plant 2. Located just to the south of Windsor Plant 3, the current minivan factory. Closed in August 1980. Built over 8 million engines. Windsor Assembly Plant (Plant 3) expanded onto the site of the old engine plant when it was being renovated for minivan production. 
|- 
|
| [[w:Detroit Assembly#LaSalle Factory/DeSoto Factory|Wyoming Ave. Assembly (DeSoto Wyoming Ave. plant)]] / Wyoming Export Plant
| [[w:Detroit|Detroit]], [[w:Michigan|Michigan]]
| United States
| 1936
| 1958 (Vehicle prod.),&lt;br&gt; 1980 (export operations)
| [[w:DeSoto (automobile)|DeSoto]] (1937-1958),&lt;br&gt; CKD Export (1960-1980)
| Located at 6000 Wyoming Avenue. Originally built to produce Liberty aircraft engines in World War I, opening in 1917. In 1919, was taken over by Saxon Motor Co., owned by Hugh Chalmers of Chalmers Motor Co. GM bought the plant in 1926 and built the LaSalle there from 1927-1933. GM sold Wyoming Assembly to Chrysler in 1934, which then used it to build its DeSoto brand. Became DeSoto's home plant. During World War II, Chrysler built wing center sections for the [[w:Curtiss SB2C Helldiver|Curtiss SB2C Helldiver]] at the Wyoming Ave. plant. For 1959, DeSoto's entry level model, the Firesweep, was moved to the Dodge plant in Hamtramck and DeSoto's other, higher end models were moved to Chrysler's Jefferson Ave. plant in Detroit. This was done so that bodies and final assembly would be done in either a single facility or a pair of connected facilities. This was part of Chrysler's move to unibody construction for 1960 for all cars except Imperial. After the DeSoto brand was discontinued in late 1960, became Wyoming Export plant which was used to prepare vehicles for export. Plant closed in 1980. Plant was demolished in 1992. Site is now occupied by Comprehensive Logistics Inc.
|}

==Non-Chrysler FCA/Stellantis Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| [[w:Fiat Cassino Plant|Cassino Plant]]
| [[w:Piedimonte San Germano|Piedimonte San Germano]], [[w:Province of Frosinone|Province of Frosinone]]
| [[w:Italy|Italy]]
| 1972
| 
| Past Chrysler Group models: [[w:Lancia Delta#|Chrysler Delta]] (UK/Ireland)
| Fiat plant. 
|-
| 3
| [[w:Alfa Romeo Pomigliano d'Arco plant|Pomigliano d'Arco plant]] (Giambattista Vico plant)
| [[w:Pomigliano d'Arco|Pomigliano d'Arco]], [[w:Metropolitan City of Naples|Metropolitan City of Naples]]
| [[w:Italy|Italy]]
| 1972
|
| Past Chrysler Group models: [[w:Dodge Hornet|Dodge Hornet]] (2023-2025). Related models:&lt;br&gt; [[w:Alfa Romeo Tonale|Alfa Romeo Tonale]] (2023-)
| Originally, an Alfa Romeo plant. Oriiginally owned by Construction Industry Neapolitan Vehicles Alfa Romeo - Alfasud S.p.A., a joint venture between Alfa Romeo (88%), Finmeccanica (10%), and IRI (2%). In 1982, Alfasud S.p.A. was renamed Inca Investments. Alfa Romeo was taken over by Fiat in 1986. Fiat merged with Chrysler to form Fiat Chrysler Automobiles (FCA) in 2014. FCA merged with PSA Group to form Stellantis in 2021. 
|}

==Non-Chrysler Group DaimlerChrysler/Daimler AG Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 5
| Mercedes-Benz Plant Düsseldorf
| [[w:Düsseldorf|Düsseldorf]], [[w:North Rhine-Westphalia|North Rhine-Westphalia]]
| [[w:Germany|Germany]]
| 1962
|
| [[w:Dodge Sprinter|Dodge Sprinter]] (2003-2009)
| Mercedes-Benz Plant. 
|-
| 9
| Mercedes-Benz Plant Ludwigsfelde
| [[w:Ludwigsfelde|Ludwigsfelde]], [[w:Brandenburg|Brandenburg]]
| [[w:Germany|Germany]]
| 1991 (Mercedes prod. began)
|
| [[w:Dodge Sprinter#Second generation (2006–2018, NCV3)|Dodge Sprinter]] chassis cab (2007-2009)
| Mercedes-Benz Plant. Originally established in 1936 by Daimler-Benz to make airplane engines. The plant was bombed by the US in 1945. After the war ended, what remained of the factory was dismantled and taken to the Soviet Union as reparations. On February 1, 1991, Mercedes-Benz took a 25% stake in the Ludwigsfelde plant, which had previously belonged to East German truckmaker VEB Automobilwerke. It became a 100% owned subsidiary of Mercedes-Benz on January 1, 1994. Sprinter production began in 2006. 
|}

==Former partner factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Prod. for Chrysler began
! style=&quot;width:10px;&quot;|Prod. for Chrysler ended
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 6
| [[w:China Motor Corporation|China Motor Corporation]]
| [[w:Yangmei District|Yangmei District]], [[w:Taoyuan, Taiwan|Taoyuan]]
| [[w:Taiwan|Taiwan]]
| 2006
| 2007
| [[w:Chrysler Town &amp; Country (minivan)#Fourth generation (2001–2007)|Chrysler Town &amp; Country]]&lt;br&gt; (Taiwan: 2006-2007),&lt;br&gt; [[w:Dodge 1000|Dodge 1000]] (Mexico: 2007-'10)
| China Motor Corporation plant. Built for Chrysler under license by China Motor Corporation of Taiwan. Production began April 18, 2006. 
|-
|
| [[w:Carrozzeria Ghia|Carrozzeria Ghia]] 
| [[w:Turin|Turin]]
| [[w:Italy|Italy]]
| 1957
| 1965
| [[w:Imperial (automobile)#Imperial Crown (1955–1965)|Imperial Crown Limousine]] (1957-1965) modified, painted limousine bodies and interiors
| 132 Imperial Crown Limousines were built by Ghia under contract for Chrysler between 1957 and 1965. The 1957-1959 models were based on modified 2-door hardtops with the more rigid chassis from the convertible. The 1960-1965 models were based on 4-door models. Ghia lengthened the frame and modified the bodywork and interiors to create the limousines. After producion ended in 1965, Ghia sold the tooling to Barreiros of Spain, which built another 10 Imperial Crown Limousines. Barreiros had been 35% owned by Chrysler since 1963. That was increased to 77% in 1967 and 100% in 1969.  
|-
| U
| [[w:Hyundai Motor Company|Hyundai Motor Co.]] - [[w:List of Hyundai Motor Company manufacturing facilities#Ulsan Plant|Ulsan plant]]
| [[w:Ulsan|Ulsan]]
| [[w:South Korea|South Korea]]
| 2000
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge Atos|Dodge Atos]] (2001-2012),&lt;br&gt;  [[w:Dodge Verna|Dodge Verna]] (2004-06),&lt;br&gt;  [[w:Dodge Attitude#First generation (MC; 2006)|Dodge Attitude (MC)]] (2007-'11), [[w:Dodge Attitude#Second generation (RB; 2011)|Dodge Attitude (RB)]] (2012-'14), [[w:Dodge H100|Dodge H100 truck]],&lt;br&gt; [[w:Hyundai Starex#Second generation (TQ; 2007)|Dodge H100 Van/Wagon]]
| Rebadged Hyundai models sold as Dodges in Mexico. 
|-
| 
| [[w:Hyundai Motor India|Hyundai Motor India]]
| [[w:Chennai|Chennai]], [[w:Tamil Nadu|Tamil Nadu]]
| [[w:India|India]]
| 2011
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge i10|Dodge i10]] (2012-2014)
| Rebadged Hyundai model sold as a Dodge in Mexico. 
|-
| X
| [[w:Karmann|Karmann Osnabrück Assembly]]
| [[w:Osnabrück|Osnabrück]], [[w:Lower Saxony|Lower Saxony]]
| [[w:Germany|Germany]]
| 2003
| 2007
| [[w:Chrysler Crossfire|Chrysler Crossfire]] (2004-2008)
| Karmann plant. Built under contract for Chrysler. 
|-
| Y
| [[w:Magna Steyr|Magna Steyr]] / Steyr-Daimler-Puch - Chrysler Steyr Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1994
| 2010
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]]&lt;br&gt; (1995-2010),&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010), [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (2003-2007),&lt;br&gt; [[w:Chrysler 300#First generation (2005)|Chrysler 300C/300C Touring]] (2005-2010)
| Originally, a Steyr-Daimler-Puch plant. Magna International acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. Production of the Chrysler Voyager and Grand Voyager minivans moved from the Eurostar plant  next door to the main Magna Steyr plant for 2003. Chrysler minivan production in Austria ended on November 30, 2007. Built under contract for Chrysler. 
|-
| B
| [[w:Maserati|Maserati]] - [[w:Innocenti|Innocenti]] plant
| [[w:Lambrate|Lambrate district]], [[w:Milan|Milan]]
| [[w:Italy|Italy]]
| 1988
| 1990
| [[w:Chrysler TC by Maserati|Chrysler TC by Maserati]]&lt;br&gt; (1989-1991)
| Developed jointly by Chrysler and Maserati, the TC was built in Italy by Maserati at the Innocenti plant in Milan. Maserati and Innocenti were both owned by DeTomaso at the time. Chrysler bought a 5% stake in Maserati in 1984 and increased its stake to 15.6% in 1986. Production ended in 1990 due to low sales.
|-
| U
| Mitsubishi - Mizushima plant (Line 1)
| [[w:Kurashiki|Kurashiki]], [[w:Okayama Prefecture|Okayama Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| [[w:Plymouth Champ|Plymouth Champ]] (1981-1982), [[w:Plymouth Colt|Plymouth Colt]] (1983-1994), [[w:Dodge Colt|Dodge Colt]] (1981-1994), [[w:Dodge Colt|Dodge/Plymouth Colt]]&lt;br&gt; (Canada only: 1995),&lt;br&gt; [[w:Eagle Summit|Eagle Summit]]&lt;br&gt; (4-d: 1989-1990, 1993-1996,&lt;br&gt; 3-d: 1991-1992, 2-d: 1993-96), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1993-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1993-96)
| Mitsubishi Motors plant.
|-
| Z
| Mitsubishi - Okazaki plant
| [[w:Okazaki, Aichi|Okazaki]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1983
| 1996
| [[w:Plymouth Conquest|Plymouth Conquest]] (1984-86), [[w:Dodge Conquest|Dodge Conquest]] (1984-1986), [[w:Chrysler Conquest|Chrysler Conquest]] (1987-1989), [[w:Dodge Colt Vista#Colt Vista|Dodge Colt Vista]] (1984-1991), [[w:Plymouth Colt Vista#Colt Vista|Plymouth Colt Vista]] (1984-91), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1992-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1992-96) [[w:Eagle Vista#Vista Wagon|Eagle Vista Wagon]]&lt;br&gt; (Canada: 1989-1991)
| Mitsubishi Motors plant.
|-
| Y (Line 1)&lt;br&gt;/&lt;br /&gt;P (Line 2)
| Mitsubishi - &lt;br&gt; Ooe plant &lt;br&gt; a.k.a. &lt;br&gt; Nagoya #1&lt;br&gt;/&lt;br&gt;Nagoya #2
| Ooe-cho, [[w:Minato-ku, Nagoya|Minato ward]], [[w:Nagoya|Nagoya]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| VIN code Y:&lt;br&gt; [[w:Plymouth Sapporo|Plymouth Sapporo]] (1981-1983), [[w:Dodge Challenger#Second generation (1978–1983)|Dodge Challenger]] (1981-1983), [[w:Plymouth Arrow Truck#Chrysler variants|Plymouth Arrow Truck]] ('81-'82), [[w:Dodge Ram 50|Dodge Ram 50]] (1981-1984), [[w:Dodge Stealth|Dodge Stealth]] (1991-1996)
VIN code P:&lt;br&gt; [[w:Dodge Ram 50|Dodge Ram 50]] (1985-1986), [[w:Dodge Ram 50#North America|Dodge Ram 50]] (1987-1993)
| Mitsubishi Motors plant. Closed in 2001. Sold to Mitsubishi Heavy Industries and now used by its Aircraft, Defense &amp; Space Business Area.
|-
| J
| Mitsubishi -  Toyo Koki/Pajero Manufacturing Co., Ltd. plant
| [[w:Sakahogi, Gifu|Sakahogi]], [[w:Gifu Prefecture|Gifu Prefecture]] 
| [[w:Japan|Japan]]
| 1986
| 1989
| [[w:Dodge Raider|Dodge Raider]] (1987-1989)
|  Originally, a Toyo Koki Co. Ltd. plant. Opened in 1976. Built vehicles under contract for Mitsubishi. Mitsubishi Motors owned 35% of Toyo Koki and increased its stake to a majority in March 1995. The plant was then renamed Pajero Manufacturing Co., Ltd. in July 1995. In March 2003, Mitsubishi bought all the remaining shares in Pajero Manufacturing Co., Ltd., making it a wholly owned subsidiary. Closed in 2021. Sold to Daio Paper in 2022.
|-
| H (Attitude), 9 (1200)
| [[w:Mitsubishi Motors (Thailand)|Mitsubishi Motors Thailand]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 2014
| 2024
| [[w:Dodge Attitude#Third generation (A10; 2015)|Dodge Attitude]]&lt;br&gt; (Mexico: 2015-2024),&lt;br&gt; [[w:Mitsubishi Triton#Fifth generation (KJ/KK/KL; 2014)|Ram 1200]]&lt;br&gt; (Middle East: 2017-2019)
| Mitsubishi Motors plant.  
|-
| ?
| [[w:MMC Automotriz|MMC Automotriz]]
| [[w:Barcelona, Venezuela|Barcelona]], [[w:Anzoátegui|Anzoátegui state]]
| [[w:Venezuela|Venezuela]]
| 2002
| 2009
| [[w:Dodge Brisa|Dodge Brisa]]&lt;br&gt; ([[w:Hyundai Accent#First generation (X3; 1994)|2002-2005]]), ([[w:Hyundai Getz|2006-2009]])
| MMC Automotriz plant. Originally, MMC Automotriz was 49% owned by Consorcio Inversionista Fabril S.A. (CIF) of Venezuela and 42% owned by Nissho Iwai Corp. The remaining 9% of the company was owned by the Japan International Development Organization Ltd., a partnership between the government-financed Overseas Economic Cooperation Fund and 98 private companies. Nissho Iwai merged with Nichimen Corp. in 2004 to form Sojitz Corp. Sojitz later increased its stake in MMC Automotriz to 98%, with the other 2% still held by CIF. MMC Automotriz was sold to the the Sylca Group (also known as Yammine Group) in 2015. MMC Automotriz produced Mitsubishi vehicles and from 1996-2012, also produced Hyundai vehicles. The Dodge Brisa was produced for DaimlerChrysler as part of its cooperation with [[w:Hyundai Motor Company|Hyundai]]. MMC Automotriz also produced Mitsubishi Fuso trucks. 
|-
| G
| [[w:Mitsubishi Motors (Thailand)|MMC Sittipol Co., Ltd.]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 1988
| 1992
| [[w:Plymouth Colt#Fifth generation (1985–1988)|Plymouth Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Dodge Colt#Fifth generation (1985–1988)|Dodge Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Eagle Vista|Eagle Vista]] (Canada: 1988-92)
| Mitsubishi Motors plant. MMC Sittipol is the predecessor company of Mitsubishi Motors (Thailand). These were the first vehicle exports from Thailand. 
|-
| 2
| [[w:Renault|Renault]] - [[w:Maubeuge Construction Automobile|Maubeuge plant]]
| [[w:Maubeuge|Maubeuge]]
| [[w:France|France]]
| 1987
| 1989
| [[w:Renault Medallion|Renault Medallion]] (1988),&lt;br&gt; [[w:Eagle Medallion|Eagle Medallion]] (1989)
| Renault plant. The Medallion was sold through Chrysler's Jeep-Eagle dealer network as a legacy of Chrysler's takeover of AMC from Renault.
|-
| 8,&lt;br&gt; 0
| [[w:Soueast|Soueast]]
| [[w:Fuzhou|Fuzhou]], [[w:Fujian|Fujian province]]
| [[w:China|China]]
| 2008
| 2010
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager]],&lt;br&gt; [[w:Dodge Caravan#Fourth generation (2001–2007)|Dodge Caravan]]
| South East (Fujian) Motor Co., Ltd. plant. Built for Chrysler under license by South East (Fujian) Motor Co., Ltd.
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{{tmbox|type=notice|text=This is a sandbox page, a place for experimenting with Wikibooks.}}

This is a history of Chrysler factories that are being or have been used to produce cars, vans, SUVs, trucks, and automobile components. 

For '''Chrysler brand''' only: Plant code in 1955-1957 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number.

For '''cars''': Plant code in 1958 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number. Plant code was the number in the 4th position of the 10-digit serial number for 1959-1965. Plant code was the number in the 7th position of the 13-digit serial number for 1966-1967. Plant code was the letter in the 7th position of the 13-digit serial number for 1968-1980.

Canadian-built cars had the plant code as the number in the 5th position of the 11-digit serial number for 1965.

For '''trucks''': The first digit of the 7-digit sequence number (4th position overall of the 10-digit serial number) indicated which plant built the truck for 1967-1968. Plant code was the number in the 4th position of the 10-digit serial number for 1969. Plant code was the letter in the 7th position of the 13-digit serial number for 1970-1980.

Canadian-built models used a different system for VINs until 1968, when Chrysler of Canada adopted the same system as the US. Plant code for trucks was the number in the 5th position of the 10-digit serial number for 1961-1967.

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

For '''AMC passenger cars from 1958 through mid-1966''': Plant code is indicated by the 2nd letter of the serial number. If there is no 2nd letter, then it was made in Kenosha, WI. If the 2nd letter is a T, then it was made in Brampton, ON, Canada. If the 2nd letter is a K, then it was a knock-down export car. 

For '''AMC passenger cars from mid-1966 through 1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. 0-6 is Kenosha, WI and 7-9 is Brampton, ON, Canada.

For '''Canadian-built Jeeps built by AMC Canada for 1979-1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. It was made in Canada if it's either an 8 (1979) or a 7 (1980). All other Jeeps from 1980 and earlier were made in Toledo. 

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

==Current factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| D (1968-),&lt;br&gt; 4 (1966-1967)
| [[w:Belvidere Assembly|Belvidere Assembly]] 
| [[w:Belvidere, Illinois|Belvidere, Illinois]]
| [[w:United States|United States]]
| 1965
| Feb. 2023
| 
| Located at 3000 West Chrysler Drive. '''Belvidere Satellite Stamping Plant''' adjoins the main assembly plant. Began production on July 7, 1965. The first vehicle produced was a 1966 Plymouth Fury four-door. In 1972, the Chrysler Town and Country station wagon was added to the Belvidere plant. In 1977, the plant was converted to build front-wheel drive subcompacts. Production of the Dodge Omni and Plymouth Horizon began on December 5, 1977. All L-body derivatives were made at Belvidere through 1987. In 1987, Belvidere was converted to build Chrysler's midsize, fwd C-body sedans. Belvidere then switched back to building small cars and began production of the Neon on November 10, 1993. Belvidere built the last Plymouth, a silver 2001 Neon LX on June 28, 2001. Neon production ended in September 2005. Dodge Caliber began production in January 2006, followed by Jeep Compass in June 2006 and Jeep Patriot in December 2006. Caliber ended production on December 19, 2011. Dodge Dart began production on April 30, 2012, and ended on October 4, 2016. Compass and Patriot production ended on December 23, 2016. Jeep Cherokee production began on June 1, 2017 and ended on February 28, 2023. Belvidere was then idled. &lt;br&gt; Past models: [[w:Plymouth Fury|Plymouth Fury]] (1966-1974), [[w:Plymouth Gran Fury#1975–1977|Plymouth Gran Fury]] (1975-1977), [[w:Dodge Monaco|Dodge Monaco]] (1966-1976), [[w:Dodge Royal Monaco#1977 (Royal Monaco)|Dodge Royal Monaco]] (1977), [[w:Dodge Polara|Dodge Polara]] (1966-1973), [[w:Chrysler Newport|Chrysler Newport]] (1977), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1973-1977), [[w:Dodge Omni|Dodge Omni]] (1978-1987), [[w:Plymouth Horizon|Plymouth Horizon]] (1978-1987), [[w:Dodge Omni 024|Dodge Omni 024]] (1979-1980), [[w:Plymouth Horizon TC3|Plymouth Horizon TC3]] (1979-1980), [[w:Dodge Omni 024|Dodge 024]] (1981-1982), [[w:Plymouth Horizon TC3|Plymouth TC3]] (1981-1982), [[w:Dodge Charger (1981)|Dodge Charger]] (1983-1987), [[w:Plymouth Turismo|Plymouth Turismo]] (1983-1987), [[w:Dodge Rampage|Dodge Rampage]] (1982-1984),  [[w:Plymouth Scamp|Plymouth Scamp]] (1983), [[w:Dodge Dynasty|Dodge Dynasty]] (1988-1993), [[w:Chrysler Dynasty|Chrysler Dynasty]] (Canada: 1988-1993), [[w:Chrysler New Yorker#1988–1993|Chrysler New Yorker]] (1988-1993), [[w:Chrysler New Yorker Fifth Avenue#1990–1993: New Yorker Fifth Avenue|Chrysler New Yorker Fifth Avenue]] (1990-1993), [[w:Chrysler Imperial#1990–1993|Chrysler Imperial]] (1990-1993), [[w:Dodge Neon|Dodge Neon]] (1995-2005), [[w:Plymouth Neon|Plymouth Neon]] (1995-2001), Chrysler Neon (Canada: 2000-02),&lt;br&gt; Dodge SX 2.0 (Canada: 2003-05),&lt;br&gt; [[w:Dodge Caliber|Dodge Caliber]] (2007-2012), [[w:Jeep Compass#First generation (MK49; 2006)|Jeep Compass]] (2007-2017), [[w:Jeep Patriot|Jeep Patriot]] (2007-2017), [[w:Dodge Dart (PF)|Dodge Dart]] (2013-2016), [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2017-2023)
|-
| H (1989-),&lt;br&gt; A (1988)
| [[w:Brampton Assembly|Brampton Assembly]] (Formerly Bramalea Assembly)
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1987
| Dec. 2023
| 
| Located at 2000 Williams Parkway East. Factory was built by AMC and Renault. The plant was acquired by Chrysler as part of its takeover of AMC. Began production on September 28, 1987. Plant was originally known as Bramalea Assembly. Plant was renamed Brampton Assembly in 1992 after Chrysler closed and sold the old AMC plant on Kennedy Road in Brampton. The attached '''Brampton Satellite Stamping Plant''' was added in December 1991 and was built for the launch of the Chrysler LH platform. On December 17, 1991, Eagle Premier and Dodge Monaco production ended. Production of the Chrysler LH platform cars began in June 1992. Production switched to the rear-wheel drive Chrysler LX platform cars in January 2004. The Chrysler 300 was also built for export to mainland Europe as the Lancia Thema from 2011-2014. Production ended on December 22, 2023 and the plant was idled. Last vehicle off the line was a Pitch-Black 2023 Dodge Challenger Demon 170. 7,147,888 vehicles were produced through 2023.&lt;br&gt; Past models: [[w:Eagle Premier|Eagle Premier]] (1988-1992), [[w:Dodge Monaco#Fifth generation (1990–1992)|Dodge Monaco]] (1990-1992),&lt;br&gt; [[w:Dodge Intrepid|Dodge Intrepid]] (1993-2004),&lt;br&gt; [[w:Chrysler Intrepid|Chrysler Intrepid]] (Canada: 1993-2004),&lt;br&gt; [[w:Chrysler Concorde|Chrysler Concorde]] (1993-2004),&lt;br&gt; [[w:Eagle Vision|Eagle Vision]] (1993-1997),&lt;br&gt; [[w:Chrysler New Yorker#1994–1996|Chrysler New Yorker]] (1994-1996),&lt;br&gt; [[w:Chrysler LHS|Chrysler LHS]] (1994-1997, 1999-2001),&lt;br&gt; [[w:Chrysler 300M|Chrysler 300M]] (1999-2004),&lt;br&gt; [[w:Chrysler 300|Chrysler 300]] (2005-2023),&lt;br&gt; [[w:Dodge Magnum#Chrysler LX platform (2005–2008)|Dodge Magnum]] (2005-2008),&lt;br&gt; [[w:Dodge Charger (2006)|Dodge Charger]] (2006-2023),&lt;br&gt; [[w:Dodge Challenger (2008)|Dodge Challenger]] (2008-2023),&lt;br&gt; [[w:Lancia Thema#Second generation (2011–2014)|Lancia Thema]] (For export: 2011-2014)
|-
| C
| [[w:Jefferson North Assembly|Detroit Assembly Complex – Jefferson]] / Jefferson North Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]] (1993-), [[w:Dodge Durango#WD|Dodge Durango]] (2011-)
| Located at 2101 Conner Street. Jefferson North replaced the previous Jefferson Assembly plant that closed in 1990 and was demolished in 1991. Jefferson North is across the street from the old Jefferson Assembly plant, on the north side of Jefferson Ave. Jefferson North was built on the site of Chrysler's old Kercheval Avenue Body Plant, which, in 1955, had been connected to the old Jefferson Assembly plant by a bridge crossing over Jefferson Ave. The Jefferson North plant site also absorbed what had been the axle plant and service parts buildings of the old [[w:Hudson Motor Car Company|Hudson]] plant, which were located at Connor St. and Vernor Hwy. The main Hudson plant is now the parking lot on the corner of Jefferson Ave. and Connor St. After 2017, the Jefferson North plant complex also absorbed the site of the former Budd Co. body- &amp; parts-making plant at Connor St. and Charlevoix Avenue, which had previously belonged to the Liberty Motor Car Co. The Budd plant extended north on Connor from Charlevoix most of the way to Mack Ave. Since the nearby Mack Ave. Assembly Plant began operations in 2021, the 2 plants have operated as the Detroit Assembly Complex. Jefferson North began production on January 14, 1992 with the original Grand Cherokee (ZJ). The 2nd gen. Grand Cherokee began production on July 17, 1998. The 3rd gen. Grand Cherokee began production on July 26, 2004 followed by the Jeep Commander on July 18, 2005. The 4th gen. Grand Cherokee began production on May 10, 2010 followed by the Dodge Durango on December 14, 2010. The 5th gen. Grand Cherokee began production in May 2022. The 4xe plug-in hybrid version of the Grand Cherokee began production at Jefferson North in March 2023. On Aug. 13, 2013, Jefferson North built its 5 millionth vehicle, a silver 2014 Grand Cherokee Overland. On May 25, 2016, Jefferson North built its 6 millionth vehicle, a Granite Crystal (silvery gray) 2016 Grand Cherokee 75th anniversary Edition.&lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010),&lt;br&gt; [[w:Jeep Grand Cherokee (WK2)#Grand Cherokee WK (2022)|Jeep Grand Cherokee WK]] (2022)&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2023-2025)
|-
| 8
| [[w:Detroit Assembly Complex – Mack|Detroit Assembly Complex – Mack]] / Mack Ave. Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2021
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2021-), [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]] (2022-)
| Located at 4000 St. Jean Avenue. The Mack Avenue Assembly Plant is built on the site of the former Mack Ave. Engine Plants I &amp; II, the New Mack Assembly Plant, &amp; the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The two plants that comprised the former Mack Avenue Engine Complex were converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North plants have operated as the Detroit Assembly Complex since 2021. Production began in March 2021 with the 3-row Grand Cherokee L. The 2-row Grand Cherokee followed in the fall of 2021. The 4xe plug-in hybrid version of the Grand Cherokee began production at Mack Ave. in August 2022. &lt;br&gt; Past models: [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2022-2025)
|-
|
| [[w:Dundee Engine Plant|Dundee Engine Plant]] (Formerly [[W:Global Engine Manufacturing Alliance|GEMA]]) 
| [[w:Dundee, Michigan|Dundee, Michigan]]
| [[w:United States|United States]]
| 2005
| 
| [[w:Prince engine#1.6-litre turbocharged (PSA)|1.6L turbo PSA/BMW Prince EP6CDTX Hybrid I4]],&lt;br&gt; [[w:FCA Global Medium Engine|2.0L turbo Hurricane4 EVO I4 (GME-T4 EVO)]],&lt;br&gt; Engine components
| Located at 5800 North Ann Arbor Road. Plant was originally part of the the Global Engine Manufacturing Alliance, a 3-way engine manufacturing joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. The North Plant launched in October 2005, followed by the South Plant in November 2006. The plant began with production of the I4 World Gasoline Engine, which was developed by the Global Engine Alliance, a 3-way engine development joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. Originally, the plant was envisioned as supplying engines to Mitsubishi and Hyundai as well as Chrysler however the plant only ever supplied engines to Chrysler. Mitsubishi and Hyundai each set up engine production at their own engine plants. On August 31, 2009, Chrysler bought Mitsubishi’s and Hyundai’s stakes in the group and now wholly owns both the Global Engine Manufacturing Alliance and its primary engine-building plant in Dundee, Michigan. In January 2012, the plant was renamed Dundee Engine Plant. Production of the Fiat 1.4-liter FIRE I4 engine began in November 2010. The Tigershark engine, an evolution of the World engine, began production in 2012 for the 2.0L and in May 2013 for the 2.4L. Tigershark engine production ended on March 16, 2023. In November 2019, the Pentastar V6 began production at Dundee, moving from the Mack Ave. Engine Plant in Detroit, which was to be converted into a vehicle assembly plant. The Pentastar V6 ended production at Dundee on August 18, 2023. In 2025, Dundee began making a North American-spec version of the European-developed Prince engine for the 2026 Jeep Cherokee Hybrid. &lt;br&gt; Past engines: [[w:World Gasoline Engine|1.8L/2.0L/2.4L/2.4L Turbo I4 World Gasoline Engine]], [[w:World Gasoline Engine#Tigershark|2.0L/2.4L I4 Tigershark Engine]], [[w:FIRE engine|Fiat 1.4L/1.4L Turbo FIRE MultiAir I4 engine]], [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]] 
|-
| 
| Etobicoke Casting Plant
| [[w:Etobicoke|Etobicoke District]], [[w:Toronto|Toronto]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1964
| 
| Aluminum die castings, Engine and Transmission Components
| Located at 15 Brown's Line. Etobicoke used to be a separate city but became part of Toronto in 1998. Factory was originally built in 1942 by the Canadian government and operated by Alcan Aluminum, which used it to make molds for military aircraft parts during World War II. It produced precision aircraft parts and other high quality aluminum castings. The aluminum foundry was purchased by Chrysler in April 1964 from Alcan Aluminum. The plant was expanded in 1965 and 1998. Etobicoke Casting is making oil pans for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee.
|-
|
| [[w:Indiana Transmission#Indiana Transmission I|Indiana Transmission Plant I]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1998
| 
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, Gear machining and final assembly of electric drive modules
| Located at 3660 North U.S. Highway 931. RFE transmission production ended in January 2025. More than 8 million RFE transmissions were produced at Indiana Transmission Plant I. Production of the nine-speed transmission began in May 2013. The 9-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. &lt;br&gt; Past transmissions:&lt;br&gt; [[w:Chrysler RFE transmission|Chrysler RFE 4-/5-/6-speed auto. trans.]]
|-
|
| [[w:Kokomo Casting|Kokomo Casting Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1965
| 
| Aluminum parts for automotive components, transmission and transaxle cases; engine block castings
| Located at 1001 East Boulevard. Kokomo Casting is the world’s largest die-cast facility. Plant was expanded in 1969, 1986, 1995 and 1997. Over 18 million four-speed transmission cases were made at Kokomo Casting from 1988 through July 2014. Kokomo Casting started making nine-speed transmission cases in 2013. Kokomo Casting is making engine blocks for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee. Kokomo Casting is making gearbox covers for the electric drive modules made at the Indiana Transmission Plant. 
|-
|
| [[w:Kokomo Engine Plant|Kokomo Engine Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2022 (as Kokomo Engine)
| 
| [[w:FCA Global Medium Engine|2.0L turbo GME-T4 I4]]
| Located at 3360 North U.S. Highway 931. Plant was previously known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Starting in 2020, the plant was converted to engine production. Engine production began in late February 2022. 
|-
|
| [[w:Kokomo Transmission|Kokomo Transmission Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1956
| 
| [[w:ZF 8HP transmission|850RE]] 8-speed auto. transmission,&lt;br&gt; [[w:ZF 8HP transmission|880RE]] 8-speed auto. transmission,&lt;br&gt; Machining of engine block castings and transmission components,&lt;br&gt; Machined components for the &lt;br&gt; 9-speed auto. transmission
| Located at 2401 South Reed Road. On October 9, 2020, Kokomo Transmission built its last 41TE 4-speed auto. transmission, assembling more than 17 million 4-speed auto. transmissions since production began in 1988. Kokomo Transmission began building 6-speed auto. transmissions in 2006. Production of the eight-speed automatic transmission began in September 2012. The 8-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. On August 8, 2023, Kokomo Transmission built its 6 millionth 8-speed transmission. Kokomo Transmission is machining gearbox covers for the electric drive modules made at the Indiana Transmission Plant. &lt;br&gt; Past transmissions: &lt;br&gt;[[w:TorqueFlite|TorqueFlite 3-/4-speed auto. trans.]],&lt;br&gt; [[w:Ultradrive|Ultradrive (TE/AE/LE/RLE/TES/TEA)&lt;br&gt; 4-/6-speed auto. trans.]],&lt;br&gt; [[w:ZF 8HP transmission|845RE]] 8-speed auto. transmission,&lt;br&gt; SI-EVT trans. (eFlite) for Pacifica Hybrid
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo North Engine Plant]] 
| [[w:Ramos Arizpe|Ramos Arizpe]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1981
| 
| [[w:Chrysler Hemi engine#Third generation: 2003–present|5.7L/6.4L/6.2L supercharged Hemi V8]], [[w:Stellantis Hurricane engine|Chrysler 3.0L twin-turbo Hurricane GME-T6 I6]]
|  Production began on May 8, 1981. Hemi V8 production began in June 2002 with the 5.7L. Production of the supercharged 6.2L Hellcat Hemi V8 began in the third quarter of 2014. Tigershark I4 production began in the first quarter of 2014. &lt;br&gt; Past engines: [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L, 2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|2.0L, 2.4L, 2.4L Turbo I4 &quot;Neon engine&quot;]],&lt;br&gt; [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4 engine]], [[w:Chrysler Hemi engine#6.1|6.1L Hemi V8]]
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo South Engine Plant]] 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]]
| Plant opened on October 29, 2010. The plant has produced over 6 million Pentastar V6 engines. &lt;br&gt; Past engines: Chrysler 3.6L Pentastar V6 engine (EH3) for Pacifica Plug-in Hybrid
|-
| 
| Saltillo Stamping Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1997
| 
| Stampings and assemblies including Body panels
| Part of the Saltillo Truck Assembly Complex.
|-
| G
| Saltillo Truck Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1995
| 
| [[w:Ram Heavy Duty (fifth generation)|Ram HD pickup &amp; chassis cab]] (2019-)
|  Production began in 1995. As of 2025, also known as Saltillo Truck Heavy Duty Plant. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1995-2012),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-2023),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram HD pickup &amp; chassis cab]] (2013-2018), [[w:Sterling Bullet|Sterling Truck Bullet]] (2008-2009)
|-
| 4
| Saltillo Truck Extension Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2025
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2025-)
| Also known as Saltillo Truck Light Duty Plant. 2 new buildings were constructed for the new light duty plant. Production began in May 2025. Initially, production was for export but production for the domestic Mexican market began in February 2026. The plant also includes a seat assembly line, the first Stellantis plant in North America to integrate this process into its own production chain.
|-
| E
| Saltillo Van Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2013
| 
| [[w:Ram ProMaster|Ram ProMaster]] (2014-),&lt;br&gt; [[w:Ram ProMaster#E-Ducato and Ram ProMaster EV (2024)|Ram ProMaster EV]] (2024-)
| Production started in July 2013. &lt;br&gt; Past models: [[w:Fiat Ducato|Fiat Ducato]] (Export to Brazil/Argentina: 2018-2022)
|-
| N
| [[w:Sterling Heights Assembly|Sterling Heights Assembly Plant]] 
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1984
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2019-)
| Located at 38111 Van Dyke Ave. The plant was originally built by the US Navy as a jet engine plant in 1953. It was called Naval Industrial Reserve Aircraft Plant and was owned by the US Navy. When the jet engine project was cancelled, the plant was transferred to the US Army, which contracted with Chrysler to build missiles at the plant, which was now known as the Michigan Ordinance Missile Plant. Chrysler began production of the PGM-11 Redstone missile at the Sterling Heights plant on September 27, 1954. The final Redstone was built in 1961. Chrysler also built the PGM-19 Jupiter missile at Sterling Heights from 1958-December 1960. Chrysler also built the first stage of the Saturn I rocket at the Sterling Heights plant. Chrysler vacated the Michigan Army Missile Plant at the end of 1969. Meanwhile, LTV Corp. (previously Ling-Temco-Vought) built the MGM-52 Lance missile at the Sterling Heights plant. The Army turned the plant over to the state of Michigan, which was then sold it to Volkswagen in 1980. VW converted the plant to automotive production and intended to make the Jetta there but VW's US sales declined and VW never ended up building anything there. VW then sold the plant to Chrysler in 1983. Chrysler LeBaron GTS and Dodge Lancer production began in September 1984 and ended on April 7, 1989. Shadow and Sundance production began on August 25, 1986 and ended on March 9, 1994. The Dodge Daytona was also moved from St. Louis to Sterling Heights in 1991 and was produced there through February 26, 1993. Chrysler then built a succession of midsize cars at Sterling Heights from June 1994. During Chrysler's bankruptcy in 2009, Sterling Heights Assembly was initially left behind in &quot;old Chrysler&quot; and was supposed to close by December 2010 but during 2010, &quot;new Chrysler&quot; changed its mind and bought the plant from &quot;old Chrysler&quot; for $20 million. The 2011 Chrysler 200 and Dodge Avenger sedans began production on December 6, 2010 followed by the Chrysler 200 Convertible in February 2011. The Chrysler 200 Convertible was also built for export to Europe as the Lancia Flavia from March 2012. The 2nd gen. Chrysler 200 sedan began production on March 14, 2014 and ended on December 2, 2016. The plant was then idled for a lengthy retooling to build body-on-frame pickups and began production of the new generation DT-series Ram 1500 in March 2018. &lt;br&gt; Past models: [[w:Dodge Lancer#1985–1989: Lancer|Dodge Lancer]] (1985-1989), [[w:Chrysler LeBaron#1985–1989 LeBaron GTS|Chrysler LeBaron GTS (H-body)]] (1985-1989), [[w:Plymouth Sundance|Plymouth Sundance]] (1987-1994), [[w:Dodge Shadow|Dodge Shadow]] (1987-1994), [[w:Dodge Daytona|Dodge Daytona]] (1992-1993), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1992-1993), [[w:Chrysler Cirrus|Chrysler Cirrus]] (1995-2000), [[w:Dodge Stratus|Dodge Stratus]] sedan (1995-2006), [[w:Plymouth Breeze|Plymouth Breeze]] (1996-2000), [[w:Chrysler Sebring|Chrysler Sebring]] sedan (2001-2010), [[w:Chrysler Sebring|Chrysler Sebring]]  convertible (2001-2006, 2008-2010), [[w:Dodge Avenger#Dodge Avenger sedan (2008–2014)|Dodge Avenger]] sedan (2008-2014), [[w:Chrysler 200|Chrysler 200]] sedan (2011-2017), [[w:Chrysler 200|Chrysler 200]] convertible (2011-2014), [[w:Chrysler 200#Lancia Flavia|Lancia Flavia]] (For export: 2012-2014)  
|-
| 
| Sterling Stamping Plant
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1965
| 
| Stampings and assemblies including hoods, roofs, liftgates, side apertures, fenders, and floorpans
| Located at 35777 Van Dyke Ave. This plant is a separate facility but is located next door to the Sterling Heights Assembly Plant. Sterling Stamping is the largest stamping plant in the world. Sterling Stamping supplies stampings to many Chrysler assembly plants, not only Sterling Heights Assembly. The first stampings were produced in January 1965. 
|-
| W
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo North Assembly Plant]] 
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]] (2018-)
| Located at 4400 Chrysler Drive. Toledo North first built the Jeep Liberty, which began in April 2001. Dodge Nitro production began in August 2006 and ended on December 16, 2011. The 2nd gen. Jeep Liberty began production in July 2007. Jeep Liberty ended production on August 16, 2012. Toledo North then closed for retooling to build the all-new 2014 Cherokee. The Jeep Cherokee began production at Toledo North on June 24, 2013 and ended on April 6, 2017. The Cherokee was then moved to Belvidere Assembly. Toledo North was then retooled to build the Jeep Wrangler, which began on November 15th, 2017. The 4xe plug-in hybrid version of the Wrangler began production in December 2020. &lt;br&gt; Past models: [[w:Jeep Liberty|Jeep Liberty]] (2002-2012), [[w:Dodge Nitro|Dodge Nitro]] (2007-2011),&lt;br&gt; [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2014-2017),&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler 4xe (JL)]] (2021-2025)
|-
| L
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo Supplier Park Plant]] (South plant)
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Gladiator (JT)|Jeep Gladiator]] (2020-)
| Located along Stickney Ave. The Toledo Supplier Park Plant is built on the site of the former Stickney Ave. plant that became part of Chrysler in 1987 as part of the acquisition of AMC. That plant was acquired by Kaiser Jeep in 1964 from Autolite and was built in 1942. The Toledo Supplier Park Plant includes body and chassis operations in partnership with Kuka and Hyundai Mobis, respectively. The paint shop was originally run in partnership with Magna but Chrysler took over the paint operation in the first quarter of 2011. The Toledo Supplier Park Plant began Wrangler production in August 2006. Wrangler production ended on April 27, 2018. Gladiator pickup production began in March 2019. &lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Wrangler (JK)|Jeep Wrangler (JK)]] (2007-2017),&lt;br&gt; [[w:Jeep Wrangler (JK)#2018 model year update|Jeep Wrangler JK]] (2018)
|-
| 
| [[w:Toledo Machining|Toledo Machining Plant]] 
| [[w:Perrysburg, Ohio|Perrysburg, Ohio]]
| [[w:United States|United States]]
| 1966
| 
| Steering Columns,&lt;br&gt; Torque Converters for 8-spd. rwd and 9-spd. fwd auto. transmissions
| Located at 8000 Chrysler Drive. Production began in 1966 and the plant was expanded in 1969. &lt;br&gt; Past products: Power Electronics module for Wrangler 4xe PHEV
|-
| T,&lt;br&gt; V (1985 only)
| [[w:Toluca Car Assembly|Toluca Car Assembly]] 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1968
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (2017-), [[w:Jeep Wagoneer S|Jeep Wagoneer S EV]] (2024-),&lt;br&gt; [[w:Jeep Cherokee (KM)|Jeep Cherokee (KM)]] (2026-), [[w:Jeep Recon|Jeep Recon EV]] (2026-)
| Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Began production on December 9, 1968. An adjacent supplier park was opened in 2007. Journey production began in early 2008. PT Cruiser production ended on July 9, 2010. Fiat 500 production began in December 2010. Journey production ended in December 2020. Jeep Compass production began on January 16, 2017. &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Dart (rwd), Dodge Dart K, Dodge Dart E, Chrysler Valiant Volaré (rwd), Chrysler Valiant Volaré K, Chrysler Valiant Volaré E, [[w:Dodge Magnum#First generation|Dodge Magnum]] [rwd] (1981-1982), [[w:Dodge Magnum#Second generation|Dodge Magnum 400/Magnum]] [fwd] (1983-1988), [[w:Chrysler Phantom|Chrysler Phantom]], [[w:Chrysler Shadow|Chrysler Shadow]], [[w:Chrysler Spirit|Chrysler Spirit]] &lt;br&gt; Export to US: [[w:Dodge Aries|Dodge Aries]] (1984-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1984-1989), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe]] (1987-1989, 1992), [[w:Dodge Shadow|Dodge Shadow]] (1988, 1991-1994), [[w:Plymouth Sundance|Plymouth Sundance]] (1988, 1992-1994), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron sedan]] (1990-1994), [[w:Dodge Spirit|Dodge Spirit]] (1991-1995), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1991-1995), [[w:Dodge Neon#First generation (1994)|Dodge Neon]] (1995-1999), [[w:Plymouth Neon#First generation (1994)|Plymouth Neon]] (1995-1999), [[w:Chrysler Sebring#Convertible (1996–2000)|Chrysler Sebring Convertible]] (1996-2000), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2001-10), [[w:Dodge Journey|Dodge Journey]] (2009-2020),&lt;br&gt; [[w:Fiat 500 (2007)|Fiat 500]] (2012-2019), [[w:Fiat 500 (2007)#Fiat 500e (2013)|Fiat 500e]] (2013-2019)&lt;br&gt; Export to Brazil/Europe/Australia:&lt;br&gt; [[w:Fiat Freemont|Fiat Freemont]] (2011-2016)
|-
| 
| Toluca Stamping Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1994
| 
| Stampings and assemblies including Body panels
| Part of the Toluca Assembly Complex.
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine South Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler Pentastar V6 engine]]
| Located at 2300 Van Horn Road. Production began in March 2010 with the 3.6L Pentastar V6. In 2022, Trenton South was upgraded with a flexible engine line that could build both the Classic and Upgrade versions of the 3.6L Pentastar V6. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production.
|-
| 
| Warren Stamping Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1949
| 
| Stampings and assemblies including roofs, tailgates, side apertures, fenders, and floorpans
| Located at 22800 Mound Road. This plant is a separate facility but is located next door to the Warren Truck Assembly Plant. The plant was expanded in 1952, 1964, 1965, and 1986. Warren Stamping supplies stampings to many Chrysler assembly plants, not only Warren Truck Assembly.
|-
| S (1970-), 1 (1967-1969),&lt;br&gt; 2 (For A-series)
| Warren Truck Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1938
| 
| [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer]] (2022-), [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer L]] (2023-)
| Located at 21500 Mound Road. Formerly known as the Dodge City Truck Plant. Also referred to as Warren Truck #1 in the 1970's when there were 2 other truck plants nearby. Began production in October 1938. The Mitsubishi Raider began production in September 2005 and ended on June 11, 2009. Dodge Dakota production ended on August 23, 2011. Full-size Jeep SUV production began in 2021. Ram 1500 Classic production ended in October 2024, ending production of Dodge/Ram trucks at Warren after 86 years. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge T-, V-, W-Series|Dodge T-, V-, W-Series]] (1939-1947), Dodge military trucks, [[w:Dodge B series#Pickup truck|Dodge B series]] (1948-1953), [[w:Dodge C series|Dodge C series]] (1954-1960),&lt;br&gt; [[w:Dodge Town Panel and Town Wagon|Dodge Town Panel/Town Wagon]] (1954-66), [[w:Dodge D series|Dodge D/W series]] (1961-1980), [[w:Dodge Ram|Dodge Ram pickup]] (1981-2012), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-24), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1977-1978, 1981-1985), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1977-1978, 1981), [[w:Dodge M-series chassis|Dodge M-series chassis]] (1968-1979),&lt;br&gt; [[w:Dodge A100|Dodge A100/A108]] (1964-1970),&lt;br&gt; [[w:Dodge Dakota|Dodge Dakota]] (1987-2011),&lt;br&gt; [[w:Mitsubishi Raider|Mitsubishi Raider]] (2006-2009),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer]] (2022-2025),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer L]] (2023-2025),&lt;br&gt; [[w:Fargo Trucks|Fargo Trucks]] (For export)
|-
| R (1968-),&lt;br&gt; 9 (1959-1967)
| [[w:Windsor Assembly|Windsor Assembly]] 
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1929
| 
| [[w:Chrysler Pacifica (minivan)|Chrysler Pacifica (minivan)]] (2017-), [[w:Chrysler Voyager#Sixth generation (2020–present)|Chrysler Voyager]] (2020-2026), Chrysler Grand Caravan (Canada: 2021-),&lt;br&gt; [[w:Dodge Charger (2024)|Dodge Charger]] (2024-)
| Located at 2199 Chrysler Centre. Today's Windsor Assembly Plant was originally Windsor Plant 3 or Windsor Car Assembly Plant. Before the 1965 US-Canada Auto Pact, Windsor Assembly made most of the cars sold by Chrysler Canada, including some unique-to-Canada variations. On June 10, 1983, Windsor ended rwd car production and was converted to build fwd minivans. Windsor began building minivans on October 7, 1983. For the first 2 generations of Chrysler minivans, Windsor only built the SWB models. For the 3rd generation, Windsor built both SWB and LWB models. For the 4th generation, Windsor only built LWB models. The minivan-based Pacifica SUV began production in January 2003 and  ended production on November 23, 2007. Production of the VW Routan began in August 2008. Production of the Ram C/V began on August 31, 2011, and ended in early 2015. Pacifica minivan production began on February 29, 2016 followed by the PHEV version on December 1, 2016. Town &amp; Country production ended on March 21, 2016 while Dodge Grand Caravan production ended on August 21, 2020. Production of the electric Charger Daytona began in December 2024 followed by the gas-powered Charger Sixpack in December 2025. &lt;br&gt; Past models: [[w:Chrysler Imperial|Chrysler Imperial]] (1929-1937) [https://www.web.imperialclub.info/registry/vin_decode.htm#1931-54], [[w:Dodge Kingsway|Dodge Kingsway]] (Canada: 1940-41, 1951-52), [[w:Dodge Regent|Dodge Regent]] (Canada: 1951-1959), [[w:Dodge Crusader|Dodge Crusader]] (Canada: 1951-1958), [[w:Dodge Mayfair|Dodge Mayfair]] (Canada: 1953-1959), [[w:Dodge Viscount|Dodge Viscount]] (Canada: 1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1961), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:Chrysler Windsor|Chrysler Windsor]] (1957-1966), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1963), [[w:Chrysler 300 non-letter series|Chrysler Saratoga 300]] (1964-1965), [[w:Chrysler Newport#1961–1964|Chrysler Newport]] (1961-1963), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1963-64, 1966), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Plymouth Fury|Plymouth Fury]] (1959-1970), [[w:Dodge Polara|Dodge Polara]] (1960, 1964-1969), Dodge 220 (Canada: 1963), [[w:Dodge 330|Dodge 330]] (1963-1965), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Monaco|Dodge Monaco]] (1967-1968), [[w:Plymouth Valiant#Canada (1960–1966)|Valiant]] (Canada: 1960-1966), [[w:Plymouth Barracuda#First generation (1964–1966)|Valiant Barracuda]] (Canada: 1964-1965), [[w:Plymouth Valiant|Plymouth Valiant]] (1970-1974), [[w:Plymouth Duster|Plymouth Duster]] (1970), [[w:Dodge Dart|Dodge Dart (compact)]] (1966, 1970-1975), [[w:Plymouth Satellite#Third generation (1971–1974)|Plymouth Satellite]] (1972-1974),  [[w:Plymouth GTX|Plymouth GTX]] (1971), [[w:Plymouth Road Runner#Second generation (1971–1974)|Plymouth Road Runner]] (1971-1974), [[w:Dodge Charger (1966)#Fourth generation|Dodge Charger]] (1975-1978), [[w:Dodge Magnum#US and Canada (1978–1979)|Dodge Magnum]] (1978-1979), [[w:Chrysler Cordoba|Chrysler Cordoba]] (1975-1983), [[w:Dodge Mirada|Dodge Mirada]] (1980-1983), [[w:Imperial (automobile)#Sixth generation (1981–1983)|Imperial]] (1981-1983), [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979), [[w:Dodge Diplomat|Dodge Diplomat]] (1981-1983), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1982-83), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle]] (Canada: 1981-1982), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1983), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron]] (1981), [[w:Chrysler New Yorker#1982|Chrysler New Yorker]] (1982), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler New Yorker Fifth Avenue]] (1983), [[w:Plymouth Voyager|Plymouth Voyager]] (1984-'00), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1984-2000), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2020), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (2001-2016), [[w:Chrysler Voyager|Chrysler Voyager]] (2000), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler minivans (S)#Cargo van|Dodge Mini Ram Van]] (1984-1988), [[w:Chrysler minivans (RT)#2011 revision|Ram C/V]] (2012-2015), [[w:Volkswagen Routan|Volkswagen Routan]] (2009-14), [[w:Chrysler Voyager#Lancia Voyager|Lancia Voyager]] (For export: 2012-2015), [[w:Chrysler Pacifica (crossover)|Chrysler Pacifica (SUV)]] (2004-2008)
|-
| 
| CpK Interior Products, Inc. - Belleville Operations
| [[w:Belleville, Ontario|Belleville]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 134 River Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. - Guelph Operations
| [[w:Guelph|Guelph]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 500 Laird Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. -&lt;br&gt; Port Hope Operations
| [[w:Port Hope, Ontario|Port Hope]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 128 Peter St. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 4
| Arab American Vehicles Company (AAV)
| [[w:Cairo|Cairo]]
| [[w:Egypt|Egypt]]
| 1978 (production began) &lt;br&gt; 1987 (became part of Chrysler)
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2025-), [[w:Citroën C4#Third generation (C41; 2020)|Citroën C4X]] (2025-)
| Arab American Vehicles Company is a joint venture between the [[w:Arab Organization for Industrialization|Arab Organization for Industrialization]], which holds 51%, and Stellantis, which holds the other 49%. AAV was originally established in 1977 as a joint venture  with [[w:American Motors Corporation|AMC]] to produce Jeeps. Production began in 1978. It became part of Chrysler when Chrysler bought AMC in 1987. It continued to be a part of subsequent corporate entities DaimlerChrysler, Chrysler LLC, Chrysler Group LLC, [[w:Fiat Chrysler Automobiles|FCA]], and [[w:Stellantis|Stellantis]]. Production of the Jeep J8, a light military vehicle based on the JK Wrangler, began on November 13, 2008. Jeep Grand Cherokee L production began in September 2024. Citroen C4X production began in April 2025. AAV has also produced vehicles for other automakers including Toyota. Toyota Fortuner SUV production began in April 2012. &lt;br&gt; Past models: Jeep CJ6, Jeep Wagoneer (SJ), Jeep AM720 military vehicle, [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]], [[w:Jeep Wrangler (TJ)|Jeep TJL]], [[w:Jeep Wrangler (JK)#Military Jeep J8 (2007–present)|Jeep J8]] (2008-2019),  [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]], [[w:Jeep Liberty (KJ)|Jeep Cherokee (KJ)]], [[w:Jeep Liberty (KK)|Jeep Cherokee (KK)]], [[w:Jeep Grand Cherokee (WK2)|Jeep Grand Cherokee (WK2)]]
|}

==Current non-Chrysler FCA/Stellantis Factories Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| Y (700),&lt;br&gt; 9 (ProMaster Rapid),&lt;br&gt; 3 (Vision)
| Betim Plant
| [[w:Betim|Betim]], [[w:Minas Gerais|Minas Gerais]]
| [[w:Brazil|Brazil]]
| 1973
| 
| [[w:RAM 700|RAM 700]] (2015-),&lt;br&gt; [[w:ProMaster Rapid|Ram V700 Rapid]] (Peru)&lt;br&gt; Related models:&lt;br&gt; [[w:Fiat Strada|Fiat Strada]] ('99-),&lt;br&gt; [[w:Fiat Fiorino#Latin America (2013–present)|Fiat Fiorino]] ('14-)
| Fiat plant. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Dodge Vision|Dodge Vision]] (Mexico: 2015-2018),&lt;br&gt; [[w:ProMaster Rapid|Ram ProMaster Rapid]] (Mexico: '18-'24)
|-
| U
| Cordoba Plant
| [[w:Córdoba, Argentina|Córdoba]], [[w:Córdoba Province, Argentina|Córdoba Province]]
| [[w:Argentina|Argentina]]
| 1995
| 
| [[w:Peugeot Landtrek|Ram Dakota]] (2026-)
| Fiat plant.
|-
| F
| [[w:FCA India Automobiles|FCA India Automobiles Private Limited]]
| [[w:Ranjangaon|Ranjangaon]], [[w:Pune district|Pune district]], [[w:Maharashtra|Maharashtra]]
| [[w:India|India]]
| 1997
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]],&lt;br&gt; [[w:Jeep Meridian|Jeep Meridian/Commander]],&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]]
| Originally established as a 50/50 joint venture between Fiat and [[w:Tata Motors|Tata Motors]] called Fiat India Automobiles Private Limited. On June 1, 2017, Jeep Compass production began in India, the first Jeep built in India under its own brand. The JL-series Wrangler began to be assembled in India in 2021. The Jeep Meridian began production in May 2022. The Meridian is also exported to Japan as the Commander. The Jeep Grand Cherokee began assembly in India in November 2022. 
|-
| K
| Goiana Plant
| [[w:Goiana|Goiana]], [[w:Pernambuco|Pernambuco]]
| [[w:Brazil|Brazil]]
| 2015
|
| [[w:Jeep Renegade|Jeep Renegade]], [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Commander (2022)|Jeep Commander]], [[w:Ram Rampage|Ram Rampage]]&lt;br&gt; Related models: [[w:Fiat Toro|Fiat Toro]]
| [[w:Fiat Chrysler Automobiles|FCA]] plant. Production at Goiana began with the Jeep Renegade in February 2015. &lt;br&gt; Past Chrysler Group models: [[w:Ram 1000|Ram 1000]]
|-
| P
| Melfi Plant (formerly SATA = Società Automobilistica Tecnologie Avanzate [Advanced Technologies Automotive Company])
| [[w:Melfi|Melfi]], [[w:Province of Potenza|Province of Potenza]]
| [[w:Italy|Italy]]
| 1993
|
| [[w:Jeep Compass#Third generation (J4U; 2025)|Jeep Compass (J4U)]]&lt;br&gt; (Europe: '26-)
| Fiat plant. Jeep production began at Melfi in 2014 with the Renegade. Renegade production at Melfi ended on October 17, 2025. Production of the 3rd gen. Compass began on October 29, 2025. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Jeep Renegade|Jeep Renegade]] (US/Can.: '15-'23, Europe: '15-'25),&lt;br&gt; [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (Europe: '20-'25) &lt;br&gt; Related models:&lt;br&gt; [[w:Fiat 500X|Fiat 500X]] (US/Can.: '16-'23,&lt;br&gt; Europe: '15-'24)
|-
| B
| Porto Real Plant
| [[w:Porto Real|Porto Real]], [[w:Rio de Janeiro (state)|Rio de Janeiro state]]
| [[w:Brazil|Brazil]]
| 2000
|
| [[w:Jeep Avenger|Jeep Avenger]]
| PSA plant. The Jeep Avenger is the first Jeep to be made in a former PSA plant. 
|-
| U (2027-), 6 (2015-2022)
| [[w:Tofaş|Tofaş]]
| [[w:Bursa|Bursa]]
| [[w:Turkey|Turkey]]
| 1971
|
| [[w:Citroën Jumpy#Ram ProMaster City|Ram ProMaster City]] (2027-)
| Originally a Fiat joint venture, it is now 37.8% owned by Stellantis, 37.8% owned by [[w:Koç Holding|Koç Holding]], and 24.3% publicly traded on the Istanbul Stock Exchange. &lt;br&gt; Past Chrysler Group models: &lt;br&gt; [[w:Fiat Doblò#Ram ProMaster City|Ram ProMaster City]] (2015-2022), [[w:Chrysler Neon#Third generation (2016)|Dodge Neon]]&lt;br&gt; (Mexico &amp; Middle East: 2017-2020), [[w:Fiat Fiorino#Europe (2007–2024)|Ram V700 City]] (Chile: 2018-2023)
|-
| J,&lt;br&gt; 5 (Ypsilon)
| Tychy Plant
| [[w:Tychy|Tychy]], [[w:Silesian Voivodeship|Silesian Voivodeship]]
| [[w:Poland|Poland]]
| 1975
|
| [[w:Jeep Avenger|Jeep Avenger]]
| Fiat plant. Production began in September 1975 when the plant was owned by Polish automaker [[w:Fabryka Samochodów Małolitrażowych|FSM]], which built Fiat-based models. Fiat took over FSM in 1992. Fiat subsequently became [[w:Fiat Chrysler Automobiles|FCA]] and then Stellantis. Jeep Avenger production began on January 31, 2023. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Chrysler Ypsilon|Chrysler Ypsilon]] (UK/Ireland/Japan)
|}

==Current partner factories making Chrysler Group vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| GAC Hangzhou plant
| [[w:Hangzhou|Hangzhou]], [[w:Zhejiang|Zhejiang]]
| [[w:China|China]]
| 2021 (production began for Chrysler)
|
| [[w:Trumpchi GS5#Dodge Journey|Dodge Journey]] (Mexico: 2022-)
| [[w:GAC Group|GAC]] plant. 
|-
| 3
| GAC Yichang plant
| [[w:Yichang|Yichang]], [[w:Hubei|Hubei]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Dodge Attitude#Fourth generation (2025)|Dodge Attitude]] (Mexico: 2025-)
| [[w:GAC Group|GAC]] plant. 
|-
| 5
| Shenzhen Baoneng Motor Co., Ltd.
| [[w:Shenzhen|Shenzhen]], [[w:Guangdong|Guangdong]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Peugeot Landtrek|Ram 1200]] (Mexico: 2025-)
| [[w:Baoneng Group#Automotive business|Shenzhen Baoneng Motor Co., Ltd.]] plant. 
|}

==Former factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Closed
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
|
| Ajax Trim plant
| [[w:Ajax, Ontario|Ajax]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1953, 1964 (became part of Chrysler)
| 2003
| Automotive Soft Trim Components, Seat Cushion and Seatback Covers, Foam-in-place Covers
| Built in 1953 by Canadian Automotive Trim. Purchased by Chrysler in 1964. Closed by December 2003.
|-
| J (1989-1992),&lt;br&gt; B (1981-1988),&lt;br&gt; 7-8 (1966-1980),&lt;br&gt; T (1958-1966) 
| [[w:Brampton Assembly (AMC)|AMC Brampton Assembly]]
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]] 
| 1961,&lt;br&gt; 1987 (became part of Chrysler)
| 1992
| [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1987-92), [[w:Jeep CJ#CJ-5|Jeep CJ-5]] (1979-1980),&lt;br&gt; [[w:Jeep CJ#CJ-7|Jeep CJ-7]] (1979-1980),&lt;br&gt; [[w:AMC Eagle|AMC Eagle]] (1981-1987), [[w:AMC Eagle|Eagle Wagon]] (1988), [[w:AMC Concord|AMC Concord]] (1978, 1981-1983), [[w:AMC Spirit|AMC Spirit]] (1983), [[w:AMC Hornet|AMC Hornet]] (1970-77), [[w:AMC Gremlin|AMC Gremlin]] (1970-1978),&lt;br&gt; [[w:AMC Rebel|AMC Rebel]] (1968-1970), [[w:Rambler Rebel#Fifth generation|Rambler Rebel]] (1967), [[w:Rambler Classic|Rambler Classic]] (1961-1966),&lt;br&gt; [[w:AMC Ambassador|AMC Ambassador]] (1966-1968), [[w:AMC Ambassador|Rambler Ambassador]] ('63-'65), [[w:Rambler American|Rambler American]] (1962-68), [[w:Rambler American|AMC Rambler]] (1969)
| [[w:American Motors Corporation|AMC]] plant. Became part of Chrysler in the 1987 buyout of AMC. Located at at the corner of Kennedy Road South and Steeles Avenue East. Production began on January 26, 1961 with the Rambler Classic. This was the last plant to produce AMC vehicles. Eagle Wagon (formerly AMC Eagle) ended production on December 11, 1987. Production ended in April 1992 and Jeep Wrangler production was moved to Toledo, OH. Buildings on the west side of the plant were demolished in 2005 and buildings on the east side were demolished in 2007. A Lowe's and a Wal-Mart now occupy some of the former plant site.
|-
| V
| [[w:Conner Avenue Assembly|Conner Avenue Assembly]] 
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1996
| 2017
| [[w:Dodge Viper|Dodge Viper]] &lt;br&gt; GTS: '96, &lt;br&gt; All models: &lt;br&gt; '97-'06, '08-'10, '13-'17, [[w:Plymouth Prowler|Plymouth Prowler]]&lt;br&gt; ('97, '99-'01),&lt;br&gt; [[w:Chrysler Prowler|Chrysler Prowler]] ('01-'02),&lt;br&gt; [[w:Viper engine|8.0L/8.3L/8.4L aluminum &lt;br&gt; Viper V10 engine]] &lt;br&gt;(5/01-2017)
| Actually located at 20000 Connor Street. The plant was originally built in 1966 to make spark plugs by Champion. After Champion was bought by Cooper Industries in 1990, the plant was closed. It remained empty until Chrysler bought it in 1995. Viper production was moved to the Connor Avenue plant from the New Mack plant beginning with the GTS coupe for 1996, followed by the RT/10 roadster for 1997. Prowler production began in May 1997 and ended on February 15, 2002. Production of the Viper's aluminum V10 engine was moved to Connor Avenue, where it was built alongside the Viper itself, in May 2001 from the Mound Road Engine Plant, which closed in 2002. Viper production ended on July 2, 2010 and the plant was dormant until production restarted in December 2012. Production ended on Aug. 31, 2017. In 2018, the plant was renamed Connor Center, a meeting and display space that will showcase Chrysler’s concept and historic vehicle collection. 
|-
|E
| [[w:Diamond-Star Motors|Diamond-Star Motors/Mitsubishi Motors Manufacturing America/Mitsubishi Motors North America Manufacturing Division]]
| [[w:Normal, Illinois|Normal, Illinois]]
| [[w:United States|United States]]
| 1988
| 2005 (end of Chrysler production)/&lt;br&gt; 2015 (end of Mitsubishi production)
| [[w:Plymouth Laser|Plymouth Laser]] (1990-1994),&lt;br&gt; [[w:Eagle Talon|Eagle Talon]] (1990-1998),&lt;br&gt; [[w:Eagle Summit#First generation (1989–1992)|Eagle Summit 4-d]] (1991-1992),&lt;br&gt; [[w:Dodge Avenger#Dodge Avenger Coupe (1995–2000)|Dodge Avenger]] (1995-2000),&lt;br&gt; [[w:Dodge Stratus#Stratus coupe (2001–2005)|Dodge Stratus Coupe]]&lt;br&gt; (2001-2005),&lt;br&gt; [[w:Chrysler Sebring|Chrysler Sebring Coupe]]&lt;br&gt; (1995-2005),&lt;br&gt;  [[w:Mitsubishi Eclipse|Mitsubishi Eclipse]] (1990-2012),&lt;br&gt; [[w:Mitsubishi Mirage#Third generation (1987)|Mirage sedan]] (1991-1992),&lt;br&gt; [[w:Mitsubishi Galant|Galant]] (1994-2012),&lt;br&gt; [[w:Mitsubishi Endeavor|Endeavor]] (2004-2011),&lt;br&gt; [[w:Mitsubishi ASX#First generation (GA; 2010)|Outlander Sport/RVR]] (2013-15)
| Originally established as Diamond-Star Motors, a 50/50 joint venture between Chrysler and Mitsubishi which assembled both Chrysler and Mitsubishi vehicles. Production began in September 1988. In October 1991, Chrysler sold its 50% stake in the plant to Mitsubishi but production for Chrysler continued under contract. On May 24, 1993, production of the Mitsubishi Galant began at the Diamond-Star plant. In July 1995, the plant was renamed Mitsubishi Motors Manufacturing America. In January 2002, the plant was renamed Mitsubishi Motors North America Manufacturing Division. In January 2003, production of the Mitsubishi Endeavor began, the first SUV built at the Mitsubishi plant. In February 2005, production of vehicles for Chrysler ended. All further production was only of Mitsubishi-branded vehicles. In mid-2012, the plant began producing the Mitsubishi Outlander Sport. The Outlander Sport is sold as the RVR in Canada. On November 30, 2015, vehicle production ended. The Mitsubishi plant produced 3,283,549 vehicles. The plant continued to produce replacement parts until May 2016, when the plant closed completely. In June 2016, the plant was sold to liquidation firm Maynards Industries. In January 2017, EV startup [[w:Rivian|Rivian Automotive]] bought the former Mitsubishi plant. Rivian began production at the Normal, IL plant in September 2021 with the [[w:Rivian R1T|R1T]] electric pickup.
|-
| U
| [[w:Eurostar Automobilwerk|Eurostar]] - Chrysler Graz Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1991
| 2002
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (1992-2002), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2002)
| Originally, a 50/50 joint venture between Chrysler and Steyr-Daimler-Puch founded in 1990. The Eurostar plant is next to the Steyr Fahrzeugtechnik plant solely owned by Steyr-Daimler-Puch. Production of the Chrysler Voyager and Grand Voyager minivans began in October 1991. This was the 2nd generation Chrysler minivan. The 3rd generation began production on September 25, 1995. In 1996, production of right-hand drive minivans began. The 4th generation began production in January 2001. Production of the PT Cruiser began in July 2001 on the same line as the Voyager minivans. In 1999, DaimlerChrysler bought the 50% stake in Eurostar held by Steyr-Daimler-Puch Fahrzeugtechnik, now majority owned by Magna International, making Eurostar a subsidiary of DaimlerChrysler. DaimlerChrysler sold 100% of Eurostar to Magna Steyr in July 2002. PT Cruiser production in Austria ended and was consolidated in Toluca, Mexico. Production of the Chrysler Voyager and Grand Voyager minivans moved next door to the main Magna Steyr plant for 2003. Magna International had acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. 
|-
| 3 (1959),&lt;br&gt; E (1958)
| Evansville Assembly
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1919, 1928 (became part of Chrysler)
| 1959 
| Graham Brothers trucks (through 1932),&lt;br&gt; Dodge Brothers trucks &lt;br&gt; (through 1932),&lt;br&gt; Plymouth cars (1936-1958),&lt;br&gt; Dodge cars (1937-1938),&lt;br&gt; [[w:Plymouth Savoy|Plymouth Savoy]] (1959), [[w:Plymouth Belvedere|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1959)
| Located at 1625 N. Garvin St. Built in 1919 by Graham Brothers Truck Company to build trucks. In 1925, Dodge Brothers bought a controlling 51% stake in Graham Brothers and then bought the rest in 1926, completely merging the 2 companies. This gave Dodge Brothers plants in Evansville, IN and Stockton, CA. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. At that point, trucks with a Dodge Brothers nameplate were rated as a half-ton; larger rated trucks were sold under the Graham Brothers name. On January 1, 1929, the Graham Brothers brand was eliminated, and all trucks produced became Dodge trucks. In 1932, Chrysler closed the Evansville plant due to the Great Depression. In 1935, as the economy improved, Chrysler reopened the Evansville plant and renovated and expanded it. It began building Plymouth cars for 1936. Dodge cars were also built for 1937 and 1938. During World War II, the plant became the Evansville Ordinance Plant, which produced more than 3.26 billion ammunition  cartridges - about 96% of all the .45 automatic ammunition produced for all the armed forces. The Evansville Ordinance Plant also rebuilt 1,600 Sherman tanks and 4,000 military trucks. After the war ended, Plymouth car production resumed. However, in the early 1950s, during the Korean War, the Evansville plant retooled and dedicated about a third of its space and manpower to building 60-foot aluminum hulls for Grumman  UF-1 Albatross air-sea rescue planes for the Navy and Coast Guard. Evansville built its 1 millionth Plymouth in March 1953. The plant was closed in 1959 and was replaced by the larger, more modern St. Louis plant in Fenton, MO, which had access to more railroad lines for shipping than Evansville did. 
|-
| 
| Evansville Stamping
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1935, 1953 (became part of Chrysler)
| 1959 
| Body panel stampings
| Opened by Briggs Manufacturing Company when Chrysler reopened Evansville Assembly in 1935 to build Plymouth cars. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Made body panels for the nearby Evansville Assembly plant. Closed when Evansville Assembly closed in 1959.
|-
| B (1968-1980),&lt;br&gt; 2 (1960-1967),&lt;br&gt; 2 (1959)
| [[w:Dodge Main|Hamtramck Assembly (Dodge Main)]]
| [[w:Hamtramck, Michigan|Hamtramck, Michigan]]
| [[w:United States|United States]]
| 1911,&lt;br&gt; 1928 (became part of Chrysler)
| 1980 
| Dodge (1914-1958),&lt;br&gt; [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1959),&lt;br&gt; [[w:Dodge Royal#Third generation (1957–1959)|Dodge Royal]] (1959),&lt;br&gt; [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959),&lt;br&gt; [[w:Dodge Matador|Dodge Matador]] (1960),&lt;br&gt; [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962),&lt;br&gt; [[w:Dodge Polara|Dodge Polara]] (1960-1964),&lt;br&gt; [[w:Dodge 330|Dodge 330]] (1963-1964),&lt;br&gt; [[w:Dodge 440|Dodge 440]] (1963-1964),&lt;br&gt; [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1975), [[w:Plymouth Duster|Plymouth Duster]] (1970-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1969, 1972-1975), [[w:Dodge_Dart#1971|Dodge Dart Demon]] (1971-1972),&lt;br&gt; [[w:Dodge Charger (1966)|Dodge Charger]] (1967-1969), [[w:Dodge Charger Daytona#First generation (1969)|Dodge Charger Daytona]] ('69), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-74), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970-1974), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volaré|Plymouth Volaré]] (1976-1980),&lt;br&gt; Engines
| Located at 7900 Joseph Campau Ave. This plant predated Dodge being part of Chrysler Corp. On November 4, 1914, the first Dodge Brothers passenger car was produced at the Hamtramck plant. Prior to that, Dodge Brothers made components for other automakers, primarily Ford. The Hamtramck plant was fully vertically integrated, capable of building almost every part needed to build a complete car. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. Even after the Chrysler takeover, Hamtramck remained Dodge's home plant. From the early 1950s, various operations were automated or moved to other plants and Hamtramck became more of an assembly plant by the early 1960s. Closed January 4, 1980. Last vehicle built was a Silver Metallic 1980 Dodge Aspen R/T 2-door. 13,943,221 vehicles were produced at the plant. Demolished in 1981. Replaced by the General Motors Detroit/Hamtramck Assembly Plant (Factory Zero), which opened in 1985. 
|-
| 
| [[w:Highland Park Chrysler Plant|Highland Park Plant]]
| [[w:Highland Park, Michigan|Highland Park, Michigan]]
| [[w:United States|United States]]
| 1909,&lt;br&gt; 1925 (became part of Chrysler)
| 1960s (end of manufacturing)
| Maxwell cars (1910-1925), Chrysler Series 50 (1926-27), Chrysler Series 52 (1928), Plymouth Model Q (1929), Chrysler Series 60 (1927), Chrysler Series 62 (1928), DeSoto Series K (1929-1930), DeSoto Series CK (1930), DeSoto Series CF (1930), Fargo Trucks (1928-1930) &lt;br&gt; Parts including fluid coupling and torque converter for [[w:Fluid Drive|Fluid Drive]]
| Located at at 12000 Chrysler Service Drive (formerly Chrysler Drive). Originally, this was [[w:Maxwell Motor Company|Maxwell Motor Company]]'s main plant. However, before Maxwell Motor Co.'s formation, parts of the site was used by several car and truck manufacturers: Grabowsky Power Wagon Company used 1 building, Brush Runabout Co. used another building, and Gray Motor Co. owned another building. Gray only used the western 1/3 of the building so they leased the middle third to Alden- Sampson Truck Co. and the eastern third to Maxwell-Briscoe Motor Co. All these companies, along with several others, combined under the [[w:United States Motor Company|United States Motor Company]] in 1910. In 1913, United States Motor Company collapsed and Maxwell was the only surviving part. The U.S. Motor Co. assets were purchased by Walter Flanders, who reorganized the company as the Maxwell Motor Co. Maxwell hired Walter P. Chrysler to turn the company around in 1921 after its finances deteriorated in the post-World War I recession in 1920. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including its Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924 at the Jefferson Ave. plant. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925 at Highland Park. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 built at Highland Park and was then reworked again into the first Plymouth in 1928, also built at Highland Park. Plymouth production was moved to the new Lynch Road Assembly plant in 1929 and DeSoto production also moved to the new Lynch Road Assembly plant in 1931. Highland Park still built parts but it no longer built vehicles. Highland Park was used more for design,  engineering, and management and served as Chrysler Corporation's headquarters through 1996. During the 1990's, Chrysler moved to its current headquarters at the Chrysler Technology Center in Auburn Hills. Much of the site has been demolished though Chrysler still has a small presence at the site with the FCA Detroit Office Warehouse. Other parts of the site are now occupied by several automotive suppliers including Magna, Valeo, Mobis, Avancez, and Yanfeng.
|-
|
| [[w:Indiana Transmission#Indiana Transmission Plant II|Indiana Transmission Plant II]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2003 (as Indiana Transmission Plant II)
| 2019 (as Indiana Transmission Plant II)
| [[w:W5A580|Mercedes W5A580 (A580) &lt;br&gt; 5-speed auto. trans.]], Transmission components
| Located at 3360 North U.S. Highway 931. Plant was originally known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Production began in November 2003. 5-speed auto. trans. production ended in August 2018 while production of components for the 8-speed auto. transmission ended in the fall of 2019. Starting in 2020, the plant was converted to engine production and is now known as Kokomo Engine Plant. Engine production began in late February 2022. 
|-
|
| Indianapolis Electrical Plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1953
| 1988
| Transmission plant: [[w:Chrysler PowerFlite transmission|Chrysler PowerFlite 2-speed auto. transmission]] &lt;br&gt; Electrical Plant: Alternators, distributors, starters, power steering units, voltage regulators, windshield wiper motors, and other electrical parts for cars
| Located at 2900 Shadeland Ave. Began making transmissions in 1953. In January 1959, Chrysler housed its new Electrical Division at the Shadeland Avenue plant, replacing transmission production. Became part of Chrysler's Acustar components subsidiary in 1987. Production ended on November 30, 1988 but shipping and other activities continued until March 1989 when the factory closed. Certain portions have been demolished and improvements were made to the remainder, which is now the Shadeland Business Center. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Naomi Street plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1946 (became part of Chrysler)
| 1970's
| Engine blocks
| Located at 1535 Naomi Street. Purchased from American Foundry Company in 1946. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Kept operating even after the Tibbs Ave. plant was launched. This location is now Wilco Gutter Supply. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Tibbs Avenue plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1950 
| 2005
| Engine heads and blocks and other components
| Located at 1100 S. Tibbs Avenue. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Production ended on September 30, 2005 and the facility closed. Demolished in 2006. 
|-
| C (1968-1990),&lt;br&gt; 3 (1960-1967),&lt;br&gt; 1 (1959)
| [[w:Detroit Assembly Complex – Jefferson#Jefferson Avenue Assembly|Jefferson Avenue Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1908,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| [[w:Chrysler Six|Chrysler Series 70 (B-70)]] (1924-1925), [[w:Chrysler Six|Chrysler Series 70 (G-70)]] (1926-1927), [[w:Chrysler Six|Chrysler Series 72]] (1928), [[w:Chrysler Royal|Chrysler Royal]] (1933, 1937-1950), DeSoto SD (1933), Chrysler Airflow (1934-1937), Chrysler Airstream (1935-36), [[w:Chrysler (brand)|Chrysler brand]] (1937-1958), Desoto Airflow (1934-1936), DeSoto Airstream (1935-1936), [[w:Chrysler Imperial|Chrysler Imperial]] (1926-1942, 1946-1954), [[w:Imperial (automobile)|Imperial]] (1955-1958, 1962-1975), [[w:Chrysler 300 letter series|Chrysler 300 letter series]] (1959-1965), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-1978), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1961), [[w:Chrysler Newport|Chrysler Newport]] (1961-1978), [[w:Chrysler 300 non-letter series|Chrysler 300 Sport Series]] (1962-1971), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1968-1972), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:DeSoto Fireflite|DeSoto Fireflite]] (1959-1960), [[w:DeSoto Adventurer|DeSoto Adventurer]] (1959-1960), [[w:DeSoto (automobile)#1961|DeSoto]] (1961), [[w:Dodge Matador|Dodge Matador]] (1960), [[w:Dodge Polara|Dodge Polara]] (1965-1966), [[w:Dodge D series|Dodge D/W series]] (1979-1980), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1979-1980), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1979-1980), [[w:Dodge Aries|Dodge Aries]] (1981-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1981-1989), [[w:Dodge 400|Dodge 400]] 4-d (1982-1983), [[w:Chrysler LeBaron|Chrysler LeBaron]] 4-d (1982-1984), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker (E-body)]] (1983-1987), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker Turbo (E-body)]] (1988), [[w:Dodge 600|Dodge 600]] 4-d (1983-1988), [[w:Chrysler E-Class|Chrysler E-Class]] (1983-1984), [[w:Plymouth Caravelle|Plymouth Caravelle]] (US: 1985-1988), [[w:Plymouth Caravelle|Plymouth Caravelle]] 4-d (Canada: 1983-1988), [[w:Dodge Omni|Dodge Omni]] (1989-1990), [[w:Plymouth Horizon|Plymouth Horizon]] (1989-1990),&lt;br&gt; Engines
| Located at 12200 East Jefferson Ave. Plant was originally opened by [[w:Chalmers Automobile|Chalmers Motor Co.]]. After falling on hard times, Chalmers agreed in 1917 to build cars for [[w:Maxwell Motor Company|Maxwell Motor Co.]] at the Jefferson Ave. plant in Detroit. In exchange, Chalmers cars would be sold through Maxwell dealers. After having its own financial problems, Maxwell stopped producing cars at the Chalmers plant in 1921. Maxwell hired Walter P. Chrysler to turn the company around in 1921. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including the Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 and was then reworked again into the first Plymouth in 1928. The Jefferson Ave. plant was the home plant of the Chrysler brand through 1978. It was also the home plant for the spin-off Imperial brand except for 1959-1961, when Imperial had its own exclusive plant on Warren Ave. in Dearborn. By the time it ended production on February 2, 1990, Jefferson Ave. Assembly had built 8,310,107 vehicles. Demolished in 1991. Replaced by the Jefferson North plant built across Jefferson Ave. from the old plant, where the Kercheval Body Plant used to be. The Jefferson North plant opened in January 1992.
|-
| W (1987-1989 Chrysler M-body) &lt;br&gt;&lt;br&gt; [K for 1981-1983 AMC and 1983-1987 Renault],&lt;br&gt; 0-6 (1966-1980 AMC)
| Kenosha I Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]]&lt;br&gt; (1987-1989),&lt;br&gt; [[w:Dodge Diplomat#Second generation (1980)|Dodge Diplomat]] (1987-1989), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1987-89), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1987-1989)
| This was the Kenosha Main Plant of [[w:American Motors Corporation|AMC]]. The Kenosha plant was the oldest still operating automobile factory in the world when it ended vehicle production in December 1988. It first built automobiles in 1902 for the Thomas B. Jeffery Company under the Rambler brand. The factory was purchased in 1900 from the Sterling Bicycle Co., which built it in 1895. In 1914, the Thomas B. Jeffery Company rebranded its vehicles under the Jeffery brand. In 1916, the Thomas B. Jeffery Company was bought by Charles Nash and renamed Nash Motors. Kenosha produced Nash vehicles from 1917-1957. Kenosha also produced Nash's entry-level Lafayette brand from 1934-1936. After Nash merged with Hudson to form American Motors in 1954, Kenosha also produced Hudson vehicles from 1955-1957. Kenosha then produced vehicles under the Rambler brand for AMC from 1958-1968 and under the AMC brand from 1966-1983. Kenosha also produced the Alliance for AMC shareholder Renault for 1983-1987 along with the Encore for 1984-1986 and the GTA for 1987. Chrysler signed a deal with AMC in September 1986 to utilize surplus capacity at AMC's Kenosha plant to build Chrysler's trio of rwd M-body sedans beginning in February 1987. Chrysler did not have any capacity left in its own plants to continue building the M-body sedans. The St. Louis North plant that had been building the M-body sedans had been converted to build the extended length minivans. This deal led to Chrysler's acquisition of AMC, announced in March 1987. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 and the M-bodies were discontinued. The site continued building engines until 2010. The plant has since been demolished.
|-
| Y
| Kenosha II Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Dodge Omni|Dodge Omni]] (1988-1989), [[w:Plymouth Horizon|Plymouth Horizon]] (1988-1989)
| This was the Kenosha Lakefront Plant of [[w:American Motors Corporation|AMC]], located on the shore of Lake Michigan. This property was originally a Simmons mattress manufacturing plant from 1870 to 1960. AMC bought it in 1960 to manufacture and paint auto bodies. Became part of Chrysler in the 1987 buyout of AMC. In September 1987, production of the Dodge Omni and Plymouth Horizon began. Production was moved to Kenosha from Chrysler's Belvidere, IL plant, which was being converted to build Chrysler's C-body sedans (Dynasty/New Yorker). Closed in December 1988. Omni &amp; Horizon production then moved to the Jefferson Ave. plant in Detroit. Demolished in 1990. In 1994, the City of Kenosha purchased the property for $1. The property was subsequently cleaned up and redeveloped into the HarborPark area, which includes a park and open space, a public museum, residential housing, and a marina. 
|-
| 
| [[w:Kenosha Engine|Kenosha Engine Plant]]
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2010
| [[w:AMC straight-4 engine|AMC straight-4 engine]],&lt;br&gt; [[w:AMC straight-6 engine|AMC straight-6 engine]],&lt;br&gt; [[w:AMC V8 engine|AMC V8 engine]],&lt;br&gt; [[w:Chrysler LH engine|Chrysler 2.7L DOHC V6]], [[w:Chrysler SOHC V6 engine#3.5|Chrysler 3.5L SOHC V6]]
| Located at 5555 30th Avenue. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 at the adjacent assembly plant but the site continued building engines until 2010. After the Chrysler buyout, the plant kept building the AMC 5.9L V8 for the SJ Jeep Grand Wagoneer through 1991, the AMC 2.5L I4 through 2002 for Jeeps and the Dodge Dakota, the AMC 4.2L I6 through 1990 for the Jeep Wrangler, and the AMC 4.0L I6 through 2006 for various Jeeps ('06 Wrangler was the last to use the 4.0L). Chrysler started building its own 2.7L V6 at Kenosha in 1997 and its own 3.5L V6 in 2003. Engine production ended in October 2010 and the plant closed. Demolished in 2012-2013.
|-
|
| Kercheval Body Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1920,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| Automobile Bodies
| Located at 12265 E Jefferson Ave., across Jefferson Ave. from Chrysler's Jefferson Ave. Assembly Plant, which had originally been the Chalmers plant. The assembly plant was on the south side of Jefferson Ave. while the body plant was on the north side. The plant was called Kercheval because the north side of the plant was bordered by Kercheval Ave. The plant was built in 1920 by Wadsworth Manufacturing Co. to replace a previous plant on the same site that burned down in 1919. That fire had also damaged the Chalmers plant across the street. In November 1920, Wadsworth Manufacturing was sold to American Motor Body Co., a division of the American Can Co. On July 1, 1923, American Motor Body Co. became American Motor Body Corp., run by Charles M. Schwab. On September 4, 1925, Chrysler Corp. bought the Detroit plant of the American Motor Body Corp. to quickly increase its manufacturing capacity. In 1955, Kercheval Body Plant was connected to the Jefferson Assembly plant by a bridge crossing over Jefferson Ave. Previously, bodies made at Kercheval had to be transported by truck across Jefferson Ave. to the assembly plant. The plant closed in Feb. 1990, at the same time the Jefferson Ave. Assembly Plant closed. The Kercheval plant was demolished and Chrysler built the new Jefferson North Assembly Plant on the site of the former Kercheval Body Plant, on the north side of Jefferson Ave. 
|-
|
| Kokomo - Home Ave. plant
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1937
| 1969
| Manual Transmissions (1937-1955), Aluminum die casting (1955-1969)
| Located at 1105 S. Home Ave. Chrysler bought this plant in 1937. This was Chrysler's first plant in Kokomo. It had previously belonged to the [[w:Haynes Automobile Company|Haynes Automobile Co.]], which went out of business in 1925. The plant had been dormant since then. 5,124,211 manual transmissions were built here from 1937-1955. Transmission production then shifted to a new plant about a mile southeast on South Reed Road. The Home Ave. plant then became an aluminum die casting plant until 1969, when that operation shifted to the new Kokomo Casting plant on East Boulevard. Chrysler subsequently sold this plant. The facility was last used by Warren's Auto Parts, an auto salvage yard, which closed in 2020 after nearly 50 years. It is currently empty though still standing as of 2025. 
|-
| M
| [[w:Lago Alberto Assembly|Lago Alberto Assembly]]
| [[w:Nuevo Polanco|Nuevo Polanco district]], [[w:Miguel Hidalgo, Mexico City|Miguel Hidalgo borough]], [[w:Mexico City|Mexico City]]
| [[w:Mexico|Mexico]]
| 1938
| 2002
| &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Savoy, Dodge Dart, Dodge 330, Dodge 440, Chrysler Valiant (1963-1969), Valiant Barracuda (1965-1969), Dodge Coronet, [[w:Dodge Ram|Dodge Ram pickup]] (1981-02), [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-96), [[w:Dodge Ramcharger#Third generation (1999–2001)|Dodge Ramcharger]] (1999-01) &lt;br&gt; Export to US:&lt;br&gt; [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-93), [[w:Dodge Ram#First generation (1981; D/W)|Dodge Ram pickup]] (1990-93), [[w:Dodge Ram#Second generation (1994; BR/BE)|Dodge Ram pickup]] (1994-02)
| Located at 320 Lago Alberto Street. Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Lago Alberto began exporting to the US with the 1986 Dodge Ramcharger, sourced exclusively from Mexico. The Lago Alberto plant was closed in 2002 and Mexican pickup production was consolidated in the newer, more modern Saltillo plant. 
|-
| E (1968-1971),&lt;br&gt; 5 (1960-1967),&lt;br&gt; 4 (1959),&lt;br&gt; L (1958),&lt;br&gt; L (1955-1957 Chrysler brand)
| [[w:Los Angeles (Maywood) Assembly|Los Angeles (Maywood) Assembly]]
| [[w:Commerce, California|City of Commerce, California]]
| [[w:United States|United States]]
| 1932
| 1971
| Plymouth (1946-1958), Dodge (1946-1953, 1955-1958), DeSoto Deluxe/Custom (1948-1952), DeSoto Powermaster Six (1953-1954), DeSoto Firedome (1952-57), DeSoto Fireflite (1955-1957), DeSoto Firesweep (1957-1958), Chrysler Windsor (1948-1958), Chrysler Royal (1949-1950), Chrysler Saratoga (1951-1952, 1957-1958), Chrysler New Yorker (1953-1958), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1960), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-60), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1957-1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Polara|Dodge Polara]] (1960-1964), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1958-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1959-1961), [[w:Plymouth Savoy|Plymouth Savoy]] (1958-1959, 1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1971), [[w:Plymouth Duster|Plymouth Duster]] (1970-1971), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1971), [[w:Dodge Dart#1971|Dodge Dart Demon]] (1971), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1966, 1970), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1971), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1971), [[w:Dodge Coronet|Dodge Coronet]] (1965-1971), [[w:Dodge Charger (1966)|Dodge Charger]] (1971), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971)
| Located at 5800 South Eastern Avenue and Slauson Avenue in Maywood, now part of City of Commerce. Across the street from the [[w:Maywood Assembly|Ford Maywood Assembly plant (Los Angeles Assembly plant No. 1)]]. 
|-
| A (1968-1981),&lt;br&gt; 1 (1960-1967),&lt;br&gt; 6 (1959)
| [[w:Lynch Road Assembly|Lynch Road Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1929
| 1981
| Plymouth (1929-1958), DeSoto (1931-1932), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-64), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (59-61), [[w:Plymouth Fury|Plymouth Fury]] (1959-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (59-61), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (62-70), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1970, 1973-1974), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1978), [[w:Plymouth GTX|Plymouth GTX]] (1967-1970), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1970, 1973, 1975), [[w:Plymouth Superbird|Plymouth Road Runner Superbird]] (1970), [[w:Dodge Coronet|Dodge Coronet]] (1965-1976), [[w:Dodge Monaco#Fourth generation (1977–1978)|Dodge Monaco]] (1977-1978), [[w:Dodge Charger (1966)#1966|Dodge Charger]] (1966), [[w:Dodge Charger (1966)#Third generation|Dodge Charger]] (1971-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971),  [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979-81), [[w:Chrysler New Yorker#1979–1981|Chrysler New Yorker]] (79-81), [[w:Dodge St. Regis|Dodge St. Regis]] (1979-81), [[w:Plymouth Gran Fury#1980–1981|Plymouth Gran Fury]] (80-81),&lt;br&gt; Engines
| Located at 6334 Lynch Road. This was originally Plymouth's home plant. At the time it opened in 1929, Lynch Road was the largest single story auto plant in the world. DeSoto production was transferred from Highland Park to Lynch Road in 1931. In June 1932, DeSoto production was moved to the Jefferson Ave. plant when the DeSoto brand moved up in the brand hierarchy to between Dodge and Chrysler. Previously, DeSoto was between Plymouth and Dodge. During World War II, Lynch Road made tank transmissions, truck parts, and uranium enrichment diffusers for the Oak Ridge Gaseous Diffusion Plant in Oak Ridge, TN to produce enriched uranium for the atomic bomb. For 1965, Lynch Road began to focus on production of Plymouth and Dodge intermediate models. For 1979, Lynch Road was switched to build Chrysler's R-body full-size cars for all 3 Chrysler car brands. Production ended on April 3, 1981 and the factory closed. Last car produced was a white Plymouth Gran Fury police car. 
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant I]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1998
| 2019
| [[w:Chrysler PowerTech engine#4.7|4.7L PowerTech SOHC V8]],&lt;br&gt; [[w:Chrysler Pentastar engine|3.0L/3.2L/3.6L Pentastar V6]]
| Located at 4000 St. Jean Avenue. Mack Ave. Engine Plant I was built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in 1998. 4.7L V8 engine production ended in April 2013. Pentastar V6 engine production ended in 2019. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant II]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2000
| 2012
| [[w:Chrysler PowerTech engine#3.7 EKG|3.7L PowerTech SOHC 90° V6]]
| Located at 4500 St. Jean Avenue. Mack Ave. Engine Plant II was built next to the Mack Ave. Engine Plant I, which had been built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in November 2000. Production ended in September 2012. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
|
| Mack Ave. Stamping Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 1979
| Stampings
| Chrysler acquired the Mack Ave. Stamping Plant from Briggs Manufacturing Company in 1953. The plant was originally built in 1916 by the Michigan Stamping Company, which was taken over by Briggs Manufacturing in 1923. The plant was closed in 1979 and the site was basically abandoned. The city of Detroit bought the plant site in 1982 but was unable to find a purchaser or afford environmental remediation for the site and returned it to Chrysler. In 1990, Chrysler began cleanup and demolition of the old plant and built a new factory on the site, which became the New Mack Assembly Plant. The site later became the Mack Ave. Engine Complex and later, the Mack Ave. Assembly Plant, which is now part of the Detroit Assembly Complex.
|-
| 
| McGraw Stamping/McGraw Glass Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 19?
| 2003
| Oil pans, valve covers, and other small stampings,&lt;br&gt; Automotive Glass (1960-)
| Located at 9400 McGraw Ave. Was around the corner and behind the Wyoming Ave. DeSoto/Export plant. Originally, a stamping plant. In 1960, switched to making automotive glass. Used Safeguard brand. Glass was DOT# 21. Demolished. 
|-
|
| [[w:Mound Road Engine|Mound Road Engine Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 2002
| [[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]],&lt;br&gt; [[w:Chrysler LA engine#239 V6|3.9L 90° V6 engine]],&lt;br&gt; [[w:Chrysler LA engine#Magnum 8.0 L V10|8.0L iron Magnum V10 engine]],&lt;br&gt; [[w:Viper engine|8.0L aluminum Viper V10 engine]] (1992-5/01)
| Located at 20300 Mound Road. One of the plants Chrysler acquired from Briggs Manufacturing Company in 1953. Chrysler used the plant to produce aircraft parts from 1953-1954 and then transferred the plant to Plymouth in 1954 to build its new A-series V8 engine for 1956 model year cars. Converted into an engine plant and enlarged by 71,000 sq. ft., it began building V8 engines for Plymouth in July 1955. Dodge later used the A engine from 1959 in the US in cars and trucks and Chrysler used the A engine from 1960 in the US. The plant was closed in 2002 and demolished in 2003. The land was then paved over and is now used as a storage lot for vehicles produced at the nearby Warren Truck Assembly Plant. Warren Truck Assembly is just to the north of where Mound Road Engine was. Mt. Elliott Tool and Die was located immediately to the south of the Mound Road Engine Plant on Outer Drive East.
|-
|
| [[w:Mount Elliott Tool and Die|Mount Elliott Tool and Die]]/Outer Drive Manufacturing Technology Center/Outer Drive Stamping
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1938, 1953 (became part of Chrysler)
| 2018
| Stamping Dies, Checking Fixtures, Stamping Fixtures 
| Located at 3675 Outer Drive East. Built in 1938 by Briggs Manufacturing Company. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Chrysler renamed it Outer Drive Stamping. Stamping operations ended in 1983 and operations from the closed Vernor Tool &amp; Die plant were moved here. The plant was then renamed Outer Drive Manufacturing Technology Center. The plant now did tool and die work as well as pilot plant operations and engineering for new stamping technologies. Once the Chrysler Technology Center in Auburn Hills was built, Pilot Operations and Advanced Stamping Manufacturing Engineering moved there and the plant was renamed Mount Elliott Tool and Die. Operations at the plant ended in 2018 and the plant was sold to German automotive supplier Laepple Automotive in 2024. Laepple Automotive is producing stamped body parts at the plant. 
|-
| V
| [[w:Detroit Assembly Complex – Mack|New Mack Assembly Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 1995
| [[w:Dodge Viper|Dodge Viper RT/10]] (1992-96)
| Located at 4000 St. Jean Avenue. The New Mack Assembly Plant is built on the site of the former Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. Viper production began in May 1992 at the New Mack Assembly Plant. After ending production in 1995, New Mack Assembly was converted into the Mack Ave. Engine Plant I. A new addition was then built to create Mack Ave. Engine Plant II. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021. 
|-
| F (1968-2009),&lt;br&gt; 6 (1960-1967),&lt;br&gt; 5 (1959),&lt;br&gt; N (1958)
| [[w:Newark Assembly|Newark Assembly]]
| [[w:Newark, Delaware|Newark, Delaware]]
| [[w:United States|United States]]
| 1951,&lt;br&gt; 1957 (Automotive prod.)
| 2008
| Plymouth (1957-1958), Dodge (1958), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1958-1959, 1961), [[w:Plymouth Fury|Plymouth Fury]] (1959-1974), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge Polara|Dodge Polara]] (1960-1966), [[w:Dodge Monaco|Dodge Monaco]] (1965-1966, 1968), [[w:Chrysler Newport|Chrysler Newport]] (1965-1970), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1965-70), [[w:Chrysler Town &amp; Country (1941–1988)|Chrysler Town &amp; Country]] (1966-1967), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1964, 1974-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1964, 1974-1975), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volare|Plymouth Volare]] (1976-1980), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1979-1980), [[w:Plymouth Reliant|Plymouth Reliant]] sedan &amp; wagon (1981-1988), [[w:Dodge Aries|Dodge Aries]] sedan &amp; wagon (1981-1988), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (sedan: 1984-1988, wagon: 1982-1988), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1989-1995), [[w:Dodge Spirit|Dodge Spirit]] (1989-1995), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron Sedan (A-body)]] (1990, 1993-1994), [[w:Chrysler Saratoga#1989–1995|Chrysler Saratoga]] (For export: 1990-1992), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe (J-body)]] (1992-1993), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron convertible (J-body)]] (1992-1995), [[w:Dodge Intrepid#First generation (1993–1997)|Dodge Intrepid]] (1994-1996), [[w:Chrysler Intrepid#First generation (1993–1997)|Chrysler Intrepid]] (Canada: 1994-1995), [[w:Chrysler Concorde#First generation (1993–1997)|Chrysler Concorde]] (1995-1996), [[w:Dodge Durango#First generation (DN; 1998)|Dodge Durango (DN)]] (1998-2003), [[w:Dodge Durango#Second generation (HB; 2004)|Dodge Durango (HB)]] (2004-2009), [[w:Chrysler Aspen|Chrysler Aspen]] (2007-2009) 
| Chrysler began construction of the Delaware Tank Plant in January 1951 to build [[w:M48 Patton|M48 Patton]] tanks. Production began in April 1952. Production ended in May 1961 and the Tank Plant was closed in October 1961. Low rate initial production of the [[w:M60 tank|M60 tank]] was also done at the Newark plant in 1959 before production was moved to the [[w:Detroit Arsenal (Warren, Michigan)|Detroit Arsenal Tank Plant]] in Warren, MI in 1960. Conversion to automotive production began in 1956. Production of Plymouth and Dodge cars began on April 30, 1957. Production ended on December 19, 2008. Sold to the University of Delaware on October 24, 2009. Most of the plant was demolished in 2010-2011 except for the Administration Building near the front of the complex. The site is now the Science, Technology, and Advanced Research (STAR) campus. The old Chrysler Administration Building has been redesigned and is now being used by the College of Health Sciences. 
|-
| K
| [[w:Pillette Road Truck Assembly|Pillette Road Truck Assembly]]
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1976
| 2003
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1976-1980),&lt;br&gt; [[w:Dodge Ram Van|Dodge Ram Van]] (1981-2003), [[w:Dodge Ram Wagon|Dodge Ram Wagon]] (1981-'02), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1976-1983)
| Located at 2935 Pillette Road. Was originally Windsor Plant 6. The Pillette Road plant was located less than a mile away from Chryslers' main plant complex in Windsor. Production began in January 1976. Production ended on June 12, 2003. 2,309,399 units were built. Demolished in 2004. Part of the site is now the Grand Central Business Park. Another part was a logistics center serving Chrysler's Windsor Assembly Plant and operated by Syncreon. The Syncreon Automotive Windsor site closed in October 2022. The parts sorting and sequencing work done there was now going to be done in-house at Chrysler's Windsor Assembly Plant. 
|-
| 
| [[w:Los Angeles (Maywood) Assembly#San Leandro Assembly|San Leandro Assembly]]
| [[w:San Leandro, California|San Leandro, California]]
| [[w:United States|United States]]
| 1948
| 1954
| Plymouth (1949-1954),&lt;br&gt; Dodge (1949-1954)
| 
|-
| B (1996-2009),&lt;br&gt; G (1968-1991),&lt;br&gt; 7 (1960-1967),&lt;br&gt; 8 (1959)
| [[w:Saint Louis Assembly|St. Louis I Assembly]] - South plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1959
| 2008
| [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1960), [[w:Plymouth Fury|Plymouth Fury]] (1960-1964), [[w:Plymouth Savoy|Plymouth Savoy]] (1960-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1961), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1965, 1976), [[w:Plymouth Duster|Plymouth Duster]] (1973-1976), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1965, 1973-1976), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1965),&lt;br&gt; [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1974), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-75), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1976), [[w:Dodge Coronet|Dodge Coronet]] (1965-1973, 75), [[w:Dodge Charger (1966)|Dodge Charger]] (1968-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971), [[w:Dodge Diplomat|Dodge Diplomat]] (1977-1981), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1977-1981), [[w:Plymouth Caravelle|Plymouth Caravelle (M-body)]] (Canada: 1978-1981), [[w:Plymouth Reliant|Plymouth Reliant]] 2-d (1982-1986), [[w:Dodge Aries|Dodge Aries]] 2-d (1982-1986), [[w:Dodge 400|Dodge 400]] 2-d &amp; convertible (1982-1983), [[w:Dodge 600|Dodge 600]] 2-d &amp; convertible (1984-1986), [[w:Plymouth Caravelle|Plymouth Caravelle]] 2-d (Canada: 1983-1986), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (2-d &amp; convertible: 1982-1986), [[w:Chrysler Executive|Chrysler Executive]] (1983-1986), [[w:Dodge Daytona|Dodge Daytona]] (1984-1991), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1984-1991), [[w:Dodge Daytona#Chrysler Laser|Chrysler Laser]] (1984-1986), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe/convertible (J-body)]] (1987-1991), [[w:Plymouth Voyager|Plymouth Voyager]] (1996-2000), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1996-2007), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2009), [[w:Chrysler Voyager|Chrysler Voyager]] (2001-2003), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1996-2001, 2004-2007)
| Located at 1001 N. Hwy Dr. St. Louis South was idled in 1991. The Dodge Daytona was moved to Sterling Heights and the J-body Chrysler LeBaron was moved to Newark, DE. St. Louis South was reopened in 1995 to build minivans, which were moved from the St. Louis North plant. For the 3rd generation, St. Louis South built both SWB and LWB models. For the 4th generation, St. Louis South built all SWB models but also built some LWB models. Closed on October 31, 2008. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1996-2009),&lt;br&gt; X (1973-1995),&lt;br&gt; U (1970-1972),&lt;br&gt; 7 (1967-1969)
| [[w:Saint Louis Assembly|St. Louis II Assembly]] - North plant / Missouri Truck Assembly Plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1966
| 2009
| [[w:Dodge D series|Dodge D/W series]] (1967-1973), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1974-1977), [[w:Plymouth Trail Duster|Plymouth Trail Duster]] (1974-1976), [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1980), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1975-1976, 1980),&lt;br&gt; [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1984-1987), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1984-1987), [[w:Dodge Diplomat|Dodge Diplomat]] (1984-1987), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]] ('84-'87), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1987-1995), [[w:Dodge Caravan|Dodge Grand Caravan]] (1987-1995), [[w:Chrysler minivans (S)#Cargo van|Dodge Extended Mini Ram Van]] (1987-1988), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1990-1995),&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1996-'09)
| Originally opened to build trucks as the Missouri Truck Assembly Plant. In 1980, the plant was idled. Plant was reopened in 1983 to build the rwd, M-body sedans, which were moved from Windsor, ON, Canada so that Windsor could be converted to build minivans. Plant was renamed St. Louis II Assembly. During 1987, the M-body sedans were moved to AMC's plant in Kenosha, WI so that St. Louis North could be converted to build the new LWB minivans. For the first 2 generations of Chrysler minivans, St. Louis North only built the LWB models. For 1996, minivan production moved to St. Louis South and the North plant was converted to build full-size pickups. Closed on July 10, 2009. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1970-1978),&lt;br&gt; 6 (1968-1969),&lt;br&gt; 9 (1961-1967)
| Tecumseh Road Truck Assembly / Windsor Truck Assembly Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1916,&lt;br&gt; 1925 (became part of Chrysler)
| 1978
| Maxwell (1924-1925),&lt;br&gt; Chrysler (1924-1929),&lt;br&gt; Dodge Trucks (1931-1960), &lt;br&gt; Dodge D-Series Trucks: &lt;br&gt; D100 (1961), D200 (1964), W200 (1965), W100 (1966), D100/W100 (1967), D200 (1970), D500 (1968),&lt;br&gt; D500/D600/D700/D800 &lt;br&gt; medium-duty trucks (1970-1972), D500/D600/W600/D700/D800 medium-duty trucks (1974-1977),&lt;br&gt; D100 (1978),&lt;br&gt; Fargo Trucks (1936-1972)
| Located at 300 Tecumseh Road East. Was originally Windsor Plant 1. Became part of Chrysler upon its founding in 1925. Was previously a Maxwell-Chalmers plant and was originally Maxwell's Canadian plant from 1916. Switched to building trucks in 1931 after Plant 3 opened in 1929. Closed in 1978. Became the Imperial Quality Assurance Centre from 1980-1983, doing extra quality control on the 1981-1983 Imperial built at the Windsor Assembly Plant (the car plant - Plant 3) about 1.5 miles east of Plant 1. The Imperial Quality Assurance Centre closed in 1983 when the Imperial was discontinued. Subsequently demolished. Is now the Plaza 300 shopping mall.
|-
|
| Tipton Transmission Plant
| [[w:Tipton, Indiana|Tipton, Indiana]]
| [[w:United States|United States]]
| 2014
| 2023
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, SI-EVT transmission
| Located at 5880 W. State Road 28. Originally, the plant was supposed to be a [[w:Getrag|Getrag]] plant focused on supplying Chrysler with dual-clutch transmissions. Chrysler withdrew from the deal in 2008 after a dispute over financing and sued Getrag. The 80% completed facility then sat dormant until Chrysler Group purchased the facility in February 2013 and completed construction. Production began in April 2014 with the ZF-designed 9-speed auto. transmission, built under license from ZF. Production ended in June 2023 and 9-speed production was consolidated into Indiana Transmission Plant I in Kokomo, IN. The SI-EVT transmission for the Pacifica Hybrid was moved to Kokomo Transmission Plant. Sold to IRH Manufacturing LLC in September 2024 to make solar cells.
|-
| L (1989-2001),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Parkway|Toledo Assembly #1 Plant]] - Jeep Parkway plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2001 (ended final assembly), 2006 (ended body assembly)
| [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]] (1984-01), [[w:Jeep Cherokee (XJ)#Wagoneer|Jeep Wagoneer (XJ)]] (1984-90), [[w:Jeep Comanche|Jeep Comanche]] (1986-1992),&lt;br&gt; Bodies for vehicles made at Stickney Ave. plant &lt;br&gt; 
Models only made before Chrysler takeover: Willys Aero (1952-1955), Kaiser Manhattan (1954-1955), Jeep CJ (1946-1986), Jeep DJ, Jeep Jeepster (1948-1950), Jeep Jeepster Commando (1967-1971), Jeep Commando (1972-1973), Willys Jeep Station Wagon (1946-1964), Willys Jeep Truck (1947-1965), Jeep Gladiator (SJ) (1963-1971), Jeep J-series pickup, Jeep Wagoneer [SJ] (1963-1981), Jeep Cherokee [SJ] (1974-1981), Jeep Forward Control [FC] (1957-1965), Jeep FJ Fleetvan (1961-1975)
| Located at 1000 Jeep Parkway. The John North Willys-owned Overland Automobile Co. purchased the plant in 1909 from Pope-Toledo, another early automaker. Overland Automobile Co. became Willys-Overland in 1912. Began building Jeeps in the 1940s. This was the original Jeep assembly plant. Willys-Overland was bought by Kaiser in 1953. Kaiser then sold its Willow Run plant in Ypsilanti, MI to GM and moved its production to the Willys plant in Toledo, OH. Kaiser and Willys production in the US ended in 1955 and Toledo focused on Jeep production going forward. Kaiser Jeep was sold to AMC in 1970. Became part of Chrysler in the 1987 buyout of AMC. Final assembly ended in 2001 when the XJ Cherokee ended production but painted body production continued until June 30, 2006, when the TJ Wrangler ended production. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant a few miles away.  The Administration Building, used from 1915 through 1974, was imploded on April 14, 1979. A third of the plant was demolished in 2002 after final assembly ended including the Jeep Museum. The remainder was demolished in 2006-2007 after body production ended. One of the three large brick smokestacks was preserved and was dedicated in 2013 to the plant's history and workforce. A bronze plaque was mounted next to the smokestack, which still says &quot;Overland&quot; on it. Over 11 million vehicles were produced at the site, including military Jeeps during World War II. The site was sold to the Toledo-Lucas County Port Authority in 2010. The site has been redeveloped into the Overland Industrial Park. Dana Inc. and Detroit Manufacturing Systems are among the tenants in the Overland Industrial Park and those plants supply the current Jeep plants elsewhere in Toledo. All-Phase Electric Supply Co. is another tenant. 
|-
| P (1989-2006),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Stickney|Toledo Assembly #2 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2006 
| [[w:Jeep Wagoneer (SJ)#1984: SJ and XJ|Jeep Grand Wagoneer (SJ)]]&lt;br&gt; (1984-1991),&lt;br&gt; [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1993-95), [[w:Jeep Wrangler (TJ)|Jeep Wrangler (TJ)]] (1997-06)
Models only made before Chrysler takeover:&lt;br&gt; Jeep Wagoneer [SJ] (1981-1983), Jeep Cherokee [SJ] (1981-1983)
| Located at 4000 Stickney Ave. Originally opened in 1942 by the Electric Auto-Lite Co., a maker of spark plugs. Sold to Kaiser-Jeep in 1964, which used it as a machining and engine plant until 1981, when AMC converted it for vehicle production. AMC had taken over Kaiser Jeep in 1970. AMC built the SJ Wagoneer and Cherokee at the Stickney Ave. plant. Body assembly was done at an SJ- or later,  Wrangler-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant. Became part of Chrysler in the 1987 buyout of AMC. Production ended in 2006 with the end of the TJ Wrangler. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant built on the site of the old Stickney Ave. plant. 
|-
| W (1994-1996)
| [[w:Toledo Complex#Stickney|Toledo Assembly #3 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1993
| 1996 
| [[w:Dodge Dakota#First generation (1987–1996)|Dodge Dakota]] (1994-1996)
| Body assembly was done at a Dakota-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant.
|-
|
| Toluca Engine Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2002
| [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Closed in 2002
|-
|
| Toluca Transmission Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2001
| Automatic Transmissions for fwd cars
| Closed in 2001
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine North Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 1952
| 2022
| [[w:Chrysler B engine|Chrysler B V8 engine]],&lt;br&gt; [[w:Chrysler B engine#RB engines|Chrysler RB V8 engine]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]],&lt;br&gt; [[w:Volkswagen EA827 engine#1.7|VW 1.7L EA827 I4 engine]] (adding Chrysler parts to already built VW engines made in W. Germany),&lt;br&gt; [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L/2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|1.8L, 2.0L I4 &quot;Neon engine&quot;]], [[w:Chrysler 3.3 &amp; 3.8 engines|3.3L/3.8L OHV V6]], [[w:Chrysler SOHC V6 engine|3.5L/3.2L/4.0L SOHC V6]], [[w:Chrysler Pentastar engine|3.2L/3.6L Pentastar V6 engine]], [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4]],&lt;br&gt; Engine components,&lt;br&gt; Air raid sirens
| Located at 2000 Van Horn Road. Trenton North began production in fall 1952 and was expanded in 1964, 1967, 1969, 1976, and 1977. At first, Trenton North began by making water pumps and air raid sirens but engines quickly followed. Trenton Engine North was Chrysler’s first dedicated engine factory in the US, separate from the assembly plants. On September 29, 1978, V8 production ended. Trenton North added Chrysler parts such as the intake and exhaust manifolds, water pump, ignition system and other major parts to already built VW 1.7L EA827 I4 engines imported from Salzgitter, W. Germany for use in the Omni/Horizon. Production of the 3.5L V6 moved to Kenosha Engine in 2003. Trenton North was idled in May 2011 when the 3.8 V6 ended production following the end of 3.3 &amp; 4.0 V6 engine production in 2010. Chrysler then announced in June 2011 it would use a fifth of the plant to make components for the Pentastar V6 being made at Trenton South. In January 2012, Trenton North began producing the 3.6L Pentastar V6 engine. Chrysler then installed a flexible production line that could build both the Pentastar V6 and the Tigershark I4. In May 2013, Trenton began producing the 3.2L Pentastar V6. Tigershark I4 production began late in 3rd quarter 2013. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production. Trenton North has been repurposed for warehousing and other non-manufacturing opportunities. 
|-
|
| [[w:Twinsburg Stamping|Twinsburg Stamping]]
| [[w:Twinsburg, Ohio|Twinsburg, Ohio]]
| [[w:United States|United States]]
| 1957
| 2010
| Stampings and assemblies
| Located at 2000 East Aurora Road. Opened in August 1957. Closed July 31, 2010. Sold in 2011. Demolished in 2012-2013. Now the Cornerstone Business Park.
|-
|
| Vernor Tool &amp; Die plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1983
| Tooling &amp; Dies
| Located at 12026 E Vernor Highway. Operations moved to Mount Elliott Tool and Die. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
|
| Vernor Trim plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1970's
| Trim
| Located at 12025 E Vernor Highway. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
| 4 (1960-1961),&lt;br&gt; 7 (1959)
| Warren Avenue Plant
| [[w:Dearborn, Michigan|Dearborn, Michigan]]
| [[w:United States|United States]]
| 1927,&lt;br&gt; 1950 (opened as part of Chrysler)
| 1960s
| DeSoto bodies (1950-1958), DeSoto engines &lt;br&gt; (1951-1958),&lt;br&gt; [[w:Imperial (automobile)#Second generation (1957–1966)|Imperial]] (1959-1961)
| Located at 8505 West Warren Avenue. This was previously the factory of Paige and Graham-Paige. Chrysler leased half the plant in 1941 to use for military production. Chrysler produced aircraft components for the [[w:Martin B-26 Marauder|B-26 Marauder]] (nose and center fuselage sections) and the [[w:Boeing B-29 Superfortress|B-29 Superfortress]] (the pressurized nose section, wing leading edges, and engine cowlings). The B-29 had a nose so large that trenches had to be dug in the floor and some of the bracing for the plant’s roof girders had to be removed to accommodate the aircraft. Chrysler bought the plant in 1947. Began building bodies for DeSoto in August 1950. Engine production began in 1951. Production of the Imperial brand moved here from the Jefferson Ave. plant in Detroit for 1959 in an attempt to give the Imperial brand its own, exclusive factory however sales weren't high enough to support its own plant so Imperial production moved back to the Jefferson Ave. plant in Detroit for 1962. Production of small parts followed for a few years as did export operations. The plant was later sold. Most of the plant has seen been demolished but the front building facing on Warren Ave. is still there and is now used as Corporate HQ by Shatila Food Products. The DeSoto logo, featuring a stylized image of Hernando de Soto, can still be seen at the top of the building above the front door. 
|-
| T (1970-), 2 (1966-1969),&lt;br&gt; 
| Warren Truck #2 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1966
| 1975
| Dodge heavy-duty trucks
| Located at 6600 East 9 Mile Road, at the corner of Sherwood Ave and East 9 Mile Road. Closed in 1975 when Dodge exited the heavy-duty truck market. Site now belongs to Sundance Beverage Co., the parent of Everfresh Juice Co.
|-
| V (1971-1979)
| Warren Truck (Compact) #3 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1970
| 1979
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1979),&lt;br&gt;[[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1974-1979)
| Located on Hoover Road between 8 Mile Road and 9 Mile Road. 
|-
| 
| Windsor Engine Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1938
| 1980
| [[w:Chrysler flathead engine#Straight-6|Chrysler flathead inline 6]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], &lt;br&gt;[[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Was originally Windsor Plant 2. Located just to the south of Windsor Plant 3, the current minivan factory. Closed in August 1980. Built over 8 million engines. Windsor Assembly Plant (Plant 3) expanded onto the site of the old engine plant when it was being renovated for minivan production. 
|- 
|
| [[w:Detroit Assembly#LaSalle Factory/DeSoto Factory|Wyoming Ave. Assembly (DeSoto Wyoming Ave. plant)]] / Wyoming Export Plant
| [[w:Detroit|Detroit]], [[w:Michigan|Michigan]]
| United States
| 1936
| 1958 (Vehicle prod.),&lt;br&gt; 1980 (export operations)
| [[w:DeSoto (automobile)|DeSoto]] (1937-1958),&lt;br&gt; CKD Export (1960-1980)
| Located at 6000 Wyoming Avenue. Originally built to produce Liberty aircraft engines in World War I, opening in 1917. In 1919, was taken over by Saxon Motor Co., owned by Hugh Chalmers of Chalmers Motor Co. GM bought the plant in 1926 and built the LaSalle there from 1927-1933. GM sold Wyoming Assembly to Chrysler in 1934, which then used it to build its DeSoto brand. Became DeSoto's home plant. During World War II, Chrysler built wing center sections for the [[w:Curtiss SB2C Helldiver|Curtiss SB2C Helldiver]] at the Wyoming Ave. plant. For 1959, DeSoto's entry level model, the Firesweep, was moved to the Dodge plant in Hamtramck and DeSoto's other, higher end models were moved to Chrysler's Jefferson Ave. plant in Detroit. This was done so that bodies and final assembly would be done in either a single facility or a pair of connected facilities. This was part of Chrysler's move to unibody construction for 1960 for all cars except Imperial. After the DeSoto brand was discontinued in late 1960, became Wyoming Export plant which was used to prepare vehicles for export. Plant closed in 1980. Plant was demolished in 1992. Site is now occupied by Comprehensive Logistics Inc.
|}

==Non-Chrysler FCA/Stellantis Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| [[w:Fiat Cassino Plant|Cassino Plant]]
| [[w:Piedimonte San Germano|Piedimonte San Germano]], [[w:Province of Frosinone|Province of Frosinone]]
| [[w:Italy|Italy]]
| 1972
| 
| Past Chrysler Group models: [[w:Lancia Delta#|Chrysler Delta]] (UK/Ireland)
| Fiat plant. 
|-
| 3
| [[w:Alfa Romeo Pomigliano d'Arco plant|Pomigliano d'Arco plant]] (Giambattista Vico plant)
| [[w:Pomigliano d'Arco|Pomigliano d'Arco]], [[w:Metropolitan City of Naples|Metropolitan City of Naples]]
| [[w:Italy|Italy]]
| 1972
|
| Past Chrysler Group models: [[w:Dodge Hornet|Dodge Hornet]] (2023-2025). Related models:&lt;br&gt; [[w:Alfa Romeo Tonale|Alfa Romeo Tonale]] (2023-)
| Originally, an Alfa Romeo plant. Oriiginally owned by Construction Industry Neapolitan Vehicles Alfa Romeo - Alfasud S.p.A., a joint venture between Alfa Romeo (88%), Finmeccanica (10%), and IRI (2%). In 1982, Alfasud S.p.A. was renamed Inca Investments. Alfa Romeo was taken over by Fiat in 1986. Fiat merged with Chrysler to form Fiat Chrysler Automobiles (FCA) in 2014. FCA merged with PSA Group to form Stellantis in 2021. 
|}

==Non-Chrysler Group DaimlerChrysler/Daimler AG Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 5
| Mercedes-Benz Plant Düsseldorf
| [[w:Düsseldorf|Düsseldorf]], [[w:North Rhine-Westphalia|North Rhine-Westphalia]]
| [[w:Germany|Germany]]
| 1962
|
| [[w:Dodge Sprinter|Dodge Sprinter]] (2003-2009)
| Mercedes-Benz Plant. 
|-
| 9
| Mercedes-Benz Plant Ludwigsfelde
| [[w:Ludwigsfelde|Ludwigsfelde]], [[w:Brandenburg|Brandenburg]]
| [[w:Germany|Germany]]
| 1991 (Mercedes prod. began)
|
| [[w:Dodge Sprinter#Second generation (2006–2018, NCV3)|Dodge Sprinter]] chassis cab (2007-2009)
| Mercedes-Benz Plant. Originally established in 1936 by Daimler-Benz to make airplane engines. The plant was bombed by the US in 1945. After the war ended, what remained of the factory was dismantled and taken to the Soviet Union as reparations. On February 1, 1991, Mercedes-Benz took a 25% stake in the Ludwigsfelde plant, which had previously belonged to East German truckmaker VEB Automobilwerke. It became a 100% owned subsidiary of Mercedes-Benz on January 1, 1994. Sprinter production began in 2006. 
|}

==Former partner factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Prod. for Chrysler began
! style=&quot;width:10px;&quot;|Prod. for Chrysler ended
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 6
| [[w:China Motor Corporation|China Motor Corporation]]
| [[w:Yangmei District|Yangmei District]], [[w:Taoyuan, Taiwan|Taoyuan]]
| [[w:Taiwan|Taiwan]]
| 2006
| 2007
| [[w:Chrysler Town &amp; Country (minivan)#Fourth generation (2001–2007)|Chrysler Town &amp; Country]]&lt;br&gt; (Taiwan: 2006-2007),&lt;br&gt; [[w:Dodge 1000|Dodge 1000]] (Mexico: 2007-'10)
| China Motor Corporation plant. Built for Chrysler under license by China Motor Corporation of Taiwan. Production began April 18, 2006. 
|-
|
| [[w:Carrozzeria Ghia|Carrozzeria Ghia]] 
| [[w:Turin|Turin]]
| [[w:Italy|Italy]]
| 1957
| 1965
| [[w:Imperial (automobile)#Imperial Crown (1955–1965)|Imperial Crown Limousine]] (1957-1965) modified, painted limousine bodies and interiors
| 132 Imperial Crown Limousines were built by Ghia under contract for Chrysler between 1957 and 1965. The 1957-1959 models were based on modified 2-door hardtops with the more rigid chassis from the convertible. The 1960-1965 models were based on 4-door models. Ghia lengthened the frame and modified the bodywork and interiors to create the limousines. After producion ended in 1965, Ghia sold the tooling to Barreiros of Spain, which built another 10 Imperial Crown Limousines. Barreiros had been 35% owned by Chrysler since 1963. That was increased to 77% in 1967 and 100% in 1969.  
|-
| U
| [[w:Hyundai Motor Company|Hyundai Motor Co.]] - [[w:List of Hyundai Motor Company manufacturing facilities#Ulsan Plant|Ulsan plant]]
| [[w:Ulsan|Ulsan]]
| [[w:South Korea|South Korea]]
| 2000
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge Atos|Dodge Atos]] (2001-2012),&lt;br&gt;  [[w:Dodge Verna|Dodge Verna]] (2004-06),&lt;br&gt;  [[w:Dodge Attitude#First generation (MC; 2006)|Dodge Attitude (MC)]] (2007-'11), [[w:Dodge Attitude#Second generation (RB; 2011)|Dodge Attitude (RB)]] (2012-'14), [[w:Dodge H100|Dodge H100 truck]],&lt;br&gt; [[w:Hyundai Starex#Second generation (TQ; 2007)|Dodge H100 Van/Wagon]]
| Rebadged Hyundai models sold as Dodges in Mexico. 
|-
| 
| [[w:Hyundai Motor India|Hyundai Motor India]]
| [[w:Chennai|Chennai]], [[w:Tamil Nadu|Tamil Nadu]]
| [[w:India|India]]
| 2011
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge i10|Dodge i10]] (2012-2014)
| Rebadged Hyundai model sold as a Dodge in Mexico. 
|-
| X
| [[w:Karmann|Karmann Osnabrück Assembly]]
| [[w:Osnabrück|Osnabrück]], [[w:Lower Saxony|Lower Saxony]]
| [[w:Germany|Germany]]
| 2003
| 2007
| [[w:Chrysler Crossfire|Chrysler Crossfire]] (2004-2008)
| Karmann plant. Built under contract for Chrysler. 
|-
| Y
| [[w:Magna Steyr|Magna Steyr]] / Steyr-Daimler-Puch - Chrysler Steyr Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1994
| 2010
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]]&lt;br&gt; (1995-2010),&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010), [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (2003-2007),&lt;br&gt; [[w:Chrysler 300#First generation (2005)|Chrysler 300C/300C Touring]] (2005-2010)
| Originally, a Steyr-Daimler-Puch plant. Magna International acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. Production of the Chrysler Voyager and Grand Voyager minivans moved from the Eurostar plant  next door to the main Magna Steyr plant for 2003. Chrysler minivan production in Austria ended on November 30, 2007. Built under contract for Chrysler. 
|-
| B
| [[w:Maserati|Maserati]] - [[w:Innocenti|Innocenti]] plant
| [[w:Lambrate|Lambrate district]], [[w:Milan|Milan]]
| [[w:Italy|Italy]]
| 1988
| 1990
| [[w:Chrysler TC by Maserati|Chrysler TC by Maserati]]&lt;br&gt; (1989-1991)
| Developed jointly by Chrysler and Maserati, the TC was built in Italy by Maserati at the Innocenti plant in Milan. Maserati and Innocenti were both owned by DeTomaso at the time. Chrysler bought a 5% stake in Maserati in 1984 and increased its stake to 15.6% in 1986. Production ended in 1990 due to low sales.
|-
| U
| Mitsubishi - Mizushima plant (Line 1)
| [[w:Kurashiki|Kurashiki]], [[w:Okayama Prefecture|Okayama Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| [[w:Plymouth Champ|Plymouth Champ]] (1981-1982), [[w:Plymouth Colt|Plymouth Colt]] (1983-1994), [[w:Dodge Colt|Dodge Colt]] (1981-1994), [[w:Dodge Colt|Dodge/Plymouth Colt]]&lt;br&gt; (Canada only: 1995),&lt;br&gt; [[w:Eagle Summit|Eagle Summit]]&lt;br&gt; (4-d: 1989-1990, 1993-1996,&lt;br&gt; 3-d: 1991-1992, 2-d: 1993-96), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1993-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1993-96)
| Mitsubishi Motors plant.
|-
| Z
| Mitsubishi - Okazaki plant
| [[w:Okazaki, Aichi|Okazaki]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1983
| 1996
| [[w:Plymouth Conquest|Plymouth Conquest]] (1984-86), [[w:Dodge Conquest|Dodge Conquest]] (1984-1986), [[w:Chrysler Conquest|Chrysler Conquest]] (1987-1989), [[w:Dodge Colt Vista#Colt Vista|Dodge Colt Vista]] (1984-1991), [[w:Plymouth Colt Vista#Colt Vista|Plymouth Colt Vista]] (1984-91), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1992-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1992-96) [[w:Eagle Vista#Vista Wagon|Eagle Vista Wagon]]&lt;br&gt; (Canada: 1989-1991)
| Mitsubishi Motors plant.
|-
| Y (Line 1)&lt;br&gt;/&lt;br /&gt;P (Line 2)
| Mitsubishi - &lt;br&gt; Ooe plant &lt;br&gt; a.k.a. &lt;br&gt; Nagoya #1&lt;br&gt;/&lt;br&gt;Nagoya #2
| Ooe-cho, [[w:Minato-ku, Nagoya|Minato ward]], [[w:Nagoya|Nagoya]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| VIN code Y:&lt;br&gt; [[w:Plymouth Sapporo|Plymouth Sapporo]] (1981-1983), [[w:Dodge Challenger#Second generation (1978–1983)|Dodge Challenger]] (1981-1983), [[w:Plymouth Arrow Truck#Chrysler variants|Plymouth Arrow Truck]] ('81-'82), [[w:Dodge Ram 50|Dodge Ram 50]] (1981-1984), [[w:Dodge Stealth|Dodge Stealth]] (1991-1996)
VIN code P:&lt;br&gt; [[w:Dodge Ram 50|Dodge Ram 50]] (1985-1986), [[w:Dodge Ram 50#North America|Dodge Ram 50]] (1987-1993)
| Mitsubishi Motors plant. Closed in 2001. Sold to Mitsubishi Heavy Industries and now used by its Aircraft, Defense &amp; Space Business Area.
|-
| J
| Mitsubishi -  Toyo Koki/Pajero Manufacturing Co., Ltd. plant
| [[w:Sakahogi, Gifu|Sakahogi]], [[w:Gifu Prefecture|Gifu Prefecture]] 
| [[w:Japan|Japan]]
| 1986
| 1989
| [[w:Dodge Raider|Dodge Raider]] (1987-1989)
|  Originally, a Toyo Koki Co. Ltd. plant. Opened in 1976. Built vehicles under contract for Mitsubishi. Mitsubishi Motors owned 35% of Toyo Koki and increased its stake to a majority in March 1995. The plant was then renamed Pajero Manufacturing Co., Ltd. in July 1995. In March 2003, Mitsubishi bought all the remaining shares in Pajero Manufacturing Co., Ltd., making it a wholly owned subsidiary. Closed in 2021. Sold to Daio Paper in 2022.
|-
| H (Attitude), 9 (1200)
| [[w:Mitsubishi Motors (Thailand)|Mitsubishi Motors Thailand]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 2014
| 2024
| [[w:Dodge Attitude#Third generation (A10; 2015)|Dodge Attitude]]&lt;br&gt; (Mexico: 2015-2024),&lt;br&gt; [[w:Mitsubishi Triton#Fifth generation (KJ/KK/KL; 2014)|Ram 1200]]&lt;br&gt; (Middle East: 2017-2019)
| Mitsubishi Motors plant.  
|-
| ?
| [[w:MMC Automotriz|MMC Automotriz]]
| [[w:Barcelona, Venezuela|Barcelona]], [[w:Anzoátegui|Anzoátegui state]]
| [[w:Venezuela|Venezuela]]
| 2002
| 2009
| [[w:Dodge Brisa|Dodge Brisa]]&lt;br&gt; ([[w:Hyundai Accent#First generation (X3; 1994)|2002-2005]]), ([[w:Hyundai Getz|2006-2009]])
| MMC Automotriz plant. Originally, MMC Automotriz was 49% owned by Consorcio Inversionista Fabril S.A. (CIF) of Venezuela and 42% owned by Nissho Iwai Corp. The remaining 9% of the company was owned by the Japan International Development Organization Ltd., a partnership between the government-financed Overseas Economic Cooperation Fund and 98 private companies. Nissho Iwai merged with Nichimen Corp. in 2004 to form Sojitz Corp. Sojitz later increased its stake in MMC Automotriz to 98%, with the other 2% still held by CIF. MMC Automotriz was sold to the the Sylca Group (also known as Yammine Group) in 2015. MMC Automotriz produced Mitsubishi vehicles and from 1996-2012, also produced Hyundai vehicles. The Dodge Brisa was produced for DaimlerChrysler as part of its cooperation with [[w:Hyundai Motor Company|Hyundai]]. MMC Automotriz also produced Mitsubishi Fuso trucks. 
|-
| G
| [[w:Mitsubishi Motors (Thailand)|MMC Sittipol Co., Ltd.]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 1988
| 1992
| [[w:Plymouth Colt#Fifth generation (1985–1988)|Plymouth Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Dodge Colt#Fifth generation (1985–1988)|Dodge Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Eagle Vista|Eagle Vista]] (Canada: 1988-92)
| Mitsubishi Motors plant. MMC Sittipol is the predecessor company of Mitsubishi Motors (Thailand). These were the first vehicle exports from Thailand. 
|-
| 2
| [[w:Renault|Renault]] - [[w:Maubeuge Construction Automobile|Maubeuge plant]]
| [[w:Maubeuge|Maubeuge]]
| [[w:France|France]]
| 1987
| 1989
| [[w:Renault Medallion|Renault Medallion]] (1988),&lt;br&gt; [[w:Eagle Medallion|Eagle Medallion]] (1989)
| Renault plant. The Medallion was sold through Chrysler's Jeep-Eagle dealer network as a legacy of Chrysler's takeover of AMC from Renault.
|-
| 8,&lt;br&gt; 0
| [[w:Soueast|Soueast]]
| [[w:Fuzhou|Fuzhou]], [[w:Fujian|Fujian province]]
| [[w:China|China]]
| 2008
| 2010
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager]],&lt;br&gt; [[w:Dodge Caravan#Fourth generation (2001–2007)|Dodge Caravan]]
| South East (Fujian) Motor Co., Ltd. plant. Built for Chrysler under license by South East (Fujian) Motor Co., Ltd.
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{{tmbox|type=notice|text=This is a sandbox page, a place for experimenting with Wikibooks.}}

This is a history of Chrysler factories that are being or have been used to produce cars, vans, SUVs, trucks, and automobile components. 

For '''Chrysler brand''' only: Plant code in 1955-1957 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number.

For '''cars''': Plant code in 1958 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number. Plant code was the number in the 4th position of the 10-digit serial number for 1959-1965. Plant code was the number in the 7th position of the 13-digit serial number for 1966-1967. Plant code was the letter in the 7th position of the 13-digit serial number for 1968-1980.

Canadian-built cars had the plant code as the number in the 5th position of the 11-digit serial number for 1965.

For '''trucks''': The first digit of the 7-digit sequence number (4th position overall of the 10-digit serial number) indicated which plant built the truck for 1967-1968. Plant code was the number in the 4th position of the 10-digit serial number for 1969. Plant code was the letter in the 7th position of the 13-digit serial number for 1970-1980.

Canadian-built models used a different system for VINs until 1968, when Chrysler of Canada adopted the same system as the US. Plant code for trucks was the number in the 5th position of the 10-digit serial number for 1961-1967.

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

For '''AMC passenger cars from 1966-1967''': Plant code is indicated by the letter in the 3rd position of the serial number. If the 3rd letter is a K, then it was made in Kenosha, WI. If the 3rd letter is a B, then it was made in Brampton, ON, Canada. 

For '''AMC passenger cars from 1968 through 1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. 1-6 is Kenosha, WI and 7-9 is Brampton, ON, Canada.

For '''Canadian-built Jeeps built by AMC Canada for 1979-1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. It was made in Canada if it's either an 8 (1979) or a 7 (1980). All other Jeeps from 1980 and earlier were made in Toledo. 

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

==Current factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| D (1968-),&lt;br&gt; 4 (1966-1967)
| [[w:Belvidere Assembly|Belvidere Assembly]] 
| [[w:Belvidere, Illinois|Belvidere, Illinois]]
| [[w:United States|United States]]
| 1965
| Feb. 2023
| 
| Located at 3000 West Chrysler Drive. '''Belvidere Satellite Stamping Plant''' adjoins the main assembly plant. Began production on July 7, 1965. The first vehicle produced was a 1966 Plymouth Fury four-door. In 1972, the Chrysler Town and Country station wagon was added to the Belvidere plant. In 1977, the plant was converted to build front-wheel drive subcompacts. Production of the Dodge Omni and Plymouth Horizon began on December 5, 1977. All L-body derivatives were made at Belvidere through 1987. In 1987, Belvidere was converted to build Chrysler's midsize, fwd C-body sedans. Belvidere then switched back to building small cars and began production of the Neon on November 10, 1993. Belvidere built the last Plymouth, a silver 2001 Neon LX on June 28, 2001. Neon production ended in September 2005. Dodge Caliber began production in January 2006, followed by Jeep Compass in June 2006 and Jeep Patriot in December 2006. Caliber ended production on December 19, 2011. Dodge Dart began production on April 30, 2012, and ended on October 4, 2016. Compass and Patriot production ended on December 23, 2016. Jeep Cherokee production began on June 1, 2017 and ended on February 28, 2023. Belvidere was then idled. &lt;br&gt; Past models: [[w:Plymouth Fury|Plymouth Fury]] (1966-1974), [[w:Plymouth Gran Fury#1975–1977|Plymouth Gran Fury]] (1975-1977), [[w:Dodge Monaco|Dodge Monaco]] (1966-1976), [[w:Dodge Royal Monaco#1977 (Royal Monaco)|Dodge Royal Monaco]] (1977), [[w:Dodge Polara|Dodge Polara]] (1966-1973), [[w:Chrysler Newport|Chrysler Newport]] (1977), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1973-1977), [[w:Dodge Omni|Dodge Omni]] (1978-1987), [[w:Plymouth Horizon|Plymouth Horizon]] (1978-1987), [[w:Dodge Omni 024|Dodge Omni 024]] (1979-1980), [[w:Plymouth Horizon TC3|Plymouth Horizon TC3]] (1979-1980), [[w:Dodge Omni 024|Dodge 024]] (1981-1982), [[w:Plymouth Horizon TC3|Plymouth TC3]] (1981-1982), [[w:Dodge Charger (1981)|Dodge Charger]] (1983-1987), [[w:Plymouth Turismo|Plymouth Turismo]] (1983-1987), [[w:Dodge Rampage|Dodge Rampage]] (1982-1984),  [[w:Plymouth Scamp|Plymouth Scamp]] (1983), [[w:Dodge Dynasty|Dodge Dynasty]] (1988-1993), [[w:Chrysler Dynasty|Chrysler Dynasty]] (Canada: 1988-1993), [[w:Chrysler New Yorker#1988–1993|Chrysler New Yorker]] (1988-1993), [[w:Chrysler New Yorker Fifth Avenue#1990–1993: New Yorker Fifth Avenue|Chrysler New Yorker Fifth Avenue]] (1990-1993), [[w:Chrysler Imperial#1990–1993|Chrysler Imperial]] (1990-1993), [[w:Dodge Neon|Dodge Neon]] (1995-2005), [[w:Plymouth Neon|Plymouth Neon]] (1995-2001), Chrysler Neon (Canada: 2000-02),&lt;br&gt; Dodge SX 2.0 (Canada: 2003-05),&lt;br&gt; [[w:Dodge Caliber|Dodge Caliber]] (2007-2012), [[w:Jeep Compass#First generation (MK49; 2006)|Jeep Compass]] (2007-2017), [[w:Jeep Patriot|Jeep Patriot]] (2007-2017), [[w:Dodge Dart (PF)|Dodge Dart]] (2013-2016), [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2017-2023)
|-
| H (1989-),&lt;br&gt; A (1988)
| [[w:Brampton Assembly|Brampton Assembly]] (Formerly Bramalea Assembly)
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1987
| Dec. 2023
| 
| Located at 2000 Williams Parkway East. Factory was built by AMC and Renault. The plant was acquired by Chrysler as part of its takeover of AMC. Began production on September 28, 1987. Plant was originally known as Bramalea Assembly. Plant was renamed Brampton Assembly in 1992 after Chrysler closed and sold the old AMC plant on Kennedy Road in Brampton. The attached '''Brampton Satellite Stamping Plant''' was added in December 1991 and was built for the launch of the Chrysler LH platform. On December 17, 1991, Eagle Premier and Dodge Monaco production ended. Production of the Chrysler LH platform cars began in June 1992. Production switched to the rear-wheel drive Chrysler LX platform cars in January 2004. The Chrysler 300 was also built for export to mainland Europe as the Lancia Thema from 2011-2014. Production ended on December 22, 2023 and the plant was idled. Last vehicle off the line was a Pitch-Black 2023 Dodge Challenger Demon 170. 7,147,888 vehicles were produced through 2023.&lt;br&gt; Past models: [[w:Eagle Premier|Eagle Premier]] (1988-1992), [[w:Dodge Monaco#Fifth generation (1990–1992)|Dodge Monaco]] (1990-1992),&lt;br&gt; [[w:Dodge Intrepid|Dodge Intrepid]] (1993-2004),&lt;br&gt; [[w:Chrysler Intrepid|Chrysler Intrepid]] (Canada: 1993-2004),&lt;br&gt; [[w:Chrysler Concorde|Chrysler Concorde]] (1993-2004),&lt;br&gt; [[w:Eagle Vision|Eagle Vision]] (1993-1997),&lt;br&gt; [[w:Chrysler New Yorker#1994–1996|Chrysler New Yorker]] (1994-1996),&lt;br&gt; [[w:Chrysler LHS|Chrysler LHS]] (1994-1997, 1999-2001),&lt;br&gt; [[w:Chrysler 300M|Chrysler 300M]] (1999-2004),&lt;br&gt; [[w:Chrysler 300|Chrysler 300]] (2005-2023),&lt;br&gt; [[w:Dodge Magnum#Chrysler LX platform (2005–2008)|Dodge Magnum]] (2005-2008),&lt;br&gt; [[w:Dodge Charger (2006)|Dodge Charger]] (2006-2023),&lt;br&gt; [[w:Dodge Challenger (2008)|Dodge Challenger]] (2008-2023),&lt;br&gt; [[w:Lancia Thema#Second generation (2011–2014)|Lancia Thema]] (For export: 2011-2014)
|-
| C
| [[w:Jefferson North Assembly|Detroit Assembly Complex – Jefferson]] / Jefferson North Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]] (1993-), [[w:Dodge Durango#WD|Dodge Durango]] (2011-)
| Located at 2101 Conner Street. Jefferson North replaced the previous Jefferson Assembly plant that closed in 1990 and was demolished in 1991. Jefferson North is across the street from the old Jefferson Assembly plant, on the north side of Jefferson Ave. Jefferson North was built on the site of Chrysler's old Kercheval Avenue Body Plant, which, in 1955, had been connected to the old Jefferson Assembly plant by a bridge crossing over Jefferson Ave. The Jefferson North plant site also absorbed what had been the axle plant and service parts buildings of the old [[w:Hudson Motor Car Company|Hudson]] plant, which were located at Connor St. and Vernor Hwy. The main Hudson plant is now the parking lot on the corner of Jefferson Ave. and Connor St. After 2017, the Jefferson North plant complex also absorbed the site of the former Budd Co. body- &amp; parts-making plant at Connor St. and Charlevoix Avenue, which had previously belonged to the Liberty Motor Car Co. The Budd plant extended north on Connor from Charlevoix most of the way to Mack Ave. Since the nearby Mack Ave. Assembly Plant began operations in 2021, the 2 plants have operated as the Detroit Assembly Complex. Jefferson North began production on January 14, 1992 with the original Grand Cherokee (ZJ). The 2nd gen. Grand Cherokee began production on July 17, 1998. The 3rd gen. Grand Cherokee began production on July 26, 2004 followed by the Jeep Commander on July 18, 2005. The 4th gen. Grand Cherokee began production on May 10, 2010 followed by the Dodge Durango on December 14, 2010. The 5th gen. Grand Cherokee began production in May 2022. The 4xe plug-in hybrid version of the Grand Cherokee began production at Jefferson North in March 2023. On Aug. 13, 2013, Jefferson North built its 5 millionth vehicle, a silver 2014 Grand Cherokee Overland. On May 25, 2016, Jefferson North built its 6 millionth vehicle, a Granite Crystal (silvery gray) 2016 Grand Cherokee 75th anniversary Edition.&lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010),&lt;br&gt; [[w:Jeep Grand Cherokee (WK2)#Grand Cherokee WK (2022)|Jeep Grand Cherokee WK]] (2022)&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2023-2025)
|-
| 8
| [[w:Detroit Assembly Complex – Mack|Detroit Assembly Complex – Mack]] / Mack Ave. Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2021
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2021-), [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]] (2022-)
| Located at 4000 St. Jean Avenue. The Mack Avenue Assembly Plant is built on the site of the former Mack Ave. Engine Plants I &amp; II, the New Mack Assembly Plant, &amp; the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The two plants that comprised the former Mack Avenue Engine Complex were converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North plants have operated as the Detroit Assembly Complex since 2021. Production began in March 2021 with the 3-row Grand Cherokee L. The 2-row Grand Cherokee followed in the fall of 2021. The 4xe plug-in hybrid version of the Grand Cherokee began production at Mack Ave. in August 2022. &lt;br&gt; Past models: [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2022-2025)
|-
|
| [[w:Dundee Engine Plant|Dundee Engine Plant]] (Formerly [[W:Global Engine Manufacturing Alliance|GEMA]]) 
| [[w:Dundee, Michigan|Dundee, Michigan]]
| [[w:United States|United States]]
| 2005
| 
| [[w:Prince engine#1.6-litre turbocharged (PSA)|1.6L turbo PSA/BMW Prince EP6CDTX Hybrid I4]],&lt;br&gt; [[w:FCA Global Medium Engine|2.0L turbo Hurricane4 EVO I4 (GME-T4 EVO)]],&lt;br&gt; Engine components
| Located at 5800 North Ann Arbor Road. Plant was originally part of the the Global Engine Manufacturing Alliance, a 3-way engine manufacturing joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. The North Plant launched in October 2005, followed by the South Plant in November 2006. The plant began with production of the I4 World Gasoline Engine, which was developed by the Global Engine Alliance, a 3-way engine development joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. Originally, the plant was envisioned as supplying engines to Mitsubishi and Hyundai as well as Chrysler however the plant only ever supplied engines to Chrysler. Mitsubishi and Hyundai each set up engine production at their own engine plants. On August 31, 2009, Chrysler bought Mitsubishi’s and Hyundai’s stakes in the group and now wholly owns both the Global Engine Manufacturing Alliance and its primary engine-building plant in Dundee, Michigan. In January 2012, the plant was renamed Dundee Engine Plant. Production of the Fiat 1.4-liter FIRE I4 engine began in November 2010. The Tigershark engine, an evolution of the World engine, began production in 2012 for the 2.0L and in May 2013 for the 2.4L. Tigershark engine production ended on March 16, 2023. In November 2019, the Pentastar V6 began production at Dundee, moving from the Mack Ave. Engine Plant in Detroit, which was to be converted into a vehicle assembly plant. The Pentastar V6 ended production at Dundee on August 18, 2023. In 2025, Dundee began making a North American-spec version of the European-developed Prince engine for the 2026 Jeep Cherokee Hybrid. &lt;br&gt; Past engines: [[w:World Gasoline Engine|1.8L/2.0L/2.4L/2.4L Turbo I4 World Gasoline Engine]], [[w:World Gasoline Engine#Tigershark|2.0L/2.4L I4 Tigershark Engine]], [[w:FIRE engine|Fiat 1.4L/1.4L Turbo FIRE MultiAir I4 engine]], [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]] 
|-
| 
| Etobicoke Casting Plant
| [[w:Etobicoke|Etobicoke District]], [[w:Toronto|Toronto]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1964
| 
| Aluminum die castings, Engine and Transmission Components
| Located at 15 Brown's Line. Etobicoke used to be a separate city but became part of Toronto in 1998. Factory was originally built in 1942 by the Canadian government and operated by Alcan Aluminum, which used it to make molds for military aircraft parts during World War II. It produced precision aircraft parts and other high quality aluminum castings. The aluminum foundry was purchased by Chrysler in April 1964 from Alcan Aluminum. The plant was expanded in 1965 and 1998. Etobicoke Casting is making oil pans for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee.
|-
|
| [[w:Indiana Transmission#Indiana Transmission I|Indiana Transmission Plant I]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1998
| 
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, Gear machining and final assembly of electric drive modules
| Located at 3660 North U.S. Highway 931. RFE transmission production ended in January 2025. More than 8 million RFE transmissions were produced at Indiana Transmission Plant I. Production of the nine-speed transmission began in May 2013. The 9-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. &lt;br&gt; Past transmissions:&lt;br&gt; [[w:Chrysler RFE transmission|Chrysler RFE 4-/5-/6-speed auto. trans.]]
|-
|
| [[w:Kokomo Casting|Kokomo Casting Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1965
| 
| Aluminum parts for automotive components, transmission and transaxle cases; engine block castings
| Located at 1001 East Boulevard. Kokomo Casting is the world’s largest die-cast facility. Plant was expanded in 1969, 1986, 1995 and 1997. Over 18 million four-speed transmission cases were made at Kokomo Casting from 1988 through July 2014. Kokomo Casting started making nine-speed transmission cases in 2013. Kokomo Casting is making engine blocks for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee. Kokomo Casting is making gearbox covers for the electric drive modules made at the Indiana Transmission Plant. 
|-
|
| [[w:Kokomo Engine Plant|Kokomo Engine Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2022 (as Kokomo Engine)
| 
| [[w:FCA Global Medium Engine|2.0L turbo GME-T4 I4]]
| Located at 3360 North U.S. Highway 931. Plant was previously known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Starting in 2020, the plant was converted to engine production. Engine production began in late February 2022. 
|-
|
| [[w:Kokomo Transmission|Kokomo Transmission Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1956
| 
| [[w:ZF 8HP transmission|850RE]] 8-speed auto. transmission,&lt;br&gt; [[w:ZF 8HP transmission|880RE]] 8-speed auto. transmission,&lt;br&gt; Machining of engine block castings and transmission components,&lt;br&gt; Machined components for the &lt;br&gt; 9-speed auto. transmission
| Located at 2401 South Reed Road. On October 9, 2020, Kokomo Transmission built its last 41TE 4-speed auto. transmission, assembling more than 17 million 4-speed auto. transmissions since production began in 1988. Kokomo Transmission began building 6-speed auto. transmissions in 2006. Production of the eight-speed automatic transmission began in September 2012. The 8-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. On August 8, 2023, Kokomo Transmission built its 6 millionth 8-speed transmission. Kokomo Transmission is machining gearbox covers for the electric drive modules made at the Indiana Transmission Plant. &lt;br&gt; Past transmissions: &lt;br&gt;[[w:TorqueFlite|TorqueFlite 3-/4-speed auto. trans.]],&lt;br&gt; [[w:Ultradrive|Ultradrive (TE/AE/LE/RLE/TES/TEA)&lt;br&gt; 4-/6-speed auto. trans.]],&lt;br&gt; [[w:ZF 8HP transmission|845RE]] 8-speed auto. transmission,&lt;br&gt; SI-EVT trans. (eFlite) for Pacifica Hybrid
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo North Engine Plant]] 
| [[w:Ramos Arizpe|Ramos Arizpe]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1981
| 
| [[w:Chrysler Hemi engine#Third generation: 2003–present|5.7L/6.4L/6.2L supercharged Hemi V8]], [[w:Stellantis Hurricane engine|Chrysler 3.0L twin-turbo Hurricane GME-T6 I6]]
|  Production began on May 8, 1981. Hemi V8 production began in June 2002 with the 5.7L. Production of the supercharged 6.2L Hellcat Hemi V8 began in the third quarter of 2014. Tigershark I4 production began in the first quarter of 2014. &lt;br&gt; Past engines: [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L, 2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|2.0L, 2.4L, 2.4L Turbo I4 &quot;Neon engine&quot;]],&lt;br&gt; [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4 engine]], [[w:Chrysler Hemi engine#6.1|6.1L Hemi V8]]
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo South Engine Plant]] 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]]
| Plant opened on October 29, 2010. The plant has produced over 6 million Pentastar V6 engines. &lt;br&gt; Past engines: Chrysler 3.6L Pentastar V6 engine (EH3) for Pacifica Plug-in Hybrid
|-
| 
| Saltillo Stamping Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1997
| 
| Stampings and assemblies including Body panels
| Part of the Saltillo Truck Assembly Complex.
|-
| G
| Saltillo Truck Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1995
| 
| [[w:Ram Heavy Duty (fifth generation)|Ram HD pickup &amp; chassis cab]] (2019-)
|  Production began in 1995. As of 2025, also known as Saltillo Truck Heavy Duty Plant. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1995-2012),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-2023),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram HD pickup &amp; chassis cab]] (2013-2018), [[w:Sterling Bullet|Sterling Truck Bullet]] (2008-2009)
|-
| 4
| Saltillo Truck Extension Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2025
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2025-)
| Also known as Saltillo Truck Light Duty Plant. 2 new buildings were constructed for the new light duty plant. Production began in May 2025. Initially, production was for export but production for the domestic Mexican market began in February 2026. The plant also includes a seat assembly line, the first Stellantis plant in North America to integrate this process into its own production chain.
|-
| E
| Saltillo Van Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2013
| 
| [[w:Ram ProMaster|Ram ProMaster]] (2014-),&lt;br&gt; [[w:Ram ProMaster#E-Ducato and Ram ProMaster EV (2024)|Ram ProMaster EV]] (2024-)
| Production started in July 2013. &lt;br&gt; Past models: [[w:Fiat Ducato|Fiat Ducato]] (Export to Brazil/Argentina: 2018-2022)
|-
| N
| [[w:Sterling Heights Assembly|Sterling Heights Assembly Plant]] 
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1984
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2019-)
| Located at 38111 Van Dyke Ave. The plant was originally built by the US Navy as a jet engine plant in 1953. It was called Naval Industrial Reserve Aircraft Plant and was owned by the US Navy. When the jet engine project was cancelled, the plant was transferred to the US Army, which contracted with Chrysler to build missiles at the plant, which was now known as the Michigan Ordinance Missile Plant. Chrysler began production of the PGM-11 Redstone missile at the Sterling Heights plant on September 27, 1954. The final Redstone was built in 1961. Chrysler also built the PGM-19 Jupiter missile at Sterling Heights from 1958-December 1960. Chrysler also built the first stage of the Saturn I rocket at the Sterling Heights plant. Chrysler vacated the Michigan Army Missile Plant at the end of 1969. Meanwhile, LTV Corp. (previously Ling-Temco-Vought) built the MGM-52 Lance missile at the Sterling Heights plant. The Army turned the plant over to the state of Michigan, which was then sold it to Volkswagen in 1980. VW converted the plant to automotive production and intended to make the Jetta there but VW's US sales declined and VW never ended up building anything there. VW then sold the plant to Chrysler in 1983. Chrysler LeBaron GTS and Dodge Lancer production began in September 1984 and ended on April 7, 1989. Shadow and Sundance production began on August 25, 1986 and ended on March 9, 1994. The Dodge Daytona was also moved from St. Louis to Sterling Heights in 1991 and was produced there through February 26, 1993. Chrysler then built a succession of midsize cars at Sterling Heights from June 1994. During Chrysler's bankruptcy in 2009, Sterling Heights Assembly was initially left behind in &quot;old Chrysler&quot; and was supposed to close by December 2010 but during 2010, &quot;new Chrysler&quot; changed its mind and bought the plant from &quot;old Chrysler&quot; for $20 million. The 2011 Chrysler 200 and Dodge Avenger sedans began production on December 6, 2010 followed by the Chrysler 200 Convertible in February 2011. The Chrysler 200 Convertible was also built for export to Europe as the Lancia Flavia from March 2012. The 2nd gen. Chrysler 200 sedan began production on March 14, 2014 and ended on December 2, 2016. The plant was then idled for a lengthy retooling to build body-on-frame pickups and began production of the new generation DT-series Ram 1500 in March 2018. &lt;br&gt; Past models: [[w:Dodge Lancer#1985–1989: Lancer|Dodge Lancer]] (1985-1989), [[w:Chrysler LeBaron#1985–1989 LeBaron GTS|Chrysler LeBaron GTS (H-body)]] (1985-1989), [[w:Plymouth Sundance|Plymouth Sundance]] (1987-1994), [[w:Dodge Shadow|Dodge Shadow]] (1987-1994), [[w:Dodge Daytona|Dodge Daytona]] (1992-1993), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1992-1993), [[w:Chrysler Cirrus|Chrysler Cirrus]] (1995-2000), [[w:Dodge Stratus|Dodge Stratus]] sedan (1995-2006), [[w:Plymouth Breeze|Plymouth Breeze]] (1996-2000), [[w:Chrysler Sebring|Chrysler Sebring]] sedan (2001-2010), [[w:Chrysler Sebring|Chrysler Sebring]]  convertible (2001-2006, 2008-2010), [[w:Dodge Avenger#Dodge Avenger sedan (2008–2014)|Dodge Avenger]] sedan (2008-2014), [[w:Chrysler 200|Chrysler 200]] sedan (2011-2017), [[w:Chrysler 200|Chrysler 200]] convertible (2011-2014), [[w:Chrysler 200#Lancia Flavia|Lancia Flavia]] (For export: 2012-2014)  
|-
| 
| Sterling Stamping Plant
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1965
| 
| Stampings and assemblies including hoods, roofs, liftgates, side apertures, fenders, and floorpans
| Located at 35777 Van Dyke Ave. This plant is a separate facility but is located next door to the Sterling Heights Assembly Plant. Sterling Stamping is the largest stamping plant in the world. Sterling Stamping supplies stampings to many Chrysler assembly plants, not only Sterling Heights Assembly. The first stampings were produced in January 1965. 
|-
| W
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo North Assembly Plant]] 
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]] (2018-)
| Located at 4400 Chrysler Drive. Toledo North first built the Jeep Liberty, which began in April 2001. Dodge Nitro production began in August 2006 and ended on December 16, 2011. The 2nd gen. Jeep Liberty began production in July 2007. Jeep Liberty ended production on August 16, 2012. Toledo North then closed for retooling to build the all-new 2014 Cherokee. The Jeep Cherokee began production at Toledo North on June 24, 2013 and ended on April 6, 2017. The Cherokee was then moved to Belvidere Assembly. Toledo North was then retooled to build the Jeep Wrangler, which began on November 15th, 2017. The 4xe plug-in hybrid version of the Wrangler began production in December 2020. &lt;br&gt; Past models: [[w:Jeep Liberty|Jeep Liberty]] (2002-2012), [[w:Dodge Nitro|Dodge Nitro]] (2007-2011),&lt;br&gt; [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2014-2017),&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler 4xe (JL)]] (2021-2025)
|-
| L
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo Supplier Park Plant]] (South plant)
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Gladiator (JT)|Jeep Gladiator]] (2020-)
| Located along Stickney Ave. The Toledo Supplier Park Plant is built on the site of the former Stickney Ave. plant that became part of Chrysler in 1987 as part of the acquisition of AMC. That plant was acquired by Kaiser Jeep in 1964 from Autolite and was built in 1942. The Toledo Supplier Park Plant includes body and chassis operations in partnership with Kuka and Hyundai Mobis, respectively. The paint shop was originally run in partnership with Magna but Chrysler took over the paint operation in the first quarter of 2011. The Toledo Supplier Park Plant began Wrangler production in August 2006. Wrangler production ended on April 27, 2018. Gladiator pickup production began in March 2019. &lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Wrangler (JK)|Jeep Wrangler (JK)]] (2007-2017),&lt;br&gt; [[w:Jeep Wrangler (JK)#2018 model year update|Jeep Wrangler JK]] (2018)
|-
| 
| [[w:Toledo Machining|Toledo Machining Plant]] 
| [[w:Perrysburg, Ohio|Perrysburg, Ohio]]
| [[w:United States|United States]]
| 1966
| 
| Steering Columns,&lt;br&gt; Torque Converters for 8-spd. rwd and 9-spd. fwd auto. transmissions
| Located at 8000 Chrysler Drive. Production began in 1966 and the plant was expanded in 1969. &lt;br&gt; Past products: Power Electronics module for Wrangler 4xe PHEV
|-
| T,&lt;br&gt; V (1985 only)
| [[w:Toluca Car Assembly|Toluca Car Assembly]] 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1968
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (2017-), [[w:Jeep Wagoneer S|Jeep Wagoneer S EV]] (2024-),&lt;br&gt; [[w:Jeep Cherokee (KM)|Jeep Cherokee (KM)]] (2026-), [[w:Jeep Recon|Jeep Recon EV]] (2026-)
| Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Began production on December 9, 1968. An adjacent supplier park was opened in 2007. Journey production began in early 2008. PT Cruiser production ended on July 9, 2010. Fiat 500 production began in December 2010. Journey production ended in December 2020. Jeep Compass production began on January 16, 2017. &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Dart (rwd), Dodge Dart K, Dodge Dart E, Chrysler Valiant Volaré (rwd), Chrysler Valiant Volaré K, Chrysler Valiant Volaré E, [[w:Dodge Magnum#First generation|Dodge Magnum]] [rwd] (1981-1982), [[w:Dodge Magnum#Second generation|Dodge Magnum 400/Magnum]] [fwd] (1983-1988), [[w:Chrysler Phantom|Chrysler Phantom]], [[w:Chrysler Shadow|Chrysler Shadow]], [[w:Chrysler Spirit|Chrysler Spirit]] &lt;br&gt; Export to US: [[w:Dodge Aries|Dodge Aries]] (1984-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1984-1989), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe]] (1987-1989, 1992), [[w:Dodge Shadow|Dodge Shadow]] (1988, 1991-1994), [[w:Plymouth Sundance|Plymouth Sundance]] (1988, 1992-1994), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron sedan]] (1990-1994), [[w:Dodge Spirit|Dodge Spirit]] (1991-1995), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1991-1995), [[w:Dodge Neon#First generation (1994)|Dodge Neon]] (1995-1999), [[w:Plymouth Neon#First generation (1994)|Plymouth Neon]] (1995-1999), [[w:Chrysler Sebring#Convertible (1996–2000)|Chrysler Sebring Convertible]] (1996-2000), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2001-10), [[w:Dodge Journey|Dodge Journey]] (2009-2020),&lt;br&gt; [[w:Fiat 500 (2007)|Fiat 500]] (2012-2019), [[w:Fiat 500 (2007)#Fiat 500e (2013)|Fiat 500e]] (2013-2019)&lt;br&gt; Export to Brazil/Europe/Australia:&lt;br&gt; [[w:Fiat Freemont|Fiat Freemont]] (2011-2016)
|-
| 
| Toluca Stamping Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1994
| 
| Stampings and assemblies including Body panels
| Part of the Toluca Assembly Complex.
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine South Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler Pentastar V6 engine]]
| Located at 2300 Van Horn Road. Production began in March 2010 with the 3.6L Pentastar V6. In 2022, Trenton South was upgraded with a flexible engine line that could build both the Classic and Upgrade versions of the 3.6L Pentastar V6. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production.
|-
| 
| Warren Stamping Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1949
| 
| Stampings and assemblies including roofs, tailgates, side apertures, fenders, and floorpans
| Located at 22800 Mound Road. This plant is a separate facility but is located next door to the Warren Truck Assembly Plant. The plant was expanded in 1952, 1964, 1965, and 1986. Warren Stamping supplies stampings to many Chrysler assembly plants, not only Warren Truck Assembly.
|-
| S (1970-), 1 (1967-1969),&lt;br&gt; 2 (For A-series)
| Warren Truck Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1938
| 
| [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer]] (2022-), [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer L]] (2023-)
| Located at 21500 Mound Road. Formerly known as the Dodge City Truck Plant. Also referred to as Warren Truck #1 in the 1970's when there were 2 other truck plants nearby. Began production in October 1938. The Mitsubishi Raider began production in September 2005 and ended on June 11, 2009. Dodge Dakota production ended on August 23, 2011. Full-size Jeep SUV production began in 2021. Ram 1500 Classic production ended in October 2024, ending production of Dodge/Ram trucks at Warren after 86 years. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge T-, V-, W-Series|Dodge T-, V-, W-Series]] (1939-1947), Dodge military trucks, [[w:Dodge B series#Pickup truck|Dodge B series]] (1948-1953), [[w:Dodge C series|Dodge C series]] (1954-1960),&lt;br&gt; [[w:Dodge Town Panel and Town Wagon|Dodge Town Panel/Town Wagon]] (1954-66), [[w:Dodge D series|Dodge D/W series]] (1961-1980), [[w:Dodge Ram|Dodge Ram pickup]] (1981-2012), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-24), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1977-1978, 1981-1985), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1977-1978, 1981), [[w:Dodge M-series chassis|Dodge M-series chassis]] (1968-1979),&lt;br&gt; [[w:Dodge A100|Dodge A100/A108]] (1964-1970),&lt;br&gt; [[w:Dodge Dakota|Dodge Dakota]] (1987-2011),&lt;br&gt; [[w:Mitsubishi Raider|Mitsubishi Raider]] (2006-2009),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer]] (2022-2025),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer L]] (2023-2025),&lt;br&gt; [[w:Fargo Trucks|Fargo Trucks]] (For export)
|-
| R (1968-),&lt;br&gt; 9 (1959-1967)
| [[w:Windsor Assembly|Windsor Assembly]] 
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1929
| 
| [[w:Chrysler Pacifica (minivan)|Chrysler Pacifica (minivan)]] (2017-), [[w:Chrysler Voyager#Sixth generation (2020–present)|Chrysler Voyager]] (2020-2026), Chrysler Grand Caravan (Canada: 2021-),&lt;br&gt; [[w:Dodge Charger (2024)|Dodge Charger]] (2024-)
| Located at 2199 Chrysler Centre. Today's Windsor Assembly Plant was originally Windsor Plant 3 or Windsor Car Assembly Plant. Before the 1965 US-Canada Auto Pact, Windsor Assembly made most of the cars sold by Chrysler Canada, including some unique-to-Canada variations. On June 10, 1983, Windsor ended rwd car production and was converted to build fwd minivans. Windsor began building minivans on October 7, 1983. For the first 2 generations of Chrysler minivans, Windsor only built the SWB models. For the 3rd generation, Windsor built both SWB and LWB models. For the 4th generation, Windsor only built LWB models. The minivan-based Pacifica SUV began production in January 2003 and  ended production on November 23, 2007. Production of the VW Routan began in August 2008. Production of the Ram C/V began on August 31, 2011, and ended in early 2015. Pacifica minivan production began on February 29, 2016 followed by the PHEV version on December 1, 2016. Town &amp; Country production ended on March 21, 2016 while Dodge Grand Caravan production ended on August 21, 2020. Production of the electric Charger Daytona began in December 2024 followed by the gas-powered Charger Sixpack in December 2025. &lt;br&gt; Past models: [[w:Chrysler Imperial|Chrysler Imperial]] (1929-1937) [https://www.web.imperialclub.info/registry/vin_decode.htm#1931-54], [[w:Dodge Kingsway|Dodge Kingsway]] (Canada: 1940-41, 1951-52), [[w:Dodge Regent|Dodge Regent]] (Canada: 1951-1959), [[w:Dodge Crusader|Dodge Crusader]] (Canada: 1951-1958), [[w:Dodge Mayfair|Dodge Mayfair]] (Canada: 1953-1959), [[w:Dodge Viscount|Dodge Viscount]] (Canada: 1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1961), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:Chrysler Windsor|Chrysler Windsor]] (1957-1966), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1963), [[w:Chrysler 300 non-letter series|Chrysler Saratoga 300]] (1964-1965), [[w:Chrysler Newport#1961–1964|Chrysler Newport]] (1961-1963), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1963-64, 1966), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Plymouth Fury|Plymouth Fury]] (1959-1970), [[w:Dodge Polara|Dodge Polara]] (1960, 1964-1969), Dodge 220 (Canada: 1963), [[w:Dodge 330|Dodge 330]] (1963-1965), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Monaco|Dodge Monaco]] (1967-1968), [[w:Plymouth Valiant#Canada (1960–1966)|Valiant]] (Canada: 1960-1966), [[w:Plymouth Barracuda#First generation (1964–1966)|Valiant Barracuda]] (Canada: 1964-1965), [[w:Plymouth Valiant|Plymouth Valiant]] (1970-1974), [[w:Plymouth Duster|Plymouth Duster]] (1970), [[w:Dodge Dart|Dodge Dart (compact)]] (1966, 1970-1975), [[w:Plymouth Satellite#Third generation (1971–1974)|Plymouth Satellite]] (1972-1974),  [[w:Plymouth GTX|Plymouth GTX]] (1971), [[w:Plymouth Road Runner#Second generation (1971–1974)|Plymouth Road Runner]] (1971-1974), [[w:Dodge Charger (1966)#Fourth generation|Dodge Charger]] (1975-1978), [[w:Dodge Magnum#US and Canada (1978–1979)|Dodge Magnum]] (1978-1979), [[w:Chrysler Cordoba|Chrysler Cordoba]] (1975-1983), [[w:Dodge Mirada|Dodge Mirada]] (1980-1983), [[w:Imperial (automobile)#Sixth generation (1981–1983)|Imperial]] (1981-1983), [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979), [[w:Dodge Diplomat|Dodge Diplomat]] (1981-1983), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1982-83), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle]] (Canada: 1981-1982), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1983), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron]] (1981), [[w:Chrysler New Yorker#1982|Chrysler New Yorker]] (1982), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler New Yorker Fifth Avenue]] (1983), [[w:Plymouth Voyager|Plymouth Voyager]] (1984-'00), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1984-2000), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2020), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (2001-2016), [[w:Chrysler Voyager|Chrysler Voyager]] (2000), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler minivans (S)#Cargo van|Dodge Mini Ram Van]] (1984-1988), [[w:Chrysler minivans (RT)#2011 revision|Ram C/V]] (2012-2015), [[w:Volkswagen Routan|Volkswagen Routan]] (2009-14), [[w:Chrysler Voyager#Lancia Voyager|Lancia Voyager]] (For export: 2012-2015), [[w:Chrysler Pacifica (crossover)|Chrysler Pacifica (SUV)]] (2004-2008)
|-
| 
| CpK Interior Products, Inc. - Belleville Operations
| [[w:Belleville, Ontario|Belleville]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 134 River Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. - Guelph Operations
| [[w:Guelph|Guelph]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 500 Laird Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. -&lt;br&gt; Port Hope Operations
| [[w:Port Hope, Ontario|Port Hope]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 128 Peter St. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 4
| Arab American Vehicles Company (AAV)
| [[w:Cairo|Cairo]]
| [[w:Egypt|Egypt]]
| 1978 (production began) &lt;br&gt; 1987 (became part of Chrysler)
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2025-), [[w:Citroën C4#Third generation (C41; 2020)|Citroën C4X]] (2025-)
| Arab American Vehicles Company is a joint venture between the [[w:Arab Organization for Industrialization|Arab Organization for Industrialization]], which holds 51%, and Stellantis, which holds the other 49%. AAV was originally established in 1977 as a joint venture  with [[w:American Motors Corporation|AMC]] to produce Jeeps. Production began in 1978. It became part of Chrysler when Chrysler bought AMC in 1987. It continued to be a part of subsequent corporate entities DaimlerChrysler, Chrysler LLC, Chrysler Group LLC, [[w:Fiat Chrysler Automobiles|FCA]], and [[w:Stellantis|Stellantis]]. Production of the Jeep J8, a light military vehicle based on the JK Wrangler, began on November 13, 2008. Jeep Grand Cherokee L production began in September 2024. Citroen C4X production began in April 2025. AAV has also produced vehicles for other automakers including Toyota. Toyota Fortuner SUV production began in April 2012. &lt;br&gt; Past models: Jeep CJ6, Jeep Wagoneer (SJ), Jeep AM720 military vehicle, [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]], [[w:Jeep Wrangler (TJ)|Jeep TJL]], [[w:Jeep Wrangler (JK)#Military Jeep J8 (2007–present)|Jeep J8]] (2008-2019),  [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]], [[w:Jeep Liberty (KJ)|Jeep Cherokee (KJ)]], [[w:Jeep Liberty (KK)|Jeep Cherokee (KK)]], [[w:Jeep Grand Cherokee (WK2)|Jeep Grand Cherokee (WK2)]]
|}

==Current non-Chrysler FCA/Stellantis Factories Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| Y (700),&lt;br&gt; 9 (ProMaster Rapid),&lt;br&gt; 3 (Vision)
| Betim Plant
| [[w:Betim|Betim]], [[w:Minas Gerais|Minas Gerais]]
| [[w:Brazil|Brazil]]
| 1973
| 
| [[w:RAM 700|RAM 700]] (2015-),&lt;br&gt; [[w:ProMaster Rapid|Ram V700 Rapid]] (Peru)&lt;br&gt; Related models:&lt;br&gt; [[w:Fiat Strada|Fiat Strada]] ('99-),&lt;br&gt; [[w:Fiat Fiorino#Latin America (2013–present)|Fiat Fiorino]] ('14-)
| Fiat plant. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Dodge Vision|Dodge Vision]] (Mexico: 2015-2018),&lt;br&gt; [[w:ProMaster Rapid|Ram ProMaster Rapid]] (Mexico: '18-'24)
|-
| U
| Cordoba Plant
| [[w:Córdoba, Argentina|Córdoba]], [[w:Córdoba Province, Argentina|Córdoba Province]]
| [[w:Argentina|Argentina]]
| 1995
| 
| [[w:Peugeot Landtrek|Ram Dakota]] (2026-)
| Fiat plant.
|-
| F
| [[w:FCA India Automobiles|FCA India Automobiles Private Limited]]
| [[w:Ranjangaon|Ranjangaon]], [[w:Pune district|Pune district]], [[w:Maharashtra|Maharashtra]]
| [[w:India|India]]
| 1997
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]],&lt;br&gt; [[w:Jeep Meridian|Jeep Meridian/Commander]],&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]]
| Originally established as a 50/50 joint venture between Fiat and [[w:Tata Motors|Tata Motors]] called Fiat India Automobiles Private Limited. On June 1, 2017, Jeep Compass production began in India, the first Jeep built in India under its own brand. The JL-series Wrangler began to be assembled in India in 2021. The Jeep Meridian began production in May 2022. The Meridian is also exported to Japan as the Commander. The Jeep Grand Cherokee began assembly in India in November 2022. 
|-
| K
| Goiana Plant
| [[w:Goiana|Goiana]], [[w:Pernambuco|Pernambuco]]
| [[w:Brazil|Brazil]]
| 2015
|
| [[w:Jeep Renegade|Jeep Renegade]], [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Commander (2022)|Jeep Commander]], [[w:Ram Rampage|Ram Rampage]]&lt;br&gt; Related models: [[w:Fiat Toro|Fiat Toro]]
| [[w:Fiat Chrysler Automobiles|FCA]] plant. Production at Goiana began with the Jeep Renegade in February 2015. &lt;br&gt; Past Chrysler Group models: [[w:Ram 1000|Ram 1000]]
|-
| P
| Melfi Plant (formerly SATA = Società Automobilistica Tecnologie Avanzate [Advanced Technologies Automotive Company])
| [[w:Melfi|Melfi]], [[w:Province of Potenza|Province of Potenza]]
| [[w:Italy|Italy]]
| 1993
|
| [[w:Jeep Compass#Third generation (J4U; 2025)|Jeep Compass (J4U)]]&lt;br&gt; (Europe: '26-)
| Fiat plant. Jeep production began at Melfi in 2014 with the Renegade. Renegade production at Melfi ended on October 17, 2025. Production of the 3rd gen. Compass began on October 29, 2025. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Jeep Renegade|Jeep Renegade]] (US/Can.: '15-'23, Europe: '15-'25),&lt;br&gt; [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (Europe: '20-'25) &lt;br&gt; Related models:&lt;br&gt; [[w:Fiat 500X|Fiat 500X]] (US/Can.: '16-'23,&lt;br&gt; Europe: '15-'24)
|-
| B
| Porto Real Plant
| [[w:Porto Real|Porto Real]], [[w:Rio de Janeiro (state)|Rio de Janeiro state]]
| [[w:Brazil|Brazil]]
| 2000
|
| [[w:Jeep Avenger|Jeep Avenger]]
| PSA plant. The Jeep Avenger is the first Jeep to be made in a former PSA plant. 
|-
| U (2027-), 6 (2015-2022)
| [[w:Tofaş|Tofaş]]
| [[w:Bursa|Bursa]]
| [[w:Turkey|Turkey]]
| 1971
|
| [[w:Citroën Jumpy#Ram ProMaster City|Ram ProMaster City]] (2027-)
| Originally a Fiat joint venture, it is now 37.8% owned by Stellantis, 37.8% owned by [[w:Koç Holding|Koç Holding]], and 24.3% publicly traded on the Istanbul Stock Exchange. &lt;br&gt; Past Chrysler Group models: &lt;br&gt; [[w:Fiat Doblò#Ram ProMaster City|Ram ProMaster City]] (2015-2022), [[w:Chrysler Neon#Third generation (2016)|Dodge Neon]]&lt;br&gt; (Mexico &amp; Middle East: 2017-2020), [[w:Fiat Fiorino#Europe (2007–2024)|Ram V700 City]] (Chile: 2018-2023)
|-
| J,&lt;br&gt; 5 (Ypsilon)
| Tychy Plant
| [[w:Tychy|Tychy]], [[w:Silesian Voivodeship|Silesian Voivodeship]]
| [[w:Poland|Poland]]
| 1975
|
| [[w:Jeep Avenger|Jeep Avenger]]
| Fiat plant. Production began in September 1975 when the plant was owned by Polish automaker [[w:Fabryka Samochodów Małolitrażowych|FSM]], which built Fiat-based models. Fiat took over FSM in 1992. Fiat subsequently became [[w:Fiat Chrysler Automobiles|FCA]] and then Stellantis. Jeep Avenger production began on January 31, 2023. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Chrysler Ypsilon|Chrysler Ypsilon]] (UK/Ireland/Japan)
|}

==Current partner factories making Chrysler Group vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| GAC Hangzhou plant
| [[w:Hangzhou|Hangzhou]], [[w:Zhejiang|Zhejiang]]
| [[w:China|China]]
| 2021 (production began for Chrysler)
|
| [[w:Trumpchi GS5#Dodge Journey|Dodge Journey]] (Mexico: 2022-)
| [[w:GAC Group|GAC]] plant. 
|-
| 3
| GAC Yichang plant
| [[w:Yichang|Yichang]], [[w:Hubei|Hubei]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Dodge Attitude#Fourth generation (2025)|Dodge Attitude]] (Mexico: 2025-)
| [[w:GAC Group|GAC]] plant. 
|-
| 5
| Shenzhen Baoneng Motor Co., Ltd.
| [[w:Shenzhen|Shenzhen]], [[w:Guangdong|Guangdong]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Peugeot Landtrek|Ram 1200]] (Mexico: 2025-)
| [[w:Baoneng Group#Automotive business|Shenzhen Baoneng Motor Co., Ltd.]] plant. 
|}

==Former factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Closed
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
|
| Ajax Trim plant
| [[w:Ajax, Ontario|Ajax]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1953, 1964 (became part of Chrysler)
| 2003
| Automotive Soft Trim Components, Seat Cushion and Seatback Covers, Foam-in-place Covers
| Built in 1953 by Canadian Automotive Trim. Purchased by Chrysler in 1964. Closed by December 2003.
|-
| J (1989-1992),&lt;br&gt; B (1981-1988),&lt;br&gt; 7-8 (1966-1980),&lt;br&gt; T (1958-1966) 
| [[w:Brampton Assembly (AMC)|AMC Brampton Assembly]]
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]] 
| 1961,&lt;br&gt; 1987 (became part of Chrysler)
| 1992
| [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1987-92), [[w:Jeep CJ#CJ-5|Jeep CJ-5]] (1979-1980),&lt;br&gt; [[w:Jeep CJ#CJ-7|Jeep CJ-7]] (1979-1980),&lt;br&gt; [[w:AMC Eagle|AMC Eagle]] (1981-1987), [[w:AMC Eagle|Eagle Wagon]] (1988), [[w:AMC Concord|AMC Concord]] (1978, 1981-1983), [[w:AMC Spirit|AMC Spirit]] (1983), [[w:AMC Hornet|AMC Hornet]] (1970-77), [[w:AMC Gremlin|AMC Gremlin]] (1970-1978),&lt;br&gt; [[w:AMC Rebel|AMC Rebel]] (1968-1970), [[w:Rambler Rebel#Fifth generation|Rambler Rebel]] (1967), [[w:Rambler Classic|Rambler Classic]] (1961-1966),&lt;br&gt; [[w:AMC Ambassador|AMC Ambassador]] (1966-1968), [[w:AMC Ambassador|Rambler Ambassador]] ('63-'65), [[w:Rambler American|Rambler American]] (1962-68), [[w:Rambler American|AMC Rambler]] (1969)
| [[w:American Motors Corporation|AMC]] plant. Became part of Chrysler in the 1987 buyout of AMC. Located at at the corner of Kennedy Road South and Steeles Avenue East. Production began on January 26, 1961 with the Rambler Classic. This was the last plant to produce AMC vehicles. Eagle Wagon (formerly AMC Eagle) ended production on December 11, 1987. Production ended in April 1992 and Jeep Wrangler production was moved to Toledo, OH. Buildings on the west side of the plant were demolished in 2005 and buildings on the east side were demolished in 2007. A Lowe's and a Wal-Mart now occupy some of the former plant site.
|-
| V
| [[w:Conner Avenue Assembly|Conner Avenue Assembly]] 
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1996
| 2017
| [[w:Dodge Viper|Dodge Viper]] &lt;br&gt; GTS: '96, &lt;br&gt; All models: &lt;br&gt; '97-'06, '08-'10, '13-'17, [[w:Plymouth Prowler|Plymouth Prowler]]&lt;br&gt; ('97, '99-'01),&lt;br&gt; [[w:Chrysler Prowler|Chrysler Prowler]] ('01-'02),&lt;br&gt; [[w:Viper engine|8.0L/8.3L/8.4L aluminum &lt;br&gt; Viper V10 engine]] &lt;br&gt;(5/01-2017)
| Actually located at 20000 Connor Street. The plant was originally built in 1966 to make spark plugs by Champion. After Champion was bought by Cooper Industries in 1990, the plant was closed. It remained empty until Chrysler bought it in 1995. Viper production was moved to the Connor Avenue plant from the New Mack plant beginning with the GTS coupe for 1996, followed by the RT/10 roadster for 1997. Prowler production began in May 1997 and ended on February 15, 2002. Production of the Viper's aluminum V10 engine was moved to Connor Avenue, where it was built alongside the Viper itself, in May 2001 from the Mound Road Engine Plant, which closed in 2002. Viper production ended on July 2, 2010 and the plant was dormant until production restarted in December 2012. Production ended on Aug. 31, 2017. In 2018, the plant was renamed Connor Center, a meeting and display space that will showcase Chrysler’s concept and historic vehicle collection. 
|-
|E
| [[w:Diamond-Star Motors|Diamond-Star Motors/Mitsubishi Motors Manufacturing America/Mitsubishi Motors North America Manufacturing Division]]
| [[w:Normal, Illinois|Normal, Illinois]]
| [[w:United States|United States]]
| 1988
| 2005 (end of Chrysler production)/&lt;br&gt; 2015 (end of Mitsubishi production)
| [[w:Plymouth Laser|Plymouth Laser]] (1990-1994),&lt;br&gt; [[w:Eagle Talon|Eagle Talon]] (1990-1998),&lt;br&gt; [[w:Eagle Summit#First generation (1989–1992)|Eagle Summit 4-d]] (1991-1992),&lt;br&gt; [[w:Dodge Avenger#Dodge Avenger Coupe (1995–2000)|Dodge Avenger]] (1995-2000),&lt;br&gt; [[w:Dodge Stratus#Stratus coupe (2001–2005)|Dodge Stratus Coupe]]&lt;br&gt; (2001-2005),&lt;br&gt; [[w:Chrysler Sebring|Chrysler Sebring Coupe]]&lt;br&gt; (1995-2005),&lt;br&gt;  [[w:Mitsubishi Eclipse|Mitsubishi Eclipse]] (1990-2012),&lt;br&gt; [[w:Mitsubishi Mirage#Third generation (1987)|Mirage sedan]] (1991-1992),&lt;br&gt; [[w:Mitsubishi Galant|Galant]] (1994-2012),&lt;br&gt; [[w:Mitsubishi Endeavor|Endeavor]] (2004-2011),&lt;br&gt; [[w:Mitsubishi ASX#First generation (GA; 2010)|Outlander Sport/RVR]] (2013-15)
| Originally established as Diamond-Star Motors, a 50/50 joint venture between Chrysler and Mitsubishi which assembled both Chrysler and Mitsubishi vehicles. Production began in September 1988. In October 1991, Chrysler sold its 50% stake in the plant to Mitsubishi but production for Chrysler continued under contract. On May 24, 1993, production of the Mitsubishi Galant began at the Diamond-Star plant. In July 1995, the plant was renamed Mitsubishi Motors Manufacturing America. In January 2002, the plant was renamed Mitsubishi Motors North America Manufacturing Division. In January 2003, production of the Mitsubishi Endeavor began, the first SUV built at the Mitsubishi plant. In February 2005, production of vehicles for Chrysler ended. All further production was only of Mitsubishi-branded vehicles. In mid-2012, the plant began producing the Mitsubishi Outlander Sport. The Outlander Sport is sold as the RVR in Canada. On November 30, 2015, vehicle production ended. The Mitsubishi plant produced 3,283,549 vehicles. The plant continued to produce replacement parts until May 2016, when the plant closed completely. In June 2016, the plant was sold to liquidation firm Maynards Industries. In January 2017, EV startup [[w:Rivian|Rivian Automotive]] bought the former Mitsubishi plant. Rivian began production at the Normal, IL plant in September 2021 with the [[w:Rivian R1T|R1T]] electric pickup.
|-
| U
| [[w:Eurostar Automobilwerk|Eurostar]] - Chrysler Graz Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1991
| 2002
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (1992-2002), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2002)
| Originally, a 50/50 joint venture between Chrysler and Steyr-Daimler-Puch founded in 1990. The Eurostar plant is next to the Steyr Fahrzeugtechnik plant solely owned by Steyr-Daimler-Puch. Production of the Chrysler Voyager and Grand Voyager minivans began in October 1991. This was the 2nd generation Chrysler minivan. The 3rd generation began production on September 25, 1995. In 1996, production of right-hand drive minivans began. The 4th generation began production in January 2001. Production of the PT Cruiser began in July 2001 on the same line as the Voyager minivans. In 1999, DaimlerChrysler bought the 50% stake in Eurostar held by Steyr-Daimler-Puch Fahrzeugtechnik, now majority owned by Magna International, making Eurostar a subsidiary of DaimlerChrysler. DaimlerChrysler sold 100% of Eurostar to Magna Steyr in July 2002. PT Cruiser production in Austria ended and was consolidated in Toluca, Mexico. Production of the Chrysler Voyager and Grand Voyager minivans moved next door to the main Magna Steyr plant for 2003. Magna International had acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. 
|-
| 3 (1959),&lt;br&gt; E (1958)
| Evansville Assembly
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1919, 1928 (became part of Chrysler)
| 1959 
| Graham Brothers trucks (through 1932),&lt;br&gt; Dodge Brothers trucks &lt;br&gt; (through 1932),&lt;br&gt; Plymouth cars (1936-1958),&lt;br&gt; Dodge cars (1937-1938),&lt;br&gt; [[w:Plymouth Savoy|Plymouth Savoy]] (1959), [[w:Plymouth Belvedere|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1959)
| Located at 1625 N. Garvin St. Built in 1919 by Graham Brothers Truck Company to build trucks. In 1925, Dodge Brothers bought a controlling 51% stake in Graham Brothers and then bought the rest in 1926, completely merging the 2 companies. This gave Dodge Brothers plants in Evansville, IN and Stockton, CA. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. At that point, trucks with a Dodge Brothers nameplate were rated as a half-ton; larger rated trucks were sold under the Graham Brothers name. On January 1, 1929, the Graham Brothers brand was eliminated, and all trucks produced became Dodge trucks. In 1932, Chrysler closed the Evansville plant due to the Great Depression. In 1935, as the economy improved, Chrysler reopened the Evansville plant and renovated and expanded it. It began building Plymouth cars for 1936. Dodge cars were also built for 1937 and 1938. During World War II, the plant became the Evansville Ordinance Plant, which produced more than 3.26 billion ammunition  cartridges - about 96% of all the .45 automatic ammunition produced for all the armed forces. The Evansville Ordinance Plant also rebuilt 1,600 Sherman tanks and 4,000 military trucks. After the war ended, Plymouth car production resumed. However, in the early 1950s, during the Korean War, the Evansville plant retooled and dedicated about a third of its space and manpower to building 60-foot aluminum hulls for Grumman  UF-1 Albatross air-sea rescue planes for the Navy and Coast Guard. Evansville built its 1 millionth Plymouth in March 1953. The plant was closed in 1959 and was replaced by the larger, more modern St. Louis plant in Fenton, MO, which had access to more railroad lines for shipping than Evansville did. 
|-
| 
| Evansville Stamping
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1935, 1953 (became part of Chrysler)
| 1959 
| Body panel stampings
| Opened by Briggs Manufacturing Company when Chrysler reopened Evansville Assembly in 1935 to build Plymouth cars. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Made body panels for the nearby Evansville Assembly plant. Closed when Evansville Assembly closed in 1959.
|-
| B (1968-1980),&lt;br&gt; 2 (1960-1967),&lt;br&gt; 2 (1959)
| [[w:Dodge Main|Hamtramck Assembly (Dodge Main)]]
| [[w:Hamtramck, Michigan|Hamtramck, Michigan]]
| [[w:United States|United States]]
| 1911,&lt;br&gt; 1928 (became part of Chrysler)
| 1980 
| Dodge (1914-1958),&lt;br&gt; [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1959),&lt;br&gt; [[w:Dodge Royal#Third generation (1957–1959)|Dodge Royal]] (1959),&lt;br&gt; [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959),&lt;br&gt; [[w:Dodge Matador|Dodge Matador]] (1960),&lt;br&gt; [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962),&lt;br&gt; [[w:Dodge Polara|Dodge Polara]] (1960-1964),&lt;br&gt; [[w:Dodge 330|Dodge 330]] (1963-1964),&lt;br&gt; [[w:Dodge 440|Dodge 440]] (1963-1964),&lt;br&gt; [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1975), [[w:Plymouth Duster|Plymouth Duster]] (1970-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1969, 1972-1975), [[w:Dodge_Dart#1971|Dodge Dart Demon]] (1971-1972),&lt;br&gt; [[w:Dodge Charger (1966)|Dodge Charger]] (1967-1969), [[w:Dodge Charger Daytona#First generation (1969)|Dodge Charger Daytona]] ('69), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-74), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970-1974), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volaré|Plymouth Volaré]] (1976-1980),&lt;br&gt; Engines
| Located at 7900 Joseph Campau Ave. This plant predated Dodge being part of Chrysler Corp. On November 4, 1914, the first Dodge Brothers passenger car was produced at the Hamtramck plant. Prior to that, Dodge Brothers made components for other automakers, primarily Ford. The Hamtramck plant was fully vertically integrated, capable of building almost every part needed to build a complete car. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. Even after the Chrysler takeover, Hamtramck remained Dodge's home plant. From the early 1950s, various operations were automated or moved to other plants and Hamtramck became more of an assembly plant by the early 1960s. Closed January 4, 1980. Last vehicle built was a Silver Metallic 1980 Dodge Aspen R/T 2-door. 13,943,221 vehicles were produced at the plant. Demolished in 1981. Replaced by the General Motors Detroit/Hamtramck Assembly Plant (Factory Zero), which opened in 1985. 
|-
| 
| [[w:Highland Park Chrysler Plant|Highland Park Plant]]
| [[w:Highland Park, Michigan|Highland Park, Michigan]]
| [[w:United States|United States]]
| 1909,&lt;br&gt; 1925 (became part of Chrysler)
| 1960s (end of manufacturing)
| Maxwell cars (1910-1925), Chrysler Series 50 (1926-27), Chrysler Series 52 (1928), Plymouth Model Q (1929), Chrysler Series 60 (1927), Chrysler Series 62 (1928), DeSoto Series K (1929-1930), DeSoto Series CK (1930), DeSoto Series CF (1930), Fargo Trucks (1928-1930) &lt;br&gt; Parts including fluid coupling and torque converter for [[w:Fluid Drive|Fluid Drive]]
| Located at at 12000 Chrysler Service Drive (formerly Chrysler Drive). Originally, this was [[w:Maxwell Motor Company|Maxwell Motor Company]]'s main plant. However, before Maxwell Motor Co.'s formation, parts of the site was used by several car and truck manufacturers: Grabowsky Power Wagon Company used 1 building, Brush Runabout Co. used another building, and Gray Motor Co. owned another building. Gray only used the western 1/3 of the building so they leased the middle third to Alden- Sampson Truck Co. and the eastern third to Maxwell-Briscoe Motor Co. All these companies, along with several others, combined under the [[w:United States Motor Company|United States Motor Company]] in 1910. In 1913, United States Motor Company collapsed and Maxwell was the only surviving part. The U.S. Motor Co. assets were purchased by Walter Flanders, who reorganized the company as the Maxwell Motor Co. Maxwell hired Walter P. Chrysler to turn the company around in 1921 after its finances deteriorated in the post-World War I recession in 1920. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including its Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924 at the Jefferson Ave. plant. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925 at Highland Park. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 built at Highland Park and was then reworked again into the first Plymouth in 1928, also built at Highland Park. Plymouth production was moved to the new Lynch Road Assembly plant in 1929 and DeSoto production also moved to the new Lynch Road Assembly plant in 1931. Highland Park still built parts but it no longer built vehicles. Highland Park was used more for design,  engineering, and management and served as Chrysler Corporation's headquarters through 1996. During the 1990's, Chrysler moved to its current headquarters at the Chrysler Technology Center in Auburn Hills. Much of the site has been demolished though Chrysler still has a small presence at the site with the FCA Detroit Office Warehouse. Other parts of the site are now occupied by several automotive suppliers including Magna, Valeo, Mobis, Avancez, and Yanfeng.
|-
|
| [[w:Indiana Transmission#Indiana Transmission Plant II|Indiana Transmission Plant II]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2003 (as Indiana Transmission Plant II)
| 2019 (as Indiana Transmission Plant II)
| [[w:W5A580|Mercedes W5A580 (A580) &lt;br&gt; 5-speed auto. trans.]], Transmission components
| Located at 3360 North U.S. Highway 931. Plant was originally known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Production began in November 2003. 5-speed auto. trans. production ended in August 2018 while production of components for the 8-speed auto. transmission ended in the fall of 2019. Starting in 2020, the plant was converted to engine production and is now known as Kokomo Engine Plant. Engine production began in late February 2022. 
|-
|
| Indianapolis Electrical Plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1953
| 1988
| Transmission plant: [[w:Chrysler PowerFlite transmission|Chrysler PowerFlite 2-speed auto. transmission]] &lt;br&gt; Electrical Plant: Alternators, distributors, starters, power steering units, voltage regulators, windshield wiper motors, and other electrical parts for cars
| Located at 2900 Shadeland Ave. Began making transmissions in 1953. In January 1959, Chrysler housed its new Electrical Division at the Shadeland Avenue plant, replacing transmission production. Became part of Chrysler's Acustar components subsidiary in 1987. Production ended on November 30, 1988 but shipping and other activities continued until March 1989 when the factory closed. Certain portions have been demolished and improvements were made to the remainder, which is now the Shadeland Business Center. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Naomi Street plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1946 (became part of Chrysler)
| 1970's
| Engine blocks
| Located at 1535 Naomi Street. Purchased from American Foundry Company in 1946. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Kept operating even after the Tibbs Ave. plant was launched. This location is now Wilco Gutter Supply. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Tibbs Avenue plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1950 
| 2005
| Engine heads and blocks and other components
| Located at 1100 S. Tibbs Avenue. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Production ended on September 30, 2005 and the facility closed. Demolished in 2006. 
|-
| C (1968-1990),&lt;br&gt; 3 (1960-1967),&lt;br&gt; 1 (1959)
| [[w:Detroit Assembly Complex – Jefferson#Jefferson Avenue Assembly|Jefferson Avenue Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1908,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| [[w:Chrysler Six|Chrysler Series 70 (B-70)]] (1924-1925), [[w:Chrysler Six|Chrysler Series 70 (G-70)]] (1926-1927), [[w:Chrysler Six|Chrysler Series 72]] (1928), [[w:Chrysler Royal|Chrysler Royal]] (1933, 1937-1950), DeSoto SD (1933), Chrysler Airflow (1934-1937), Chrysler Airstream (1935-36), [[w:Chrysler (brand)|Chrysler brand]] (1937-1958), Desoto Airflow (1934-1936), DeSoto Airstream (1935-1936), [[w:Chrysler Imperial|Chrysler Imperial]] (1926-1942, 1946-1954), [[w:Imperial (automobile)|Imperial]] (1955-1958, 1962-1975), [[w:Chrysler 300 letter series|Chrysler 300 letter series]] (1959-1965), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-1978), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1961), [[w:Chrysler Newport|Chrysler Newport]] (1961-1978), [[w:Chrysler 300 non-letter series|Chrysler 300 Sport Series]] (1962-1971), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1968-1972), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:DeSoto Fireflite|DeSoto Fireflite]] (1959-1960), [[w:DeSoto Adventurer|DeSoto Adventurer]] (1959-1960), [[w:DeSoto (automobile)#1961|DeSoto]] (1961), [[w:Dodge Matador|Dodge Matador]] (1960), [[w:Dodge Polara|Dodge Polara]] (1965-1966), [[w:Dodge D series|Dodge D/W series]] (1979-1980), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1979-1980), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1979-1980), [[w:Dodge Aries|Dodge Aries]] (1981-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1981-1989), [[w:Dodge 400|Dodge 400]] 4-d (1982-1983), [[w:Chrysler LeBaron|Chrysler LeBaron]] 4-d (1982-1984), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker (E-body)]] (1983-1987), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker Turbo (E-body)]] (1988), [[w:Dodge 600|Dodge 600]] 4-d (1983-1988), [[w:Chrysler E-Class|Chrysler E-Class]] (1983-1984), [[w:Plymouth Caravelle|Plymouth Caravelle]] (US: 1985-1988), [[w:Plymouth Caravelle|Plymouth Caravelle]] 4-d (Canada: 1983-1988), [[w:Dodge Omni|Dodge Omni]] (1989-1990), [[w:Plymouth Horizon|Plymouth Horizon]] (1989-1990),&lt;br&gt; Engines
| Located at 12200 East Jefferson Ave. Plant was originally opened by [[w:Chalmers Automobile|Chalmers Motor Co.]]. After falling on hard times, Chalmers agreed in 1917 to build cars for [[w:Maxwell Motor Company|Maxwell Motor Co.]] at the Jefferson Ave. plant in Detroit. In exchange, Chalmers cars would be sold through Maxwell dealers. After having its own financial problems, Maxwell stopped producing cars at the Chalmers plant in 1921. Maxwell hired Walter P. Chrysler to turn the company around in 1921. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including the Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 and was then reworked again into the first Plymouth in 1928. The Jefferson Ave. plant was the home plant of the Chrysler brand through 1978. It was also the home plant for the spin-off Imperial brand except for 1959-1961, when Imperial had its own exclusive plant on Warren Ave. in Dearborn. By the time it ended production on February 2, 1990, Jefferson Ave. Assembly had built 8,310,107 vehicles. Demolished in 1991. Replaced by the Jefferson North plant built across Jefferson Ave. from the old plant, where the Kercheval Body Plant used to be. The Jefferson North plant opened in January 1992.
|-
| W (1987-1989 Chrysler M-body) &lt;br&gt;&lt;br&gt; [K for 1981-1983 AMC and 1983-1987 Renault],&lt;br&gt; 0-6 (1966-1980 AMC)
| Kenosha I Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]]&lt;br&gt; (1987-1989),&lt;br&gt; [[w:Dodge Diplomat#Second generation (1980)|Dodge Diplomat]] (1987-1989), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1987-89), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1987-1989)
| This was the Kenosha Main Plant of [[w:American Motors Corporation|AMC]]. The Kenosha plant was the oldest still operating automobile factory in the world when it ended vehicle production in December 1988. It first built automobiles in 1902 for the Thomas B. Jeffery Company under the Rambler brand. The factory was purchased in 1900 from the Sterling Bicycle Co., which built it in 1895. In 1914, the Thomas B. Jeffery Company rebranded its vehicles under the Jeffery brand. In 1916, the Thomas B. Jeffery Company was bought by Charles Nash and renamed Nash Motors. Kenosha produced Nash vehicles from 1917-1957. Kenosha also produced Nash's entry-level Lafayette brand from 1934-1936. After Nash merged with Hudson to form American Motors in 1954, Kenosha also produced Hudson vehicles from 1955-1957. Kenosha then produced vehicles under the Rambler brand for AMC from 1958-1968 and under the AMC brand from 1966-1983. Kenosha also produced the Alliance for AMC shareholder Renault for 1983-1987 along with the Encore for 1984-1986 and the GTA for 1987. Chrysler signed a deal with AMC in September 1986 to utilize surplus capacity at AMC's Kenosha plant to build Chrysler's trio of rwd M-body sedans beginning in February 1987. Chrysler did not have any capacity left in its own plants to continue building the M-body sedans. The St. Louis North plant that had been building the M-body sedans had been converted to build the extended length minivans. This deal led to Chrysler's acquisition of AMC, announced in March 1987. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 and the M-bodies were discontinued. The site continued building engines until 2010. The plant has since been demolished.
|-
| Y
| Kenosha II Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Dodge Omni|Dodge Omni]] (1988-1989), [[w:Plymouth Horizon|Plymouth Horizon]] (1988-1989)
| This was the Kenosha Lakefront Plant of [[w:American Motors Corporation|AMC]], located on the shore of Lake Michigan. This property was originally a Simmons mattress manufacturing plant from 1870 to 1960. AMC bought it in 1960 to manufacture and paint auto bodies. Became part of Chrysler in the 1987 buyout of AMC. In September 1987, production of the Dodge Omni and Plymouth Horizon began. Production was moved to Kenosha from Chrysler's Belvidere, IL plant, which was being converted to build Chrysler's C-body sedans (Dynasty/New Yorker). Closed in December 1988. Omni &amp; Horizon production then moved to the Jefferson Ave. plant in Detroit. Demolished in 1990. In 1994, the City of Kenosha purchased the property for $1. The property was subsequently cleaned up and redeveloped into the HarborPark area, which includes a park and open space, a public museum, residential housing, and a marina. 
|-
| 
| [[w:Kenosha Engine|Kenosha Engine Plant]]
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2010
| [[w:AMC straight-4 engine|AMC straight-4 engine]],&lt;br&gt; [[w:AMC straight-6 engine|AMC straight-6 engine]],&lt;br&gt; [[w:AMC V8 engine|AMC V8 engine]],&lt;br&gt; [[w:Chrysler LH engine|Chrysler 2.7L DOHC V6]], [[w:Chrysler SOHC V6 engine#3.5|Chrysler 3.5L SOHC V6]]
| Located at 5555 30th Avenue. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 at the adjacent assembly plant but the site continued building engines until 2010. After the Chrysler buyout, the plant kept building the AMC 5.9L V8 for the SJ Jeep Grand Wagoneer through 1991, the AMC 2.5L I4 through 2002 for Jeeps and the Dodge Dakota, the AMC 4.2L I6 through 1990 for the Jeep Wrangler, and the AMC 4.0L I6 through 2006 for various Jeeps ('06 Wrangler was the last to use the 4.0L). Chrysler started building its own 2.7L V6 at Kenosha in 1997 and its own 3.5L V6 in 2003. Engine production ended in October 2010 and the plant closed. Demolished in 2012-2013.
|-
|
| Kercheval Body Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1920,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| Automobile Bodies
| Located at 12265 E Jefferson Ave., across Jefferson Ave. from Chrysler's Jefferson Ave. Assembly Plant, which had originally been the Chalmers plant. The assembly plant was on the south side of Jefferson Ave. while the body plant was on the north side. The plant was called Kercheval because the north side of the plant was bordered by Kercheval Ave. The plant was built in 1920 by Wadsworth Manufacturing Co. to replace a previous plant on the same site that burned down in 1919. That fire had also damaged the Chalmers plant across the street. In November 1920, Wadsworth Manufacturing was sold to American Motor Body Co., a division of the American Can Co. On July 1, 1923, American Motor Body Co. became American Motor Body Corp., run by Charles M. Schwab. On September 4, 1925, Chrysler Corp. bought the Detroit plant of the American Motor Body Corp. to quickly increase its manufacturing capacity. In 1955, Kercheval Body Plant was connected to the Jefferson Assembly plant by a bridge crossing over Jefferson Ave. Previously, bodies made at Kercheval had to be transported by truck across Jefferson Ave. to the assembly plant. The plant closed in Feb. 1990, at the same time the Jefferson Ave. Assembly Plant closed. The Kercheval plant was demolished and Chrysler built the new Jefferson North Assembly Plant on the site of the former Kercheval Body Plant, on the north side of Jefferson Ave. 
|-
|
| Kokomo - Home Ave. plant
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1937
| 1969
| Manual Transmissions (1937-1955), Aluminum die casting (1955-1969)
| Located at 1105 S. Home Ave. Chrysler bought this plant in 1937. This was Chrysler's first plant in Kokomo. It had previously belonged to the [[w:Haynes Automobile Company|Haynes Automobile Co.]], which went out of business in 1925. The plant had been dormant since then. 5,124,211 manual transmissions were built here from 1937-1955. Transmission production then shifted to a new plant about a mile southeast on South Reed Road. The Home Ave. plant then became an aluminum die casting plant until 1969, when that operation shifted to the new Kokomo Casting plant on East Boulevard. Chrysler subsequently sold this plant. The facility was last used by Warren's Auto Parts, an auto salvage yard, which closed in 2020 after nearly 50 years. It is currently empty though still standing as of 2025. 
|-
| M
| [[w:Lago Alberto Assembly|Lago Alberto Assembly]]
| [[w:Nuevo Polanco|Nuevo Polanco district]], [[w:Miguel Hidalgo, Mexico City|Miguel Hidalgo borough]], [[w:Mexico City|Mexico City]]
| [[w:Mexico|Mexico]]
| 1938
| 2002
| &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Savoy, Dodge Dart, Dodge 330, Dodge 440, Chrysler Valiant (1963-1969), Valiant Barracuda (1965-1969), Dodge Coronet, [[w:Dodge Ram|Dodge Ram pickup]] (1981-02), [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-96), [[w:Dodge Ramcharger#Third generation (1999–2001)|Dodge Ramcharger]] (1999-01) &lt;br&gt; Export to US:&lt;br&gt; [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-93), [[w:Dodge Ram#First generation (1981; D/W)|Dodge Ram pickup]] (1990-93), [[w:Dodge Ram#Second generation (1994; BR/BE)|Dodge Ram pickup]] (1994-02)
| Located at 320 Lago Alberto Street. Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Lago Alberto began exporting to the US with the 1986 Dodge Ramcharger, sourced exclusively from Mexico. The Lago Alberto plant was closed in 2002 and Mexican pickup production was consolidated in the newer, more modern Saltillo plant. 
|-
| E (1968-1971),&lt;br&gt; 5 (1960-1967),&lt;br&gt; 4 (1959),&lt;br&gt; L (1958),&lt;br&gt; L (1955-1957 Chrysler brand)
| [[w:Los Angeles (Maywood) Assembly|Los Angeles (Maywood) Assembly]]
| [[w:Commerce, California|City of Commerce, California]]
| [[w:United States|United States]]
| 1932
| 1971
| Plymouth (1946-1958), Dodge (1946-1953, 1955-1958), DeSoto Deluxe/Custom (1948-1952), DeSoto Powermaster Six (1953-1954), DeSoto Firedome (1952-57), DeSoto Fireflite (1955-1957), DeSoto Firesweep (1957-1958), Chrysler Windsor (1948-1958), Chrysler Royal (1949-1950), Chrysler Saratoga (1951-1952, 1957-1958), Chrysler New Yorker (1953-1958), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1960), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-60), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1957-1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Polara|Dodge Polara]] (1960-1964), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1958-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1959-1961), [[w:Plymouth Savoy|Plymouth Savoy]] (1958-1959, 1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1971), [[w:Plymouth Duster|Plymouth Duster]] (1970-1971), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1971), [[w:Dodge Dart#1971|Dodge Dart Demon]] (1971), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1966, 1970), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1971), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1971), [[w:Dodge Coronet|Dodge Coronet]] (1965-1971), [[w:Dodge Charger (1966)|Dodge Charger]] (1971), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971)
| Located at 5800 South Eastern Avenue and Slauson Avenue in Maywood, now part of City of Commerce. Across the street from the [[w:Maywood Assembly|Ford Maywood Assembly plant (Los Angeles Assembly plant No. 1)]]. 
|-
| A (1968-1981),&lt;br&gt; 1 (1960-1967),&lt;br&gt; 6 (1959)
| [[w:Lynch Road Assembly|Lynch Road Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1929
| 1981
| Plymouth (1929-1958), DeSoto (1931-1932), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-64), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (59-61), [[w:Plymouth Fury|Plymouth Fury]] (1959-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (59-61), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (62-70), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1970, 1973-1974), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1978), [[w:Plymouth GTX|Plymouth GTX]] (1967-1970), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1970, 1973, 1975), [[w:Plymouth Superbird|Plymouth Road Runner Superbird]] (1970), [[w:Dodge Coronet|Dodge Coronet]] (1965-1976), [[w:Dodge Monaco#Fourth generation (1977–1978)|Dodge Monaco]] (1977-1978), [[w:Dodge Charger (1966)#1966|Dodge Charger]] (1966), [[w:Dodge Charger (1966)#Third generation|Dodge Charger]] (1971-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971),  [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979-81), [[w:Chrysler New Yorker#1979–1981|Chrysler New Yorker]] (79-81), [[w:Dodge St. Regis|Dodge St. Regis]] (1979-81), [[w:Plymouth Gran Fury#1980–1981|Plymouth Gran Fury]] (80-81),&lt;br&gt; Engines
| Located at 6334 Lynch Road. This was originally Plymouth's home plant. At the time it opened in 1929, Lynch Road was the largest single story auto plant in the world. DeSoto production was transferred from Highland Park to Lynch Road in 1931. In June 1932, DeSoto production was moved to the Jefferson Ave. plant when the DeSoto brand moved up in the brand hierarchy to between Dodge and Chrysler. Previously, DeSoto was between Plymouth and Dodge. During World War II, Lynch Road made tank transmissions, truck parts, and uranium enrichment diffusers for the Oak Ridge Gaseous Diffusion Plant in Oak Ridge, TN to produce enriched uranium for the atomic bomb. For 1965, Lynch Road began to focus on production of Plymouth and Dodge intermediate models. For 1979, Lynch Road was switched to build Chrysler's R-body full-size cars for all 3 Chrysler car brands. Production ended on April 3, 1981 and the factory closed. Last car produced was a white Plymouth Gran Fury police car. 
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant I]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1998
| 2019
| [[w:Chrysler PowerTech engine#4.7|4.7L PowerTech SOHC V8]],&lt;br&gt; [[w:Chrysler Pentastar engine|3.0L/3.2L/3.6L Pentastar V6]]
| Located at 4000 St. Jean Avenue. Mack Ave. Engine Plant I was built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in 1998. 4.7L V8 engine production ended in April 2013. Pentastar V6 engine production ended in 2019. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant II]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2000
| 2012
| [[w:Chrysler PowerTech engine#3.7 EKG|3.7L PowerTech SOHC 90° V6]]
| Located at 4500 St. Jean Avenue. Mack Ave. Engine Plant II was built next to the Mack Ave. Engine Plant I, which had been built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in November 2000. Production ended in September 2012. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
|
| Mack Ave. Stamping Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 1979
| Stampings
| Chrysler acquired the Mack Ave. Stamping Plant from Briggs Manufacturing Company in 1953. The plant was originally built in 1916 by the Michigan Stamping Company, which was taken over by Briggs Manufacturing in 1923. The plant was closed in 1979 and the site was basically abandoned. The city of Detroit bought the plant site in 1982 but was unable to find a purchaser or afford environmental remediation for the site and returned it to Chrysler. In 1990, Chrysler began cleanup and demolition of the old plant and built a new factory on the site, which became the New Mack Assembly Plant. The site later became the Mack Ave. Engine Complex and later, the Mack Ave. Assembly Plant, which is now part of the Detroit Assembly Complex.
|-
| 
| McGraw Stamping/McGraw Glass Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 19?
| 2003
| Oil pans, valve covers, and other small stampings,&lt;br&gt; Automotive Glass (1960-)
| Located at 9400 McGraw Ave. Was around the corner and behind the Wyoming Ave. DeSoto/Export plant. Originally, a stamping plant. In 1960, switched to making automotive glass. Used Safeguard brand. Glass was DOT# 21. Demolished. 
|-
|
| [[w:Mound Road Engine|Mound Road Engine Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 2002
| [[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]],&lt;br&gt; [[w:Chrysler LA engine#239 V6|3.9L 90° V6 engine]],&lt;br&gt; [[w:Chrysler LA engine#Magnum 8.0 L V10|8.0L iron Magnum V10 engine]],&lt;br&gt; [[w:Viper engine|8.0L aluminum Viper V10 engine]] (1992-5/01)
| Located at 20300 Mound Road. One of the plants Chrysler acquired from Briggs Manufacturing Company in 1953. Chrysler used the plant to produce aircraft parts from 1953-1954 and then transferred the plant to Plymouth in 1954 to build its new A-series V8 engine for 1956 model year cars. Converted into an engine plant and enlarged by 71,000 sq. ft., it began building V8 engines for Plymouth in July 1955. Dodge later used the A engine from 1959 in the US in cars and trucks and Chrysler used the A engine from 1960 in the US. The plant was closed in 2002 and demolished in 2003. The land was then paved over and is now used as a storage lot for vehicles produced at the nearby Warren Truck Assembly Plant. Warren Truck Assembly is just to the north of where Mound Road Engine was. Mt. Elliott Tool and Die was located immediately to the south of the Mound Road Engine Plant on Outer Drive East.
|-
|
| [[w:Mount Elliott Tool and Die|Mount Elliott Tool and Die]]/Outer Drive Manufacturing Technology Center/Outer Drive Stamping
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1938, 1953 (became part of Chrysler)
| 2018
| Stamping Dies, Checking Fixtures, Stamping Fixtures 
| Located at 3675 Outer Drive East. Built in 1938 by Briggs Manufacturing Company. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Chrysler renamed it Outer Drive Stamping. Stamping operations ended in 1983 and operations from the closed Vernor Tool &amp; Die plant were moved here. The plant was then renamed Outer Drive Manufacturing Technology Center. The plant now did tool and die work as well as pilot plant operations and engineering for new stamping technologies. Once the Chrysler Technology Center in Auburn Hills was built, Pilot Operations and Advanced Stamping Manufacturing Engineering moved there and the plant was renamed Mount Elliott Tool and Die. Operations at the plant ended in 2018 and the plant was sold to German automotive supplier Laepple Automotive in 2024. Laepple Automotive is producing stamped body parts at the plant. 
|-
| V
| [[w:Detroit Assembly Complex – Mack|New Mack Assembly Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 1995
| [[w:Dodge Viper|Dodge Viper RT/10]] (1992-96)
| Located at 4000 St. Jean Avenue. The New Mack Assembly Plant is built on the site of the former Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. Viper production began in May 1992 at the New Mack Assembly Plant. After ending production in 1995, New Mack Assembly was converted into the Mack Ave. Engine Plant I. A new addition was then built to create Mack Ave. Engine Plant II. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021. 
|-
| F (1968-2009),&lt;br&gt; 6 (1960-1967),&lt;br&gt; 5 (1959),&lt;br&gt; N (1958)
| [[w:Newark Assembly|Newark Assembly]]
| [[w:Newark, Delaware|Newark, Delaware]]
| [[w:United States|United States]]
| 1951,&lt;br&gt; 1957 (Automotive prod.)
| 2008
| Plymouth (1957-1958), Dodge (1958), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1958-1959, 1961), [[w:Plymouth Fury|Plymouth Fury]] (1959-1974), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge Polara|Dodge Polara]] (1960-1966), [[w:Dodge Monaco|Dodge Monaco]] (1965-1966, 1968), [[w:Chrysler Newport|Chrysler Newport]] (1965-1970), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1965-70), [[w:Chrysler Town &amp; Country (1941–1988)|Chrysler Town &amp; Country]] (1966-1967), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1964, 1974-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1964, 1974-1975), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volare|Plymouth Volare]] (1976-1980), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1979-1980), [[w:Plymouth Reliant|Plymouth Reliant]] sedan &amp; wagon (1981-1988), [[w:Dodge Aries|Dodge Aries]] sedan &amp; wagon (1981-1988), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (sedan: 1984-1988, wagon: 1982-1988), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1989-1995), [[w:Dodge Spirit|Dodge Spirit]] (1989-1995), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron Sedan (A-body)]] (1990, 1993-1994), [[w:Chrysler Saratoga#1989–1995|Chrysler Saratoga]] (For export: 1990-1992), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe (J-body)]] (1992-1993), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron convertible (J-body)]] (1992-1995), [[w:Dodge Intrepid#First generation (1993–1997)|Dodge Intrepid]] (1994-1996), [[w:Chrysler Intrepid#First generation (1993–1997)|Chrysler Intrepid]] (Canada: 1994-1995), [[w:Chrysler Concorde#First generation (1993–1997)|Chrysler Concorde]] (1995-1996), [[w:Dodge Durango#First generation (DN; 1998)|Dodge Durango (DN)]] (1998-2003), [[w:Dodge Durango#Second generation (HB; 2004)|Dodge Durango (HB)]] (2004-2009), [[w:Chrysler Aspen|Chrysler Aspen]] (2007-2009) 
| Chrysler began construction of the Delaware Tank Plant in January 1951 to build [[w:M48 Patton|M48 Patton]] tanks. Production began in April 1952. Production ended in May 1961 and the Tank Plant was closed in October 1961. Low rate initial production of the [[w:M60 tank|M60 tank]] was also done at the Newark plant in 1959 before production was moved to the [[w:Detroit Arsenal (Warren, Michigan)|Detroit Arsenal Tank Plant]] in Warren, MI in 1960. Conversion to automotive production began in 1956. Production of Plymouth and Dodge cars began on April 30, 1957. Production ended on December 19, 2008. Sold to the University of Delaware on October 24, 2009. Most of the plant was demolished in 2010-2011 except for the Administration Building near the front of the complex. The site is now the Science, Technology, and Advanced Research (STAR) campus. The old Chrysler Administration Building has been redesigned and is now being used by the College of Health Sciences. 
|-
| K
| [[w:Pillette Road Truck Assembly|Pillette Road Truck Assembly]]
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1976
| 2003
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1976-1980),&lt;br&gt; [[w:Dodge Ram Van|Dodge Ram Van]] (1981-2003), [[w:Dodge Ram Wagon|Dodge Ram Wagon]] (1981-'02), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1976-1983)
| Located at 2935 Pillette Road. Was originally Windsor Plant 6. The Pillette Road plant was located less than a mile away from Chryslers' main plant complex in Windsor. Production began in January 1976. Production ended on June 12, 2003. 2,309,399 units were built. Demolished in 2004. Part of the site is now the Grand Central Business Park. Another part was a logistics center serving Chrysler's Windsor Assembly Plant and operated by Syncreon. The Syncreon Automotive Windsor site closed in October 2022. The parts sorting and sequencing work done there was now going to be done in-house at Chrysler's Windsor Assembly Plant. 
|-
| 
| [[w:Los Angeles (Maywood) Assembly#San Leandro Assembly|San Leandro Assembly]]
| [[w:San Leandro, California|San Leandro, California]]
| [[w:United States|United States]]
| 1948
| 1954
| Plymouth (1949-1954),&lt;br&gt; Dodge (1949-1954)
| 
|-
| B (1996-2009),&lt;br&gt; G (1968-1991),&lt;br&gt; 7 (1960-1967),&lt;br&gt; 8 (1959)
| [[w:Saint Louis Assembly|St. Louis I Assembly]] - South plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1959
| 2008
| [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1960), [[w:Plymouth Fury|Plymouth Fury]] (1960-1964), [[w:Plymouth Savoy|Plymouth Savoy]] (1960-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1961), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1965, 1976), [[w:Plymouth Duster|Plymouth Duster]] (1973-1976), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1965, 1973-1976), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1965),&lt;br&gt; [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1974), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-75), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1976), [[w:Dodge Coronet|Dodge Coronet]] (1965-1973, 75), [[w:Dodge Charger (1966)|Dodge Charger]] (1968-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971), [[w:Dodge Diplomat|Dodge Diplomat]] (1977-1981), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1977-1981), [[w:Plymouth Caravelle|Plymouth Caravelle (M-body)]] (Canada: 1978-1981), [[w:Plymouth Reliant|Plymouth Reliant]] 2-d (1982-1986), [[w:Dodge Aries|Dodge Aries]] 2-d (1982-1986), [[w:Dodge 400|Dodge 400]] 2-d &amp; convertible (1982-1983), [[w:Dodge 600|Dodge 600]] 2-d &amp; convertible (1984-1986), [[w:Plymouth Caravelle|Plymouth Caravelle]] 2-d (Canada: 1983-1986), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (2-d &amp; convertible: 1982-1986), [[w:Chrysler Executive|Chrysler Executive]] (1983-1986), [[w:Dodge Daytona|Dodge Daytona]] (1984-1991), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1984-1991), [[w:Dodge Daytona#Chrysler Laser|Chrysler Laser]] (1984-1986), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe/convertible (J-body)]] (1987-1991), [[w:Plymouth Voyager|Plymouth Voyager]] (1996-2000), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1996-2007), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2009), [[w:Chrysler Voyager|Chrysler Voyager]] (2001-2003), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1996-2001, 2004-2007)
| Located at 1001 N. Hwy Dr. St. Louis South was idled in 1991. The Dodge Daytona was moved to Sterling Heights and the J-body Chrysler LeBaron was moved to Newark, DE. St. Louis South was reopened in 1995 to build minivans, which were moved from the St. Louis North plant. For the 3rd generation, St. Louis South built both SWB and LWB models. For the 4th generation, St. Louis South built all SWB models but also built some LWB models. Closed on October 31, 2008. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1996-2009),&lt;br&gt; X (1973-1995),&lt;br&gt; U (1970-1972),&lt;br&gt; 7 (1967-1969)
| [[w:Saint Louis Assembly|St. Louis II Assembly]] - North plant / Missouri Truck Assembly Plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1966
| 2009
| [[w:Dodge D series|Dodge D/W series]] (1967-1973), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1974-1977), [[w:Plymouth Trail Duster|Plymouth Trail Duster]] (1974-1976), [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1980), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1975-1976, 1980),&lt;br&gt; [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1984-1987), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1984-1987), [[w:Dodge Diplomat|Dodge Diplomat]] (1984-1987), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]] ('84-'87), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1987-1995), [[w:Dodge Caravan|Dodge Grand Caravan]] (1987-1995), [[w:Chrysler minivans (S)#Cargo van|Dodge Extended Mini Ram Van]] (1987-1988), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1990-1995),&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1996-'09)
| Originally opened to build trucks as the Missouri Truck Assembly Plant. In 1980, the plant was idled. Plant was reopened in 1983 to build the rwd, M-body sedans, which were moved from Windsor, ON, Canada so that Windsor could be converted to build minivans. Plant was renamed St. Louis II Assembly. During 1987, the M-body sedans were moved to AMC's plant in Kenosha, WI so that St. Louis North could be converted to build the new LWB minivans. For the first 2 generations of Chrysler minivans, St. Louis North only built the LWB models. For 1996, minivan production moved to St. Louis South and the North plant was converted to build full-size pickups. Closed on July 10, 2009. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1970-1978),&lt;br&gt; 6 (1968-1969),&lt;br&gt; 9 (1961-1967)
| Tecumseh Road Truck Assembly / Windsor Truck Assembly Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1916,&lt;br&gt; 1925 (became part of Chrysler)
| 1978
| Maxwell (1924-1925),&lt;br&gt; Chrysler (1924-1929),&lt;br&gt; Dodge Trucks (1931-1960), &lt;br&gt; Dodge D-Series Trucks: &lt;br&gt; D100 (1961), D200 (1964), W200 (1965), W100 (1966), D100/W100 (1967), D200 (1970), D500 (1968),&lt;br&gt; D500/D600/D700/D800 &lt;br&gt; medium-duty trucks (1970-1972), D500/D600/W600/D700/D800 medium-duty trucks (1974-1977),&lt;br&gt; D100 (1978),&lt;br&gt; Fargo Trucks (1936-1972)
| Located at 300 Tecumseh Road East. Was originally Windsor Plant 1. Became part of Chrysler upon its founding in 1925. Was previously a Maxwell-Chalmers plant and was originally Maxwell's Canadian plant from 1916. Switched to building trucks in 1931 after Plant 3 opened in 1929. Closed in 1978. Became the Imperial Quality Assurance Centre from 1980-1983, doing extra quality control on the 1981-1983 Imperial built at the Windsor Assembly Plant (the car plant - Plant 3) about 1.5 miles east of Plant 1. The Imperial Quality Assurance Centre closed in 1983 when the Imperial was discontinued. Subsequently demolished. Is now the Plaza 300 shopping mall.
|-
|
| Tipton Transmission Plant
| [[w:Tipton, Indiana|Tipton, Indiana]]
| [[w:United States|United States]]
| 2014
| 2023
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, SI-EVT transmission
| Located at 5880 W. State Road 28. Originally, the plant was supposed to be a [[w:Getrag|Getrag]] plant focused on supplying Chrysler with dual-clutch transmissions. Chrysler withdrew from the deal in 2008 after a dispute over financing and sued Getrag. The 80% completed facility then sat dormant until Chrysler Group purchased the facility in February 2013 and completed construction. Production began in April 2014 with the ZF-designed 9-speed auto. transmission, built under license from ZF. Production ended in June 2023 and 9-speed production was consolidated into Indiana Transmission Plant I in Kokomo, IN. The SI-EVT transmission for the Pacifica Hybrid was moved to Kokomo Transmission Plant. Sold to IRH Manufacturing LLC in September 2024 to make solar cells.
|-
| L (1989-2001),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Parkway|Toledo Assembly #1 Plant]] - Jeep Parkway plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2001 (ended final assembly), 2006 (ended body assembly)
| [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]] (1984-01), [[w:Jeep Cherokee (XJ)#Wagoneer|Jeep Wagoneer (XJ)]] (1984-90), [[w:Jeep Comanche|Jeep Comanche]] (1986-1992),&lt;br&gt; Bodies for vehicles made at Stickney Ave. plant &lt;br&gt; 
Models only made before Chrysler takeover: Willys Aero (1952-1955), Kaiser Manhattan (1954-1955), Jeep CJ (1946-1986), Jeep DJ, Jeep Jeepster (1948-1950), Jeep Jeepster Commando (1967-1971), Jeep Commando (1972-1973), Willys Jeep Station Wagon (1946-1964), Willys Jeep Truck (1947-1965), Jeep Gladiator (SJ) (1963-1971), Jeep J-series pickup, Jeep Wagoneer [SJ] (1963-1981), Jeep Cherokee [SJ] (1974-1981), Jeep Forward Control [FC] (1957-1965), Jeep FJ Fleetvan (1961-1975)
| Located at 1000 Jeep Parkway. The John North Willys-owned Overland Automobile Co. purchased the plant in 1909 from Pope-Toledo, another early automaker. Overland Automobile Co. became Willys-Overland in 1912. Began building Jeeps in the 1940s. This was the original Jeep assembly plant. Willys-Overland was bought by Kaiser in 1953. Kaiser then sold its Willow Run plant in Ypsilanti, MI to GM and moved its production to the Willys plant in Toledo, OH. Kaiser and Willys production in the US ended in 1955 and Toledo focused on Jeep production going forward. Kaiser Jeep was sold to AMC in 1970. Became part of Chrysler in the 1987 buyout of AMC. Final assembly ended in 2001 when the XJ Cherokee ended production but painted body production continued until June 30, 2006, when the TJ Wrangler ended production. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant a few miles away.  The Administration Building, used from 1915 through 1974, was imploded on April 14, 1979. A third of the plant was demolished in 2002 after final assembly ended including the Jeep Museum. The remainder was demolished in 2006-2007 after body production ended. One of the three large brick smokestacks was preserved and was dedicated in 2013 to the plant's history and workforce. A bronze plaque was mounted next to the smokestack, which still says &quot;Overland&quot; on it. Over 11 million vehicles were produced at the site, including military Jeeps during World War II. The site was sold to the Toledo-Lucas County Port Authority in 2010. The site has been redeveloped into the Overland Industrial Park. Dana Inc. and Detroit Manufacturing Systems are among the tenants in the Overland Industrial Park and those plants supply the current Jeep plants elsewhere in Toledo. All-Phase Electric Supply Co. is another tenant. 
|-
| P (1989-2006),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Stickney|Toledo Assembly #2 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2006 
| [[w:Jeep Wagoneer (SJ)#1984: SJ and XJ|Jeep Grand Wagoneer (SJ)]]&lt;br&gt; (1984-1991),&lt;br&gt; [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1993-95), [[w:Jeep Wrangler (TJ)|Jeep Wrangler (TJ)]] (1997-06)
Models only made before Chrysler takeover:&lt;br&gt; Jeep Wagoneer [SJ] (1981-1983), Jeep Cherokee [SJ] (1981-1983)
| Located at 4000 Stickney Ave. Originally opened in 1942 by the Electric Auto-Lite Co., a maker of spark plugs. Sold to Kaiser-Jeep in 1964, which used it as a machining and engine plant until 1981, when AMC converted it for vehicle production. AMC had taken over Kaiser Jeep in 1970. AMC built the SJ Wagoneer and Cherokee at the Stickney Ave. plant. Body assembly was done at an SJ- or later,  Wrangler-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant. Became part of Chrysler in the 1987 buyout of AMC. Production ended in 2006 with the end of the TJ Wrangler. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant built on the site of the old Stickney Ave. plant. 
|-
| W (1994-1996)
| [[w:Toledo Complex#Stickney|Toledo Assembly #3 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1993
| 1996 
| [[w:Dodge Dakota#First generation (1987–1996)|Dodge Dakota]] (1994-1996)
| Body assembly was done at a Dakota-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant.
|-
|
| Toluca Engine Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2002
| [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Closed in 2002
|-
|
| Toluca Transmission Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2001
| Automatic Transmissions for fwd cars
| Closed in 2001
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine North Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 1952
| 2022
| [[w:Chrysler B engine|Chrysler B V8 engine]],&lt;br&gt; [[w:Chrysler B engine#RB engines|Chrysler RB V8 engine]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]],&lt;br&gt; [[w:Volkswagen EA827 engine#1.7|VW 1.7L EA827 I4 engine]] (adding Chrysler parts to already built VW engines made in W. Germany),&lt;br&gt; [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L/2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|1.8L, 2.0L I4 &quot;Neon engine&quot;]], [[w:Chrysler 3.3 &amp; 3.8 engines|3.3L/3.8L OHV V6]], [[w:Chrysler SOHC V6 engine|3.5L/3.2L/4.0L SOHC V6]], [[w:Chrysler Pentastar engine|3.2L/3.6L Pentastar V6 engine]], [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4]],&lt;br&gt; Engine components,&lt;br&gt; Air raid sirens
| Located at 2000 Van Horn Road. Trenton North began production in fall 1952 and was expanded in 1964, 1967, 1969, 1976, and 1977. At first, Trenton North began by making water pumps and air raid sirens but engines quickly followed. Trenton Engine North was Chrysler’s first dedicated engine factory in the US, separate from the assembly plants. On September 29, 1978, V8 production ended. Trenton North added Chrysler parts such as the intake and exhaust manifolds, water pump, ignition system and other major parts to already built VW 1.7L EA827 I4 engines imported from Salzgitter, W. Germany for use in the Omni/Horizon. Production of the 3.5L V6 moved to Kenosha Engine in 2003. Trenton North was idled in May 2011 when the 3.8 V6 ended production following the end of 3.3 &amp; 4.0 V6 engine production in 2010. Chrysler then announced in June 2011 it would use a fifth of the plant to make components for the Pentastar V6 being made at Trenton South. In January 2012, Trenton North began producing the 3.6L Pentastar V6 engine. Chrysler then installed a flexible production line that could build both the Pentastar V6 and the Tigershark I4. In May 2013, Trenton began producing the 3.2L Pentastar V6. Tigershark I4 production began late in 3rd quarter 2013. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production. Trenton North has been repurposed for warehousing and other non-manufacturing opportunities. 
|-
|
| [[w:Twinsburg Stamping|Twinsburg Stamping]]
| [[w:Twinsburg, Ohio|Twinsburg, Ohio]]
| [[w:United States|United States]]
| 1957
| 2010
| Stampings and assemblies
| Located at 2000 East Aurora Road. Opened in August 1957. Closed July 31, 2010. Sold in 2011. Demolished in 2012-2013. Now the Cornerstone Business Park.
|-
|
| Vernor Tool &amp; Die plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1983
| Tooling &amp; Dies
| Located at 12026 E Vernor Highway. Operations moved to Mount Elliott Tool and Die. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
|
| Vernor Trim plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1970's
| Trim
| Located at 12025 E Vernor Highway. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
| 4 (1960-1961),&lt;br&gt; 7 (1959)
| Warren Avenue Plant
| [[w:Dearborn, Michigan|Dearborn, Michigan]]
| [[w:United States|United States]]
| 1927,&lt;br&gt; 1950 (opened as part of Chrysler)
| 1960s
| DeSoto bodies (1950-1958), DeSoto engines &lt;br&gt; (1951-1958),&lt;br&gt; [[w:Imperial (automobile)#Second generation (1957–1966)|Imperial]] (1959-1961)
| Located at 8505 West Warren Avenue. This was previously the factory of Paige and Graham-Paige. Chrysler leased half the plant in 1941 to use for military production. Chrysler produced aircraft components for the [[w:Martin B-26 Marauder|B-26 Marauder]] (nose and center fuselage sections) and the [[w:Boeing B-29 Superfortress|B-29 Superfortress]] (the pressurized nose section, wing leading edges, and engine cowlings). The B-29 had a nose so large that trenches had to be dug in the floor and some of the bracing for the plant’s roof girders had to be removed to accommodate the aircraft. Chrysler bought the plant in 1947. Began building bodies for DeSoto in August 1950. Engine production began in 1951. Production of the Imperial brand moved here from the Jefferson Ave. plant in Detroit for 1959 in an attempt to give the Imperial brand its own, exclusive factory however sales weren't high enough to support its own plant so Imperial production moved back to the Jefferson Ave. plant in Detroit for 1962. Production of small parts followed for a few years as did export operations. The plant was later sold. Most of the plant has seen been demolished but the front building facing on Warren Ave. is still there and is now used as Corporate HQ by Shatila Food Products. The DeSoto logo, featuring a stylized image of Hernando de Soto, can still be seen at the top of the building above the front door. 
|-
| T (1970-), 2 (1966-1969),&lt;br&gt; 
| Warren Truck #2 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1966
| 1975
| Dodge heavy-duty trucks
| Located at 6600 East 9 Mile Road, at the corner of Sherwood Ave and East 9 Mile Road. Closed in 1975 when Dodge exited the heavy-duty truck market. Site now belongs to Sundance Beverage Co., the parent of Everfresh Juice Co.
|-
| V (1971-1979)
| Warren Truck (Compact) #3 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1970
| 1979
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1979),&lt;br&gt;[[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1974-1979)
| Located on Hoover Road between 8 Mile Road and 9 Mile Road. 
|-
| 
| Windsor Engine Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1938
| 1980
| [[w:Chrysler flathead engine#Straight-6|Chrysler flathead inline 6]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], &lt;br&gt;[[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Was originally Windsor Plant 2. Located just to the south of Windsor Plant 3, the current minivan factory. Closed in August 1980. Built over 8 million engines. Windsor Assembly Plant (Plant 3) expanded onto the site of the old engine plant when it was being renovated for minivan production. 
|- 
|
| [[w:Detroit Assembly#LaSalle Factory/DeSoto Factory|Wyoming Ave. Assembly (DeSoto Wyoming Ave. plant)]] / Wyoming Export Plant
| [[w:Detroit|Detroit]], [[w:Michigan|Michigan]]
| United States
| 1936
| 1958 (Vehicle prod.),&lt;br&gt; 1980 (export operations)
| [[w:DeSoto (automobile)|DeSoto]] (1937-1958),&lt;br&gt; CKD Export (1960-1980)
| Located at 6000 Wyoming Avenue. Originally built to produce Liberty aircraft engines in World War I, opening in 1917. In 1919, was taken over by Saxon Motor Co., owned by Hugh Chalmers of Chalmers Motor Co. GM bought the plant in 1926 and built the LaSalle there from 1927-1933. GM sold Wyoming Assembly to Chrysler in 1934, which then used it to build its DeSoto brand. Became DeSoto's home plant. During World War II, Chrysler built wing center sections for the [[w:Curtiss SB2C Helldiver|Curtiss SB2C Helldiver]] at the Wyoming Ave. plant. For 1959, DeSoto's entry level model, the Firesweep, was moved to the Dodge plant in Hamtramck and DeSoto's other, higher end models were moved to Chrysler's Jefferson Ave. plant in Detroit. This was done so that bodies and final assembly would be done in either a single facility or a pair of connected facilities. This was part of Chrysler's move to unibody construction for 1960 for all cars except Imperial. After the DeSoto brand was discontinued in late 1960, became Wyoming Export plant which was used to prepare vehicles for export. Plant closed in 1980. Plant was demolished in 1992. Site is now occupied by Comprehensive Logistics Inc.
|}

==Non-Chrysler FCA/Stellantis Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| [[w:Fiat Cassino Plant|Cassino Plant]]
| [[w:Piedimonte San Germano|Piedimonte San Germano]], [[w:Province of Frosinone|Province of Frosinone]]
| [[w:Italy|Italy]]
| 1972
| 
| Past Chrysler Group models: [[w:Lancia Delta#|Chrysler Delta]] (UK/Ireland)
| Fiat plant. 
|-
| 3
| [[w:Alfa Romeo Pomigliano d'Arco plant|Pomigliano d'Arco plant]] (Giambattista Vico plant)
| [[w:Pomigliano d'Arco|Pomigliano d'Arco]], [[w:Metropolitan City of Naples|Metropolitan City of Naples]]
| [[w:Italy|Italy]]
| 1972
|
| Past Chrysler Group models: [[w:Dodge Hornet|Dodge Hornet]] (2023-2025). Related models:&lt;br&gt; [[w:Alfa Romeo Tonale|Alfa Romeo Tonale]] (2023-)
| Originally, an Alfa Romeo plant. Oriiginally owned by Construction Industry Neapolitan Vehicles Alfa Romeo - Alfasud S.p.A., a joint venture between Alfa Romeo (88%), Finmeccanica (10%), and IRI (2%). In 1982, Alfasud S.p.A. was renamed Inca Investments. Alfa Romeo was taken over by Fiat in 1986. Fiat merged with Chrysler to form Fiat Chrysler Automobiles (FCA) in 2014. FCA merged with PSA Group to form Stellantis in 2021. 
|}

==Non-Chrysler Group DaimlerChrysler/Daimler AG Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 5
| Mercedes-Benz Plant Düsseldorf
| [[w:Düsseldorf|Düsseldorf]], [[w:North Rhine-Westphalia|North Rhine-Westphalia]]
| [[w:Germany|Germany]]
| 1962
|
| [[w:Dodge Sprinter|Dodge Sprinter]] (2003-2009)
| Mercedes-Benz Plant. 
|-
| 9
| Mercedes-Benz Plant Ludwigsfelde
| [[w:Ludwigsfelde|Ludwigsfelde]], [[w:Brandenburg|Brandenburg]]
| [[w:Germany|Germany]]
| 1991 (Mercedes prod. began)
|
| [[w:Dodge Sprinter#Second generation (2006–2018, NCV3)|Dodge Sprinter]] chassis cab (2007-2009)
| Mercedes-Benz Plant. Originally established in 1936 by Daimler-Benz to make airplane engines. The plant was bombed by the US in 1945. After the war ended, what remained of the factory was dismantled and taken to the Soviet Union as reparations. On February 1, 1991, Mercedes-Benz took a 25% stake in the Ludwigsfelde plant, which had previously belonged to East German truckmaker VEB Automobilwerke. It became a 100% owned subsidiary of Mercedes-Benz on January 1, 1994. Sprinter production began in 2006. 
|}

==Former partner factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Prod. for Chrysler began
! style=&quot;width:10px;&quot;|Prod. for Chrysler ended
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 6
| [[w:China Motor Corporation|China Motor Corporation]]
| [[w:Yangmei District|Yangmei District]], [[w:Taoyuan, Taiwan|Taoyuan]]
| [[w:Taiwan|Taiwan]]
| 2006
| 2007
| [[w:Chrysler Town &amp; Country (minivan)#Fourth generation (2001–2007)|Chrysler Town &amp; Country]]&lt;br&gt; (Taiwan: 2006-2007),&lt;br&gt; [[w:Dodge 1000|Dodge 1000]] (Mexico: 2007-'10)
| China Motor Corporation plant. Built for Chrysler under license by China Motor Corporation of Taiwan. Production began April 18, 2006. 
|-
|
| [[w:Carrozzeria Ghia|Carrozzeria Ghia]] 
| [[w:Turin|Turin]]
| [[w:Italy|Italy]]
| 1957
| 1965
| [[w:Imperial (automobile)#Imperial Crown (1955–1965)|Imperial Crown Limousine]] (1957-1965) modified, painted limousine bodies and interiors
| 132 Imperial Crown Limousines were built by Ghia under contract for Chrysler between 1957 and 1965. The 1957-1959 models were based on modified 2-door hardtops with the more rigid chassis from the convertible. The 1960-1965 models were based on 4-door models. Ghia lengthened the frame and modified the bodywork and interiors to create the limousines. After producion ended in 1965, Ghia sold the tooling to Barreiros of Spain, which built another 10 Imperial Crown Limousines. Barreiros had been 35% owned by Chrysler since 1963. That was increased to 77% in 1967 and 100% in 1969.  
|-
| U
| [[w:Hyundai Motor Company|Hyundai Motor Co.]] - [[w:List of Hyundai Motor Company manufacturing facilities#Ulsan Plant|Ulsan plant]]
| [[w:Ulsan|Ulsan]]
| [[w:South Korea|South Korea]]
| 2000
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge Atos|Dodge Atos]] (2001-2012),&lt;br&gt;  [[w:Dodge Verna|Dodge Verna]] (2004-06),&lt;br&gt;  [[w:Dodge Attitude#First generation (MC; 2006)|Dodge Attitude (MC)]] (2007-'11), [[w:Dodge Attitude#Second generation (RB; 2011)|Dodge Attitude (RB)]] (2012-'14), [[w:Dodge H100|Dodge H100 truck]],&lt;br&gt; [[w:Hyundai Starex#Second generation (TQ; 2007)|Dodge H100 Van/Wagon]]
| Rebadged Hyundai models sold as Dodges in Mexico. 
|-
| 
| [[w:Hyundai Motor India|Hyundai Motor India]]
| [[w:Chennai|Chennai]], [[w:Tamil Nadu|Tamil Nadu]]
| [[w:India|India]]
| 2011
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge i10|Dodge i10]] (2012-2014)
| Rebadged Hyundai model sold as a Dodge in Mexico. 
|-
| X
| [[w:Karmann|Karmann Osnabrück Assembly]]
| [[w:Osnabrück|Osnabrück]], [[w:Lower Saxony|Lower Saxony]]
| [[w:Germany|Germany]]
| 2003
| 2007
| [[w:Chrysler Crossfire|Chrysler Crossfire]] (2004-2008)
| Karmann plant. Built under contract for Chrysler. 
|-
| Y
| [[w:Magna Steyr|Magna Steyr]] / Steyr-Daimler-Puch - Chrysler Steyr Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1994
| 2010
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]]&lt;br&gt; (1995-2010),&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010), [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (2003-2007),&lt;br&gt; [[w:Chrysler 300#First generation (2005)|Chrysler 300C/300C Touring]] (2005-2010)
| Originally, a Steyr-Daimler-Puch plant. Magna International acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. Production of the Chrysler Voyager and Grand Voyager minivans moved from the Eurostar plant  next door to the main Magna Steyr plant for 2003. Chrysler minivan production in Austria ended on November 30, 2007. Built under contract for Chrysler. 
|-
| B
| [[w:Maserati|Maserati]] - [[w:Innocenti|Innocenti]] plant
| [[w:Lambrate|Lambrate district]], [[w:Milan|Milan]]
| [[w:Italy|Italy]]
| 1988
| 1990
| [[w:Chrysler TC by Maserati|Chrysler TC by Maserati]]&lt;br&gt; (1989-1991)
| Developed jointly by Chrysler and Maserati, the TC was built in Italy by Maserati at the Innocenti plant in Milan. Maserati and Innocenti were both owned by DeTomaso at the time. Chrysler bought a 5% stake in Maserati in 1984 and increased its stake to 15.6% in 1986. Production ended in 1990 due to low sales.
|-
| U
| Mitsubishi - Mizushima plant (Line 1)
| [[w:Kurashiki|Kurashiki]], [[w:Okayama Prefecture|Okayama Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| [[w:Plymouth Champ|Plymouth Champ]] (1981-1982), [[w:Plymouth Colt|Plymouth Colt]] (1983-1994), [[w:Dodge Colt|Dodge Colt]] (1981-1994), [[w:Dodge Colt|Dodge/Plymouth Colt]]&lt;br&gt; (Canada only: 1995),&lt;br&gt; [[w:Eagle Summit|Eagle Summit]]&lt;br&gt; (4-d: 1989-1990, 1993-1996,&lt;br&gt; 3-d: 1991-1992, 2-d: 1993-96), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1993-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1993-96)
| Mitsubishi Motors plant.
|-
| Z
| Mitsubishi - Okazaki plant
| [[w:Okazaki, Aichi|Okazaki]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1983
| 1996
| [[w:Plymouth Conquest|Plymouth Conquest]] (1984-86), [[w:Dodge Conquest|Dodge Conquest]] (1984-1986), [[w:Chrysler Conquest|Chrysler Conquest]] (1987-1989), [[w:Dodge Colt Vista#Colt Vista|Dodge Colt Vista]] (1984-1991), [[w:Plymouth Colt Vista#Colt Vista|Plymouth Colt Vista]] (1984-91), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1992-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1992-96) [[w:Eagle Vista#Vista Wagon|Eagle Vista Wagon]]&lt;br&gt; (Canada: 1989-1991)
| Mitsubishi Motors plant.
|-
| Y (Line 1)&lt;br&gt;/&lt;br /&gt;P (Line 2)
| Mitsubishi - &lt;br&gt; Ooe plant &lt;br&gt; a.k.a. &lt;br&gt; Nagoya #1&lt;br&gt;/&lt;br&gt;Nagoya #2
| Ooe-cho, [[w:Minato-ku, Nagoya|Minato ward]], [[w:Nagoya|Nagoya]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| VIN code Y:&lt;br&gt; [[w:Plymouth Sapporo|Plymouth Sapporo]] (1981-1983), [[w:Dodge Challenger#Second generation (1978–1983)|Dodge Challenger]] (1981-1983), [[w:Plymouth Arrow Truck#Chrysler variants|Plymouth Arrow Truck]] ('81-'82), [[w:Dodge Ram 50|Dodge Ram 50]] (1981-1984), [[w:Dodge Stealth|Dodge Stealth]] (1991-1996)
VIN code P:&lt;br&gt; [[w:Dodge Ram 50|Dodge Ram 50]] (1985-1986), [[w:Dodge Ram 50#North America|Dodge Ram 50]] (1987-1993)
| Mitsubishi Motors plant. Closed in 2001. Sold to Mitsubishi Heavy Industries and now used by its Aircraft, Defense &amp; Space Business Area.
|-
| J
| Mitsubishi -  Toyo Koki/Pajero Manufacturing Co., Ltd. plant
| [[w:Sakahogi, Gifu|Sakahogi]], [[w:Gifu Prefecture|Gifu Prefecture]] 
| [[w:Japan|Japan]]
| 1986
| 1989
| [[w:Dodge Raider|Dodge Raider]] (1987-1989)
|  Originally, a Toyo Koki Co. Ltd. plant. Opened in 1976. Built vehicles under contract for Mitsubishi. Mitsubishi Motors owned 35% of Toyo Koki and increased its stake to a majority in March 1995. The plant was then renamed Pajero Manufacturing Co., Ltd. in July 1995. In March 2003, Mitsubishi bought all the remaining shares in Pajero Manufacturing Co., Ltd., making it a wholly owned subsidiary. Closed in 2021. Sold to Daio Paper in 2022.
|-
| H (Attitude), 9 (1200)
| [[w:Mitsubishi Motors (Thailand)|Mitsubishi Motors Thailand]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 2014
| 2024
| [[w:Dodge Attitude#Third generation (A10; 2015)|Dodge Attitude]]&lt;br&gt; (Mexico: 2015-2024),&lt;br&gt; [[w:Mitsubishi Triton#Fifth generation (KJ/KK/KL; 2014)|Ram 1200]]&lt;br&gt; (Middle East: 2017-2019)
| Mitsubishi Motors plant.  
|-
| ?
| [[w:MMC Automotriz|MMC Automotriz]]
| [[w:Barcelona, Venezuela|Barcelona]], [[w:Anzoátegui|Anzoátegui state]]
| [[w:Venezuela|Venezuela]]
| 2002
| 2009
| [[w:Dodge Brisa|Dodge Brisa]]&lt;br&gt; ([[w:Hyundai Accent#First generation (X3; 1994)|2002-2005]]), ([[w:Hyundai Getz|2006-2009]])
| MMC Automotriz plant. Originally, MMC Automotriz was 49% owned by Consorcio Inversionista Fabril S.A. (CIF) of Venezuela and 42% owned by Nissho Iwai Corp. The remaining 9% of the company was owned by the Japan International Development Organization Ltd., a partnership between the government-financed Overseas Economic Cooperation Fund and 98 private companies. Nissho Iwai merged with Nichimen Corp. in 2004 to form Sojitz Corp. Sojitz later increased its stake in MMC Automotriz to 98%, with the other 2% still held by CIF. MMC Automotriz was sold to the the Sylca Group (also known as Yammine Group) in 2015. MMC Automotriz produced Mitsubishi vehicles and from 1996-2012, also produced Hyundai vehicles. The Dodge Brisa was produced for DaimlerChrysler as part of its cooperation with [[w:Hyundai Motor Company|Hyundai]]. MMC Automotriz also produced Mitsubishi Fuso trucks. 
|-
| G
| [[w:Mitsubishi Motors (Thailand)|MMC Sittipol Co., Ltd.]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 1988
| 1992
| [[w:Plymouth Colt#Fifth generation (1985–1988)|Plymouth Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Dodge Colt#Fifth generation (1985–1988)|Dodge Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Eagle Vista|Eagle Vista]] (Canada: 1988-92)
| Mitsubishi Motors plant. MMC Sittipol is the predecessor company of Mitsubishi Motors (Thailand). These were the first vehicle exports from Thailand. 
|-
| 2
| [[w:Renault|Renault]] - [[w:Maubeuge Construction Automobile|Maubeuge plant]]
| [[w:Maubeuge|Maubeuge]]
| [[w:France|France]]
| 1987
| 1989
| [[w:Renault Medallion|Renault Medallion]] (1988),&lt;br&gt; [[w:Eagle Medallion|Eagle Medallion]] (1989)
| Renault plant. The Medallion was sold through Chrysler's Jeep-Eagle dealer network as a legacy of Chrysler's takeover of AMC from Renault.
|-
| 8,&lt;br&gt; 0
| [[w:Soueast|Soueast]]
| [[w:Fuzhou|Fuzhou]], [[w:Fujian|Fujian province]]
| [[w:China|China]]
| 2008
| 2010
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager]],&lt;br&gt; [[w:Dodge Caravan#Fourth generation (2001–2007)|Dodge Caravan]]
| South East (Fujian) Motor Co., Ltd. plant. Built for Chrysler under license by South East (Fujian) Motor Co., Ltd.
|}</text>
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{{tmbox|type=notice|text=This is a sandbox page, a place for experimenting with Wikibooks.}}

This is a history of Chrysler factories that are being or have been used to produce cars, vans, SUVs, trucks, and automobile components. 

For '''Chrysler brand''' only: Plant code in 1955-1957 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number.

For '''cars''': Plant code in 1958 was not indicated if it was made in the home plant in Michigan but if it was made in a different plant, then it was indicated by a letter in the 4th position of the serial number. Plant code was the number in the 4th position of the 10-digit serial number for 1959-1965. Plant code was the number in the 7th position of the 13-digit serial number for 1966-1967. Plant code was the letter in the 7th position of the 13-digit serial number for 1968-1980.

Canadian-built cars had the plant code as the number in the 5th position of the 11-digit serial number for 1965.

For '''trucks''': The first digit of the 7-digit sequence number (4th position overall of the 10-digit serial number) indicated which plant built the truck for 1967-1968. Plant code was the number in the 4th position of the 10-digit serial number for 1969. Plant code was the letter in the 7th position of the 13-digit serial number for 1970-1980.

Canadian-built models used a different system for VINs until 1968, when Chrysler of Canada adopted the same system as the US. Plant code for trucks was the number in the 5th position of the 10-digit serial number for 1961-1967.

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

For '''AMC passenger cars from 1966-1967''': Plant code is indicated by the letter in the 3rd position of the 13-digit vehicle number. If the 3rd letter is a K, then it was made in Kenosha, WI. If the 3rd letter is a B, then it was made in Brampton, ON, Canada. 

For '''AMC passenger cars from 1968 through 1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. 1-6 is Kenosha, WI and 7-9 is Brampton, ON, Canada.

For '''Canadian-built Jeeps built by AMC Canada for 1979-1980''': Plant code is indicated by the 8th digit (the first of the sequential serial number) of the 13-digit vehicle number. It was made in Canada if it's either an 8 (1979) or a 7 (1980). All other Jeeps from 1980 and earlier were made in Toledo. 

All models from 1981 on have the plant code in the 11th position as per standardized VIN regulations.

==Current factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| D (1968-),&lt;br&gt; 4 (1966-1967)
| [[w:Belvidere Assembly|Belvidere Assembly]] 
| [[w:Belvidere, Illinois|Belvidere, Illinois]]
| [[w:United States|United States]]
| 1965
| Feb. 2023
| 
| Located at 3000 West Chrysler Drive. '''Belvidere Satellite Stamping Plant''' adjoins the main assembly plant. Began production on July 7, 1965. The first vehicle produced was a 1966 Plymouth Fury four-door. In 1972, the Chrysler Town and Country station wagon was added to the Belvidere plant. In 1977, the plant was converted to build front-wheel drive subcompacts. Production of the Dodge Omni and Plymouth Horizon began on December 5, 1977. All L-body derivatives were made at Belvidere through 1987. In 1987, Belvidere was converted to build Chrysler's midsize, fwd C-body sedans. Belvidere then switched back to building small cars and began production of the Neon on November 10, 1993. Belvidere built the last Plymouth, a silver 2001 Neon LX on June 28, 2001. Neon production ended in September 2005. Dodge Caliber began production in January 2006, followed by Jeep Compass in June 2006 and Jeep Patriot in December 2006. Caliber ended production on December 19, 2011. Dodge Dart began production on April 30, 2012, and ended on October 4, 2016. Compass and Patriot production ended on December 23, 2016. Jeep Cherokee production began on June 1, 2017 and ended on February 28, 2023. Belvidere was then idled. &lt;br&gt; Past models: [[w:Plymouth Fury|Plymouth Fury]] (1966-1974), [[w:Plymouth Gran Fury#1975–1977|Plymouth Gran Fury]] (1975-1977), [[w:Dodge Monaco|Dodge Monaco]] (1966-1976), [[w:Dodge Royal Monaco#1977 (Royal Monaco)|Dodge Royal Monaco]] (1977), [[w:Dodge Polara|Dodge Polara]] (1966-1973), [[w:Chrysler Newport|Chrysler Newport]] (1977), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1973-1977), [[w:Dodge Omni|Dodge Omni]] (1978-1987), [[w:Plymouth Horizon|Plymouth Horizon]] (1978-1987), [[w:Dodge Omni 024|Dodge Omni 024]] (1979-1980), [[w:Plymouth Horizon TC3|Plymouth Horizon TC3]] (1979-1980), [[w:Dodge Omni 024|Dodge 024]] (1981-1982), [[w:Plymouth Horizon TC3|Plymouth TC3]] (1981-1982), [[w:Dodge Charger (1981)|Dodge Charger]] (1983-1987), [[w:Plymouth Turismo|Plymouth Turismo]] (1983-1987), [[w:Dodge Rampage|Dodge Rampage]] (1982-1984),  [[w:Plymouth Scamp|Plymouth Scamp]] (1983), [[w:Dodge Dynasty|Dodge Dynasty]] (1988-1993), [[w:Chrysler Dynasty|Chrysler Dynasty]] (Canada: 1988-1993), [[w:Chrysler New Yorker#1988–1993|Chrysler New Yorker]] (1988-1993), [[w:Chrysler New Yorker Fifth Avenue#1990–1993: New Yorker Fifth Avenue|Chrysler New Yorker Fifth Avenue]] (1990-1993), [[w:Chrysler Imperial#1990–1993|Chrysler Imperial]] (1990-1993), [[w:Dodge Neon|Dodge Neon]] (1995-2005), [[w:Plymouth Neon|Plymouth Neon]] (1995-2001), Chrysler Neon (Canada: 2000-02),&lt;br&gt; Dodge SX 2.0 (Canada: 2003-05),&lt;br&gt; [[w:Dodge Caliber|Dodge Caliber]] (2007-2012), [[w:Jeep Compass#First generation (MK49; 2006)|Jeep Compass]] (2007-2017), [[w:Jeep Patriot|Jeep Patriot]] (2007-2017), [[w:Dodge Dart (PF)|Dodge Dart]] (2013-2016), [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2017-2023)
|-
| H (1989-),&lt;br&gt; A (1988)
| [[w:Brampton Assembly|Brampton Assembly]] (Formerly Bramalea Assembly)
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1987
| Dec. 2023
| 
| Located at 2000 Williams Parkway East. Factory was built by AMC and Renault. The plant was acquired by Chrysler as part of its takeover of AMC. Began production on September 28, 1987. Plant was originally known as Bramalea Assembly. Plant was renamed Brampton Assembly in 1992 after Chrysler closed and sold the old AMC plant on Kennedy Road in Brampton. The attached '''Brampton Satellite Stamping Plant''' was added in December 1991 and was built for the launch of the Chrysler LH platform. On December 17, 1991, Eagle Premier and Dodge Monaco production ended. Production of the Chrysler LH platform cars began in June 1992. Production switched to the rear-wheel drive Chrysler LX platform cars in January 2004. The Chrysler 300 was also built for export to mainland Europe as the Lancia Thema from 2011-2014. Production ended on December 22, 2023 and the plant was idled. Last vehicle off the line was a Pitch-Black 2023 Dodge Challenger Demon 170. 7,147,888 vehicles were produced through 2023.&lt;br&gt; Past models: [[w:Eagle Premier|Eagle Premier]] (1988-1992), [[w:Dodge Monaco#Fifth generation (1990–1992)|Dodge Monaco]] (1990-1992),&lt;br&gt; [[w:Dodge Intrepid|Dodge Intrepid]] (1993-2004),&lt;br&gt; [[w:Chrysler Intrepid|Chrysler Intrepid]] (Canada: 1993-2004),&lt;br&gt; [[w:Chrysler Concorde|Chrysler Concorde]] (1993-2004),&lt;br&gt; [[w:Eagle Vision|Eagle Vision]] (1993-1997),&lt;br&gt; [[w:Chrysler New Yorker#1994–1996|Chrysler New Yorker]] (1994-1996),&lt;br&gt; [[w:Chrysler LHS|Chrysler LHS]] (1994-1997, 1999-2001),&lt;br&gt; [[w:Chrysler 300M|Chrysler 300M]] (1999-2004),&lt;br&gt; [[w:Chrysler 300|Chrysler 300]] (2005-2023),&lt;br&gt; [[w:Dodge Magnum#Chrysler LX platform (2005–2008)|Dodge Magnum]] (2005-2008),&lt;br&gt; [[w:Dodge Charger (2006)|Dodge Charger]] (2006-2023),&lt;br&gt; [[w:Dodge Challenger (2008)|Dodge Challenger]] (2008-2023),&lt;br&gt; [[w:Lancia Thema#Second generation (2011–2014)|Lancia Thema]] (For export: 2011-2014)
|-
| C
| [[w:Jefferson North Assembly|Detroit Assembly Complex – Jefferson]] / Jefferson North Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]] (1993-), [[w:Dodge Durango#WD|Dodge Durango]] (2011-)
| Located at 2101 Conner Street. Jefferson North replaced the previous Jefferson Assembly plant that closed in 1990 and was demolished in 1991. Jefferson North is across the street from the old Jefferson Assembly plant, on the north side of Jefferson Ave. Jefferson North was built on the site of Chrysler's old Kercheval Avenue Body Plant, which, in 1955, had been connected to the old Jefferson Assembly plant by a bridge crossing over Jefferson Ave. The Jefferson North plant site also absorbed what had been the axle plant and service parts buildings of the old [[w:Hudson Motor Car Company|Hudson]] plant, which were located at Connor St. and Vernor Hwy. The main Hudson plant is now the parking lot on the corner of Jefferson Ave. and Connor St. After 2017, the Jefferson North plant complex also absorbed the site of the former Budd Co. body- &amp; parts-making plant at Connor St. and Charlevoix Avenue, which had previously belonged to the Liberty Motor Car Co. The Budd plant extended north on Connor from Charlevoix most of the way to Mack Ave. Since the nearby Mack Ave. Assembly Plant began operations in 2021, the 2 plants have operated as the Detroit Assembly Complex. Jefferson North began production on January 14, 1992 with the original Grand Cherokee (ZJ). The 2nd gen. Grand Cherokee began production on July 17, 1998. The 3rd gen. Grand Cherokee began production on July 26, 2004 followed by the Jeep Commander on July 18, 2005. The 4th gen. Grand Cherokee began production on May 10, 2010 followed by the Dodge Durango on December 14, 2010. The 5th gen. Grand Cherokee began production in May 2022. The 4xe plug-in hybrid version of the Grand Cherokee began production at Jefferson North in March 2023. On Aug. 13, 2013, Jefferson North built its 5 millionth vehicle, a silver 2014 Grand Cherokee Overland. On May 25, 2016, Jefferson North built its 6 millionth vehicle, a Granite Crystal (silvery gray) 2016 Grand Cherokee 75th anniversary Edition.&lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010),&lt;br&gt; [[w:Jeep Grand Cherokee (WK2)#Grand Cherokee WK (2022)|Jeep Grand Cherokee WK]] (2022)&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2023-2025)
|-
| 8
| [[w:Detroit Assembly Complex – Mack|Detroit Assembly Complex – Mack]] / Mack Ave. Assembly Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2021
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2021-), [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]] (2022-)
| Located at 4000 St. Jean Avenue. The Mack Avenue Assembly Plant is built on the site of the former Mack Ave. Engine Plants I &amp; II, the New Mack Assembly Plant, &amp; the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The two plants that comprised the former Mack Avenue Engine Complex were converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North plants have operated as the Detroit Assembly Complex since 2021. Production began in March 2021 with the 3-row Grand Cherokee L. The 2-row Grand Cherokee followed in the fall of 2021. The 4xe plug-in hybrid version of the Grand Cherokee began production at Mack Ave. in August 2022. &lt;br&gt; Past models: [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee 4xe]] (2022-2025)
|-
|
| [[w:Dundee Engine Plant|Dundee Engine Plant]] (Formerly [[W:Global Engine Manufacturing Alliance|GEMA]]) 
| [[w:Dundee, Michigan|Dundee, Michigan]]
| [[w:United States|United States]]
| 2005
| 
| [[w:Prince engine#1.6-litre turbocharged (PSA)|1.6L turbo PSA/BMW Prince EP6CDTX Hybrid I4]],&lt;br&gt; [[w:FCA Global Medium Engine|2.0L turbo Hurricane4 EVO I4 (GME-T4 EVO)]],&lt;br&gt; Engine components
| Located at 5800 North Ann Arbor Road. Plant was originally part of the the Global Engine Manufacturing Alliance, a 3-way engine manufacturing joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. The North Plant launched in October 2005, followed by the South Plant in November 2006. The plant began with production of the I4 World Gasoline Engine, which was developed by the Global Engine Alliance, a 3-way engine development joint venture between DaimlerChrysler, Mitsubishi, and Hyundai. Originally, the plant was envisioned as supplying engines to Mitsubishi and Hyundai as well as Chrysler however the plant only ever supplied engines to Chrysler. Mitsubishi and Hyundai each set up engine production at their own engine plants. On August 31, 2009, Chrysler bought Mitsubishi’s and Hyundai’s stakes in the group and now wholly owns both the Global Engine Manufacturing Alliance and its primary engine-building plant in Dundee, Michigan. In January 2012, the plant was renamed Dundee Engine Plant. Production of the Fiat 1.4-liter FIRE I4 engine began in November 2010. The Tigershark engine, an evolution of the World engine, began production in 2012 for the 2.0L and in May 2013 for the 2.4L. Tigershark engine production ended on March 16, 2023. In November 2019, the Pentastar V6 began production at Dundee, moving from the Mack Ave. Engine Plant in Detroit, which was to be converted into a vehicle assembly plant. The Pentastar V6 ended production at Dundee on August 18, 2023. In 2025, Dundee began making a North American-spec version of the European-developed Prince engine for the 2026 Jeep Cherokee Hybrid. &lt;br&gt; Past engines: [[w:World Gasoline Engine|1.8L/2.0L/2.4L/2.4L Turbo I4 World Gasoline Engine]], [[w:World Gasoline Engine#Tigershark|2.0L/2.4L I4 Tigershark Engine]], [[w:FIRE engine|Fiat 1.4L/1.4L Turbo FIRE MultiAir I4 engine]], [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]] 
|-
| 
| Etobicoke Casting Plant
| [[w:Etobicoke|Etobicoke District]], [[w:Toronto|Toronto]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1964
| 
| Aluminum die castings, Engine and Transmission Components
| Located at 15 Brown's Line. Etobicoke used to be a separate city but became part of Toronto in 1998. Factory was originally built in 1942 by the Canadian government and operated by Alcan Aluminum, which used it to make molds for military aircraft parts during World War II. It produced precision aircraft parts and other high quality aluminum castings. The aluminum foundry was purchased by Chrysler in April 1964 from Alcan Aluminum. The plant was expanded in 1965 and 1998. Etobicoke Casting is making oil pans for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee.
|-
|
| [[w:Indiana Transmission#Indiana Transmission I|Indiana Transmission Plant I]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1998
| 
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, Gear machining and final assembly of electric drive modules
| Located at 3660 North U.S. Highway 931. RFE transmission production ended in January 2025. More than 8 million RFE transmissions were produced at Indiana Transmission Plant I. Production of the nine-speed transmission began in May 2013. The 9-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. &lt;br&gt; Past transmissions:&lt;br&gt; [[w:Chrysler RFE transmission|Chrysler RFE 4-/5-/6-speed auto. trans.]]
|-
|
| [[w:Kokomo Casting|Kokomo Casting Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1965
| 
| Aluminum parts for automotive components, transmission and transaxle cases; engine block castings
| Located at 1001 East Boulevard. Kokomo Casting is the world’s largest die-cast facility. Plant was expanded in 1969, 1986, 1995 and 1997. Over 18 million four-speed transmission cases were made at Kokomo Casting from 1988 through July 2014. Kokomo Casting started making nine-speed transmission cases in 2013. Kokomo Casting is making engine blocks for the 1.6 liter turbo I4 in the 2026 Jeep Cherokee. Kokomo Casting is making gearbox covers for the electric drive modules made at the Indiana Transmission Plant. 
|-
|
| [[w:Kokomo Engine Plant|Kokomo Engine Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2022 (as Kokomo Engine)
| 
| [[w:FCA Global Medium Engine|2.0L turbo GME-T4 I4]]
| Located at 3360 North U.S. Highway 931. Plant was previously known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Starting in 2020, the plant was converted to engine production. Engine production began in late February 2022. 
|-
|
| [[w:Kokomo Transmission|Kokomo Transmission Plant]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1956
| 
| [[w:ZF 8HP transmission|850RE]] 8-speed auto. transmission,&lt;br&gt; [[w:ZF 8HP transmission|880RE]] 8-speed auto. transmission,&lt;br&gt; Machining of engine block castings and transmission components,&lt;br&gt; Machined components for the &lt;br&gt; 9-speed auto. transmission
| Located at 2401 South Reed Road. On October 9, 2020, Kokomo Transmission built its last 41TE 4-speed auto. transmission, assembling more than 17 million 4-speed auto. transmissions since production began in 1988. Kokomo Transmission began building 6-speed auto. transmissions in 2006. Production of the eight-speed automatic transmission began in September 2012. The 8-speed is built under license from [[w:ZF Friedrichshafen|ZF]]. On August 8, 2023, Kokomo Transmission built its 6 millionth 8-speed transmission. Kokomo Transmission is machining gearbox covers for the electric drive modules made at the Indiana Transmission Plant. &lt;br&gt; Past transmissions: &lt;br&gt;[[w:TorqueFlite|TorqueFlite 3-/4-speed auto. trans.]],&lt;br&gt; [[w:Ultradrive|Ultradrive (TE/AE/LE/RLE/TES/TEA)&lt;br&gt; 4-/6-speed auto. trans.]],&lt;br&gt; [[w:ZF 8HP transmission|845RE]] 8-speed auto. transmission,&lt;br&gt; SI-EVT trans. (eFlite) for Pacifica Hybrid
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo North Engine Plant]] 
| [[w:Ramos Arizpe|Ramos Arizpe]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1981
| 
| [[w:Chrysler Hemi engine#Third generation: 2003–present|5.7L/6.4L/6.2L supercharged Hemi V8]], [[w:Stellantis Hurricane engine|Chrysler 3.0L twin-turbo Hurricane GME-T6 I6]]
|  Production began on May 8, 1981. Hemi V8 production began in June 2002 with the 5.7L. Production of the supercharged 6.2L Hellcat Hemi V8 began in the third quarter of 2014. Tigershark I4 production began in the first quarter of 2014. &lt;br&gt; Past engines: [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L, 2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|2.0L, 2.4L, 2.4L Turbo I4 &quot;Neon engine&quot;]],&lt;br&gt; [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4 engine]], [[w:Chrysler Hemi engine#6.1|6.1L Hemi V8]]
|-
| 
| [[w:Saltillo Engine Plant#South Engine Plant|Saltillo South Engine Plant]] 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler 3.6L Pentastar V6 engine]]
| Plant opened on October 29, 2010. The plant has produced over 6 million Pentastar V6 engines. &lt;br&gt; Past engines: Chrysler 3.6L Pentastar V6 engine (EH3) for Pacifica Plug-in Hybrid
|-
| 
| Saltillo Stamping Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1997
| 
| Stampings and assemblies including Body panels
| Part of the Saltillo Truck Assembly Complex.
|-
| G
| Saltillo Truck Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 1995
| 
| [[w:Ram Heavy Duty (fifth generation)|Ram HD pickup &amp; chassis cab]] (2019-)
|  Production began in 1995. As of 2025, also known as Saltillo Truck Heavy Duty Plant. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1995-2012),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-2023),&lt;br&gt; [[w:Ram pickup#Fourth generation (2009; DS)|Ram HD pickup &amp; chassis cab]] (2013-2018), [[w:Sterling Bullet|Sterling Truck Bullet]] (2008-2009)
|-
| 4
| Saltillo Truck Extension Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2025
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2025-)
| Also known as Saltillo Truck Light Duty Plant. 2 new buildings were constructed for the new light duty plant. Production began in May 2025. Initially, production was for export but production for the domestic Mexican market began in February 2026. The plant also includes a seat assembly line, the first Stellantis plant in North America to integrate this process into its own production chain.
|-
| E
| Saltillo Van Assembly Plant 
| [[w:Saltillo|Saltillo]], [[w:Coahuila|Coahuila]]
| [[w:Mexico|Mexico]]
| 2013
| 
| [[w:Ram ProMaster|Ram ProMaster]] (2014-),&lt;br&gt; [[w:Ram ProMaster#E-Ducato and Ram ProMaster EV (2024)|Ram ProMaster EV]] (2024-)
| Production started in July 2013. &lt;br&gt; Past models: [[w:Fiat Ducato|Fiat Ducato]] (Export to Brazil/Argentina: 2018-2022)
|-
| N
| [[w:Sterling Heights Assembly|Sterling Heights Assembly Plant]] 
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1984
| 
| [[w:Ram 1500 (DT)|Ram 1500]] (DT) (2019-)
| Located at 38111 Van Dyke Ave. The plant was originally built by the US Navy as a jet engine plant in 1953. It was called Naval Industrial Reserve Aircraft Plant and was owned by the US Navy. When the jet engine project was cancelled, the plant was transferred to the US Army, which contracted with Chrysler to build missiles at the plant, which was now known as the Michigan Ordinance Missile Plant. Chrysler began production of the PGM-11 Redstone missile at the Sterling Heights plant on September 27, 1954. The final Redstone was built in 1961. Chrysler also built the PGM-19 Jupiter missile at Sterling Heights from 1958-December 1960. Chrysler also built the first stage of the Saturn I rocket at the Sterling Heights plant. Chrysler vacated the Michigan Army Missile Plant at the end of 1969. Meanwhile, LTV Corp. (previously Ling-Temco-Vought) built the MGM-52 Lance missile at the Sterling Heights plant. The Army turned the plant over to the state of Michigan, which was then sold it to Volkswagen in 1980. VW converted the plant to automotive production and intended to make the Jetta there but VW's US sales declined and VW never ended up building anything there. VW then sold the plant to Chrysler in 1983. Chrysler LeBaron GTS and Dodge Lancer production began in September 1984 and ended on April 7, 1989. Shadow and Sundance production began on August 25, 1986 and ended on March 9, 1994. The Dodge Daytona was also moved from St. Louis to Sterling Heights in 1991 and was produced there through February 26, 1993. Chrysler then built a succession of midsize cars at Sterling Heights from June 1994. During Chrysler's bankruptcy in 2009, Sterling Heights Assembly was initially left behind in &quot;old Chrysler&quot; and was supposed to close by December 2010 but during 2010, &quot;new Chrysler&quot; changed its mind and bought the plant from &quot;old Chrysler&quot; for $20 million. The 2011 Chrysler 200 and Dodge Avenger sedans began production on December 6, 2010 followed by the Chrysler 200 Convertible in February 2011. The Chrysler 200 Convertible was also built for export to Europe as the Lancia Flavia from March 2012. The 2nd gen. Chrysler 200 sedan began production on March 14, 2014 and ended on December 2, 2016. The plant was then idled for a lengthy retooling to build body-on-frame pickups and began production of the new generation DT-series Ram 1500 in March 2018. &lt;br&gt; Past models: [[w:Dodge Lancer#1985–1989: Lancer|Dodge Lancer]] (1985-1989), [[w:Chrysler LeBaron#1985–1989 LeBaron GTS|Chrysler LeBaron GTS (H-body)]] (1985-1989), [[w:Plymouth Sundance|Plymouth Sundance]] (1987-1994), [[w:Dodge Shadow|Dodge Shadow]] (1987-1994), [[w:Dodge Daytona|Dodge Daytona]] (1992-1993), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1992-1993), [[w:Chrysler Cirrus|Chrysler Cirrus]] (1995-2000), [[w:Dodge Stratus|Dodge Stratus]] sedan (1995-2006), [[w:Plymouth Breeze|Plymouth Breeze]] (1996-2000), [[w:Chrysler Sebring|Chrysler Sebring]] sedan (2001-2010), [[w:Chrysler Sebring|Chrysler Sebring]]  convertible (2001-2006, 2008-2010), [[w:Dodge Avenger#Dodge Avenger sedan (2008–2014)|Dodge Avenger]] sedan (2008-2014), [[w:Chrysler 200|Chrysler 200]] sedan (2011-2017), [[w:Chrysler 200|Chrysler 200]] convertible (2011-2014), [[w:Chrysler 200#Lancia Flavia|Lancia Flavia]] (For export: 2012-2014)  
|-
| 
| Sterling Stamping Plant
| [[w:Sterling Heights, Michigan|Sterling Heights, Michigan]]
| [[w:United States|United States]]
| 1965
| 
| Stampings and assemblies including hoods, roofs, liftgates, side apertures, fenders, and floorpans
| Located at 35777 Van Dyke Ave. This plant is a separate facility but is located next door to the Sterling Heights Assembly Plant. Sterling Stamping is the largest stamping plant in the world. Sterling Stamping supplies stampings to many Chrysler assembly plants, not only Sterling Heights Assembly. The first stampings were produced in January 1965. 
|-
| W
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo North Assembly Plant]] 
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]] (2018-)
| Located at 4400 Chrysler Drive. Toledo North first built the Jeep Liberty, which began in April 2001. Dodge Nitro production began in August 2006 and ended on December 16, 2011. The 2nd gen. Jeep Liberty began production in July 2007. Jeep Liberty ended production on August 16, 2012. Toledo North then closed for retooling to build the all-new 2014 Cherokee. The Jeep Cherokee began production at Toledo North on June 24, 2013 and ended on April 6, 2017. The Cherokee was then moved to Belvidere Assembly. Toledo North was then retooled to build the Jeep Wrangler, which began on November 15th, 2017. The 4xe plug-in hybrid version of the Wrangler began production in December 2020. &lt;br&gt; Past models: [[w:Jeep Liberty|Jeep Liberty]] (2002-2012), [[w:Dodge Nitro|Dodge Nitro]] (2007-2011),&lt;br&gt; [[w:Jeep Cherokee (KL)|Jeep Cherokee]] (2014-2017),&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler 4xe (JL)]] (2021-2025)
|-
| L
| [[w:Toledo Complex|Toledo Assembly Complex - Toledo Supplier Park Plant]] (South plant)
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 2001
| 
| [[w:Jeep Gladiator (JT)|Jeep Gladiator]] (2020-)
| Located along Stickney Ave. The Toledo Supplier Park Plant is built on the site of the former Stickney Ave. plant that became part of Chrysler in 1987 as part of the acquisition of AMC. That plant was acquired by Kaiser Jeep in 1964 from Autolite and was built in 1942. The Toledo Supplier Park Plant includes body and chassis operations in partnership with Kuka and Hyundai Mobis, respectively. The paint shop was originally run in partnership with Magna but Chrysler took over the paint operation in the first quarter of 2011. The Toledo Supplier Park Plant began Wrangler production in August 2006. Wrangler production ended on April 27, 2018. Gladiator pickup production began in March 2019. &lt;br&gt; Past models:&lt;br&gt; [[w:Jeep Wrangler (JK)|Jeep Wrangler (JK)]] (2007-2017),&lt;br&gt; [[w:Jeep Wrangler (JK)#2018 model year update|Jeep Wrangler JK]] (2018)
|-
| 
| [[w:Toledo Machining|Toledo Machining Plant]] 
| [[w:Perrysburg, Ohio|Perrysburg, Ohio]]
| [[w:United States|United States]]
| 1966
| 
| Steering Columns,&lt;br&gt; Torque Converters for 8-spd. rwd and 9-spd. fwd auto. transmissions
| Located at 8000 Chrysler Drive. Production began in 1966 and the plant was expanded in 1969. &lt;br&gt; Past products: Power Electronics module for Wrangler 4xe PHEV
|-
| T,&lt;br&gt; V (1985 only)
| [[w:Toluca Car Assembly|Toluca Car Assembly]] 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1968
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (2017-), [[w:Jeep Wagoneer S|Jeep Wagoneer S EV]] (2024-),&lt;br&gt; [[w:Jeep Cherokee (KM)|Jeep Cherokee (KM)]] (2026-), [[w:Jeep Recon|Jeep Recon EV]] (2026-)
| Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Began production on December 9, 1968. An adjacent supplier park was opened in 2007. Journey production began in early 2008. PT Cruiser production ended on July 9, 2010. Fiat 500 production began in December 2010. Journey production ended in December 2020. Jeep Compass production began on January 16, 2017. &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Dart (rwd), Dodge Dart K, Dodge Dart E, Chrysler Valiant Volaré (rwd), Chrysler Valiant Volaré K, Chrysler Valiant Volaré E, [[w:Dodge Magnum#First generation|Dodge Magnum]] [rwd] (1981-1982), [[w:Dodge Magnum#Second generation|Dodge Magnum 400/Magnum]] [fwd] (1983-1988), [[w:Chrysler Phantom|Chrysler Phantom]], [[w:Chrysler Shadow|Chrysler Shadow]], [[w:Chrysler Spirit|Chrysler Spirit]] &lt;br&gt; Export to US: [[w:Dodge Aries|Dodge Aries]] (1984-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1984-1989), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe]] (1987-1989, 1992), [[w:Dodge Shadow|Dodge Shadow]] (1988, 1991-1994), [[w:Plymouth Sundance|Plymouth Sundance]] (1988, 1992-1994), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron sedan]] (1990-1994), [[w:Dodge Spirit|Dodge Spirit]] (1991-1995), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1991-1995), [[w:Dodge Neon#First generation (1994)|Dodge Neon]] (1995-1999), [[w:Plymouth Neon#First generation (1994)|Plymouth Neon]] (1995-1999), [[w:Chrysler Sebring#Convertible (1996–2000)|Chrysler Sebring Convertible]] (1996-2000), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2001-10), [[w:Dodge Journey|Dodge Journey]] (2009-2020),&lt;br&gt; [[w:Fiat 500 (2007)|Fiat 500]] (2012-2019), [[w:Fiat 500 (2007)#Fiat 500e (2013)|Fiat 500e]] (2013-2019)&lt;br&gt; Export to Brazil/Europe/Australia:&lt;br&gt; [[w:Fiat Freemont|Fiat Freemont]] (2011-2016)
|-
| 
| Toluca Stamping Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| 1994
| 
| Stampings and assemblies including Body panels
| Part of the Toluca Assembly Complex.
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine South Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 2010
| 
| [[w:Chrysler Pentastar engine|Chrysler Pentastar V6 engine]]
| Located at 2300 Van Horn Road. Production began in March 2010 with the 3.6L Pentastar V6. In 2022, Trenton South was upgraded with a flexible engine line that could build both the Classic and Upgrade versions of the 3.6L Pentastar V6. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production.
|-
| 
| Warren Stamping Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1949
| 
| Stampings and assemblies including roofs, tailgates, side apertures, fenders, and floorpans
| Located at 22800 Mound Road. This plant is a separate facility but is located next door to the Warren Truck Assembly Plant. The plant was expanded in 1952, 1964, 1965, and 1986. Warren Stamping supplies stampings to many Chrysler assembly plants, not only Warren Truck Assembly.
|-
| S (1970-), 1 (1967-1969),&lt;br&gt; 2 (For A-series)
| Warren Truck Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1938
| 
| [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer]] (2022-), [[w:Jeep Wagoneer (WS)|Jeep Grand Wagoneer L]] (2023-)
| Located at 21500 Mound Road. Formerly known as the Dodge City Truck Plant. Also referred to as Warren Truck #1 in the 1970's when there were 2 other truck plants nearby. Began production in October 1938. The Mitsubishi Raider began production in September 2005 and ended on June 11, 2009. Dodge Dakota production ended on August 23, 2011. Full-size Jeep SUV production began in 2021. Ram 1500 Classic production ended in October 2024, ending production of Dodge/Ram trucks at Warren after 86 years. &lt;br&gt; Past models:&lt;br&gt; [[w:Dodge T-, V-, W-Series|Dodge T-, V-, W-Series]] (1939-1947), Dodge military trucks, [[w:Dodge B series#Pickup truck|Dodge B series]] (1948-1953), [[w:Dodge C series|Dodge C series]] (1954-1960),&lt;br&gt; [[w:Dodge Town Panel and Town Wagon|Dodge Town Panel/Town Wagon]] (1954-66), [[w:Dodge D series|Dodge D/W series]] (1961-1980), [[w:Dodge Ram|Dodge Ram pickup]] (1981-2012), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 pickup]] (2013-2018), [[w:Ram pickup#Fourth generation (2009; DS)|Ram 1500 Classic pickup]] (2019-24), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1977-1978, 1981-1985), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1977-1978, 1981), [[w:Dodge M-series chassis|Dodge M-series chassis]] (1968-1979),&lt;br&gt; [[w:Dodge A100|Dodge A100/A108]] (1964-1970),&lt;br&gt; [[w:Dodge Dakota|Dodge Dakota]] (1987-2011),&lt;br&gt; [[w:Mitsubishi Raider|Mitsubishi Raider]] (2006-2009),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer]] (2022-2025),&lt;br&gt; [[w:Jeep Wagoneer (WS)|Jeep Wagoneer L]] (2023-2025),&lt;br&gt; [[w:Fargo Trucks|Fargo Trucks]] (For export)
|-
| R (1968-),&lt;br&gt; 9 (1959-1967)
| [[w:Windsor Assembly|Windsor Assembly]] 
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1929
| 
| [[w:Chrysler Pacifica (minivan)|Chrysler Pacifica (minivan)]] (2017-), [[w:Chrysler Voyager#Sixth generation (2020–present)|Chrysler Voyager]] (2020-2026), Chrysler Grand Caravan (Canada: 2021-),&lt;br&gt; [[w:Dodge Charger (2024)|Dodge Charger]] (2024-)
| Located at 2199 Chrysler Centre. Today's Windsor Assembly Plant was originally Windsor Plant 3 or Windsor Car Assembly Plant. Before the 1965 US-Canada Auto Pact, Windsor Assembly made most of the cars sold by Chrysler Canada, including some unique-to-Canada variations. On June 10, 1983, Windsor ended rwd car production and was converted to build fwd minivans. Windsor began building minivans on October 7, 1983. For the first 2 generations of Chrysler minivans, Windsor only built the SWB models. For the 3rd generation, Windsor built both SWB and LWB models. For the 4th generation, Windsor only built LWB models. The minivan-based Pacifica SUV began production in January 2003 and  ended production on November 23, 2007. Production of the VW Routan began in August 2008. Production of the Ram C/V began on August 31, 2011, and ended in early 2015. Pacifica minivan production began on February 29, 2016 followed by the PHEV version on December 1, 2016. Town &amp; Country production ended on March 21, 2016 while Dodge Grand Caravan production ended on August 21, 2020. Production of the electric Charger Daytona began in December 2024 followed by the gas-powered Charger Sixpack in December 2025. &lt;br&gt; Past models: [[w:Chrysler Imperial|Chrysler Imperial]] (1929-1937) [https://www.web.imperialclub.info/registry/vin_decode.htm#1931-54], [[w:Dodge Kingsway|Dodge Kingsway]] (Canada: 1940-41, 1951-52), [[w:Dodge Regent|Dodge Regent]] (Canada: 1951-1959), [[w:Dodge Crusader|Dodge Crusader]] (Canada: 1951-1958), [[w:Dodge Mayfair|Dodge Mayfair]] (Canada: 1953-1959), [[w:Dodge Viscount|Dodge Viscount]] (Canada: 1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1961), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:Chrysler Windsor|Chrysler Windsor]] (1957-1966), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1963), [[w:Chrysler 300 non-letter series|Chrysler Saratoga 300]] (1964-1965), [[w:Chrysler Newport#1961–1964|Chrysler Newport]] (1961-1963), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1963-64, 1966), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Plymouth Fury|Plymouth Fury]] (1959-1970), [[w:Dodge Polara|Dodge Polara]] (1960, 1964-1969), Dodge 220 (Canada: 1963), [[w:Dodge 330|Dodge 330]] (1963-1965), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Monaco|Dodge Monaco]] (1967-1968), [[w:Plymouth Valiant#Canada (1960–1966)|Valiant]] (Canada: 1960-1966), [[w:Plymouth Barracuda#First generation (1964–1966)|Valiant Barracuda]] (Canada: 1964-1965), [[w:Plymouth Valiant|Plymouth Valiant]] (1970-1974), [[w:Plymouth Duster|Plymouth Duster]] (1970), [[w:Dodge Dart|Dodge Dart (compact)]] (1966, 1970-1975), [[w:Plymouth Satellite#Third generation (1971–1974)|Plymouth Satellite]] (1972-1974),  [[w:Plymouth GTX|Plymouth GTX]] (1971), [[w:Plymouth Road Runner#Second generation (1971–1974)|Plymouth Road Runner]] (1971-1974), [[w:Dodge Charger (1966)#Fourth generation|Dodge Charger]] (1975-1978), [[w:Dodge Magnum#US and Canada (1978–1979)|Dodge Magnum]] (1978-1979), [[w:Chrysler Cordoba|Chrysler Cordoba]] (1975-1983), [[w:Dodge Mirada|Dodge Mirada]] (1980-1983), [[w:Imperial (automobile)#Sixth generation (1981–1983)|Imperial]] (1981-1983), [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979), [[w:Dodge Diplomat|Dodge Diplomat]] (1981-1983), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1982-83), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle]] (Canada: 1981-1982), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1983), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron]] (1981), [[w:Chrysler New Yorker#1982|Chrysler New Yorker]] (1982), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler New Yorker Fifth Avenue]] (1983), [[w:Plymouth Voyager|Plymouth Voyager]] (1984-'00), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1984-2000), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2020), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (2001-2016), [[w:Chrysler Voyager|Chrysler Voyager]] (2000), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler minivans (S)#Cargo van|Dodge Mini Ram Van]] (1984-1988), [[w:Chrysler minivans (RT)#2011 revision|Ram C/V]] (2012-2015), [[w:Volkswagen Routan|Volkswagen Routan]] (2009-14), [[w:Chrysler Voyager#Lancia Voyager|Lancia Voyager]] (For export: 2012-2015), [[w:Chrysler Pacifica (crossover)|Chrysler Pacifica (SUV)]] (2004-2008)
|-
| 
| CpK Interior Products, Inc. - Belleville Operations
| [[w:Belleville, Ontario|Belleville]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 134 River Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. - Guelph Operations
| [[w:Guelph|Guelph]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 500 Laird Rd. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 
| CpK Interior Products, Inc. -&lt;br&gt; Port Hope Operations
| [[w:Port Hope, Ontario|Port Hope]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 2010 (became part of Chrysler)
| 
| Components for Automotive Interiors
| Located at 128 Peter St. Formed in 2010 as a subsidiary of Chrysler through the buyout of 3 Collins and Aikman plants in Canada after  Collins and Aikman went bankrupt.
|-
| 4
| Arab American Vehicles Company (AAV)
| [[w:Cairo|Cairo]]
| [[w:Egypt|Egypt]]
| 1978 (production began) &lt;br&gt; 1987 (became part of Chrysler)
| 
| [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee L]] (2025-), [[w:Citroën C4#Third generation (C41; 2020)|Citroën C4X]] (2025-)
| Arab American Vehicles Company is a joint venture between the [[w:Arab Organization for Industrialization|Arab Organization for Industrialization]], which holds 51%, and Stellantis, which holds the other 49%. AAV was originally established in 1977 as a joint venture  with [[w:American Motors Corporation|AMC]] to produce Jeeps. Production began in 1978. It became part of Chrysler when Chrysler bought AMC in 1987. It continued to be a part of subsequent corporate entities DaimlerChrysler, Chrysler LLC, Chrysler Group LLC, [[w:Fiat Chrysler Automobiles|FCA]], and [[w:Stellantis|Stellantis]]. Production of the Jeep J8, a light military vehicle based on the JK Wrangler, began on November 13, 2008. Jeep Grand Cherokee L production began in September 2024. Citroen C4X production began in April 2025. AAV has also produced vehicles for other automakers including Toyota. Toyota Fortuner SUV production began in April 2012. &lt;br&gt; Past models: Jeep CJ6, Jeep Wagoneer (SJ), Jeep AM720 military vehicle, [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]], [[w:Jeep Wrangler (TJ)|Jeep TJL]], [[w:Jeep Wrangler (JK)#Military Jeep J8 (2007–present)|Jeep J8]] (2008-2019),  [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]], [[w:Jeep Liberty (KJ)|Jeep Cherokee (KJ)]], [[w:Jeep Liberty (KK)|Jeep Cherokee (KK)]], [[w:Jeep Grand Cherokee (WK2)|Jeep Grand Cherokee (WK2)]]
|}

==Current non-Chrysler FCA/Stellantis Factories Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| Y (700),&lt;br&gt; 9 (ProMaster Rapid),&lt;br&gt; 3 (Vision)
| Betim Plant
| [[w:Betim|Betim]], [[w:Minas Gerais|Minas Gerais]]
| [[w:Brazil|Brazil]]
| 1973
| 
| [[w:RAM 700|RAM 700]] (2015-),&lt;br&gt; [[w:ProMaster Rapid|Ram V700 Rapid]] (Peru)&lt;br&gt; Related models:&lt;br&gt; [[w:Fiat Strada|Fiat Strada]] ('99-),&lt;br&gt; [[w:Fiat Fiorino#Latin America (2013–present)|Fiat Fiorino]] ('14-)
| Fiat plant. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Dodge Vision|Dodge Vision]] (Mexico: 2015-2018),&lt;br&gt; [[w:ProMaster Rapid|Ram ProMaster Rapid]] (Mexico: '18-'24)
|-
| U
| Cordoba Plant
| [[w:Córdoba, Argentina|Córdoba]], [[w:Córdoba Province, Argentina|Córdoba Province]]
| [[w:Argentina|Argentina]]
| 1995
| 
| [[w:Peugeot Landtrek|Ram Dakota]] (2026-)
| Fiat plant.
|-
| F
| [[w:FCA India Automobiles|FCA India Automobiles Private Limited]]
| [[w:Ranjangaon|Ranjangaon]], [[w:Pune district|Pune district]], [[w:Maharashtra|Maharashtra]]
| [[w:India|India]]
| 1997
| 
| [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Wrangler (JL)|Jeep Wrangler (JL)]],&lt;br&gt; [[w:Jeep Meridian|Jeep Meridian/Commander]],&lt;br&gt; [[w:Jeep Grand Cherokee#Fifth generation (WL; 2021)|Jeep Grand Cherokee]]
| Originally established as a 50/50 joint venture between Fiat and [[w:Tata Motors|Tata Motors]] called Fiat India Automobiles Private Limited. On June 1, 2017, Jeep Compass production began in India, the first Jeep built in India under its own brand. The JL-series Wrangler began to be assembled in India in 2021. The Jeep Meridian began production in May 2022. The Meridian is also exported to Japan as the Commander. The Jeep Grand Cherokee began assembly in India in November 2022. 
|-
| K
| Goiana Plant
| [[w:Goiana|Goiana]], [[w:Pernambuco|Pernambuco]]
| [[w:Brazil|Brazil]]
| 2015
|
| [[w:Jeep Renegade|Jeep Renegade]], [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass]],&lt;br&gt; [[w:Jeep Commander (2022)|Jeep Commander]], [[w:Ram Rampage|Ram Rampage]]&lt;br&gt; Related models: [[w:Fiat Toro|Fiat Toro]]
| [[w:Fiat Chrysler Automobiles|FCA]] plant. Production at Goiana began with the Jeep Renegade in February 2015. &lt;br&gt; Past Chrysler Group models: [[w:Ram 1000|Ram 1000]]
|-
| P
| Melfi Plant (formerly SATA = Società Automobilistica Tecnologie Avanzate [Advanced Technologies Automotive Company])
| [[w:Melfi|Melfi]], [[w:Province of Potenza|Province of Potenza]]
| [[w:Italy|Italy]]
| 1993
|
| [[w:Jeep Compass#Third generation (J4U; 2025)|Jeep Compass (J4U)]]&lt;br&gt; (Europe: '26-)
| Fiat plant. Jeep production began at Melfi in 2014 with the Renegade. Renegade production at Melfi ended on October 17, 2025. Production of the 3rd gen. Compass began on October 29, 2025. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Jeep Renegade|Jeep Renegade]] (US/Can.: '15-'23, Europe: '15-'25),&lt;br&gt; [[w:Jeep Compass#Second generation (MP/552; 2016)|Jeep Compass (MP)]] (Europe: '20-'25) &lt;br&gt; Related models:&lt;br&gt; [[w:Fiat 500X|Fiat 500X]] (US/Can.: '16-'23,&lt;br&gt; Europe: '15-'24)
|-
| B
| Porto Real Plant
| [[w:Porto Real|Porto Real]], [[w:Rio de Janeiro (state)|Rio de Janeiro state]]
| [[w:Brazil|Brazil]]
| 2000
|
| [[w:Jeep Avenger|Jeep Avenger]]
| PSA plant. The Jeep Avenger is the first Jeep to be made in a former PSA plant. 
|-
| U (2027-), 6 (2015-2022)
| [[w:Tofaş|Tofaş]]
| [[w:Bursa|Bursa]]
| [[w:Turkey|Turkey]]
| 1971
|
| [[w:Citroën Jumpy#Ram ProMaster City|Ram ProMaster City]] (2027-)
| Originally a Fiat joint venture, it is now 37.8% owned by Stellantis, 37.8% owned by [[w:Koç Holding|Koç Holding]], and 24.3% publicly traded on the Istanbul Stock Exchange. &lt;br&gt; Past Chrysler Group models: &lt;br&gt; [[w:Fiat Doblò#Ram ProMaster City|Ram ProMaster City]] (2015-2022), [[w:Chrysler Neon#Third generation (2016)|Dodge Neon]]&lt;br&gt; (Mexico &amp; Middle East: 2017-2020), [[w:Fiat Fiorino#Europe (2007–2024)|Ram V700 City]] (Chile: 2018-2023)
|-
| J,&lt;br&gt; 5 (Ypsilon)
| Tychy Plant
| [[w:Tychy|Tychy]], [[w:Silesian Voivodeship|Silesian Voivodeship]]
| [[w:Poland|Poland]]
| 1975
|
| [[w:Jeep Avenger|Jeep Avenger]]
| Fiat plant. Production began in September 1975 when the plant was owned by Polish automaker [[w:Fabryka Samochodów Małolitrażowych|FSM]], which built Fiat-based models. Fiat took over FSM in 1992. Fiat subsequently became [[w:Fiat Chrysler Automobiles|FCA]] and then Stellantis. Jeep Avenger production began on January 31, 2023. &lt;br&gt; Past Chrysler Group models:&lt;br&gt; [[w:Chrysler Ypsilon|Chrysler Ypsilon]] (UK/Ireland/Japan)
|}

==Current partner factories making Chrysler Group vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Current Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| GAC Hangzhou plant
| [[w:Hangzhou|Hangzhou]], [[w:Zhejiang|Zhejiang]]
| [[w:China|China]]
| 2021 (production began for Chrysler)
|
| [[w:Trumpchi GS5#Dodge Journey|Dodge Journey]] (Mexico: 2022-)
| [[w:GAC Group|GAC]] plant. 
|-
| 3
| GAC Yichang plant
| [[w:Yichang|Yichang]], [[w:Hubei|Hubei]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Dodge Attitude#Fourth generation (2025)|Dodge Attitude]] (Mexico: 2025-)
| [[w:GAC Group|GAC]] plant. 
|-
| 5
| Shenzhen Baoneng Motor Co., Ltd.
| [[w:Shenzhen|Shenzhen]], [[w:Guangdong|Guangdong]]
| [[w:China|China]]
| 2024 (production began for Chrysler)
|
| [[w:Peugeot Landtrek|Ram 1200]] (Mexico: 2025-)
| [[w:Baoneng Group#Automotive business|Shenzhen Baoneng Motor Co., Ltd.]] plant. 
|}

==Former factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Closed
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
|
| Ajax Trim plant
| [[w:Ajax, Ontario|Ajax]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1953, 1964 (became part of Chrysler)
| 2003
| Automotive Soft Trim Components, Seat Cushion and Seatback Covers, Foam-in-place Covers
| Built in 1953 by Canadian Automotive Trim. Purchased by Chrysler in 1964. Closed by December 2003.
|-
| J (1989-1992),&lt;br&gt; B (1981-1988),&lt;br&gt; 7-8 (1966-1980),&lt;br&gt; T (1958-1966) 
| [[w:Brampton Assembly (AMC)|AMC Brampton Assembly]]
| [[w:Brampton|Brampton]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]] 
| 1961,&lt;br&gt; 1987 (became part of Chrysler)
| 1992
| [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1987-92), [[w:Jeep CJ#CJ-5|Jeep CJ-5]] (1979-1980),&lt;br&gt; [[w:Jeep CJ#CJ-7|Jeep CJ-7]] (1979-1980),&lt;br&gt; [[w:AMC Eagle|AMC Eagle]] (1981-1987), [[w:AMC Eagle|Eagle Wagon]] (1988), [[w:AMC Concord|AMC Concord]] (1978, 1981-1983), [[w:AMC Spirit|AMC Spirit]] (1983), [[w:AMC Hornet|AMC Hornet]] (1970-77), [[w:AMC Gremlin|AMC Gremlin]] (1970-1978),&lt;br&gt; [[w:AMC Rebel|AMC Rebel]] (1968-1970), [[w:Rambler Rebel#Fifth generation|Rambler Rebel]] (1967), [[w:Rambler Classic|Rambler Classic]] (1961-1966),&lt;br&gt; [[w:AMC Ambassador|AMC Ambassador]] (1966-1968), [[w:AMC Ambassador|Rambler Ambassador]] ('63-'65), [[w:Rambler American|Rambler American]] (1962-68), [[w:Rambler American|AMC Rambler]] (1969)
| [[w:American Motors Corporation|AMC]] plant. Became part of Chrysler in the 1987 buyout of AMC. Located at at the corner of Kennedy Road South and Steeles Avenue East. Production began on January 26, 1961 with the Rambler Classic. This was the last plant to produce AMC vehicles. Eagle Wagon (formerly AMC Eagle) ended production on December 11, 1987. Production ended in April 1992 and Jeep Wrangler production was moved to Toledo, OH. Buildings on the west side of the plant were demolished in 2005 and buildings on the east side were demolished in 2007. A Lowe's and a Wal-Mart now occupy some of the former plant site.
|-
| V
| [[w:Conner Avenue Assembly|Conner Avenue Assembly]] 
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1996
| 2017
| [[w:Dodge Viper|Dodge Viper]] &lt;br&gt; GTS: '96, &lt;br&gt; All models: &lt;br&gt; '97-'06, '08-'10, '13-'17, [[w:Plymouth Prowler|Plymouth Prowler]]&lt;br&gt; ('97, '99-'01),&lt;br&gt; [[w:Chrysler Prowler|Chrysler Prowler]] ('01-'02),&lt;br&gt; [[w:Viper engine|8.0L/8.3L/8.4L aluminum &lt;br&gt; Viper V10 engine]] &lt;br&gt;(5/01-2017)
| Actually located at 20000 Connor Street. The plant was originally built in 1966 to make spark plugs by Champion. After Champion was bought by Cooper Industries in 1990, the plant was closed. It remained empty until Chrysler bought it in 1995. Viper production was moved to the Connor Avenue plant from the New Mack plant beginning with the GTS coupe for 1996, followed by the RT/10 roadster for 1997. Prowler production began in May 1997 and ended on February 15, 2002. Production of the Viper's aluminum V10 engine was moved to Connor Avenue, where it was built alongside the Viper itself, in May 2001 from the Mound Road Engine Plant, which closed in 2002. Viper production ended on July 2, 2010 and the plant was dormant until production restarted in December 2012. Production ended on Aug. 31, 2017. In 2018, the plant was renamed Connor Center, a meeting and display space that will showcase Chrysler’s concept and historic vehicle collection. 
|-
|E
| [[w:Diamond-Star Motors|Diamond-Star Motors/Mitsubishi Motors Manufacturing America/Mitsubishi Motors North America Manufacturing Division]]
| [[w:Normal, Illinois|Normal, Illinois]]
| [[w:United States|United States]]
| 1988
| 2005 (end of Chrysler production)/&lt;br&gt; 2015 (end of Mitsubishi production)
| [[w:Plymouth Laser|Plymouth Laser]] (1990-1994),&lt;br&gt; [[w:Eagle Talon|Eagle Talon]] (1990-1998),&lt;br&gt; [[w:Eagle Summit#First generation (1989–1992)|Eagle Summit 4-d]] (1991-1992),&lt;br&gt; [[w:Dodge Avenger#Dodge Avenger Coupe (1995–2000)|Dodge Avenger]] (1995-2000),&lt;br&gt; [[w:Dodge Stratus#Stratus coupe (2001–2005)|Dodge Stratus Coupe]]&lt;br&gt; (2001-2005),&lt;br&gt; [[w:Chrysler Sebring|Chrysler Sebring Coupe]]&lt;br&gt; (1995-2005),&lt;br&gt;  [[w:Mitsubishi Eclipse|Mitsubishi Eclipse]] (1990-2012),&lt;br&gt; [[w:Mitsubishi Mirage#Third generation (1987)|Mirage sedan]] (1991-1992),&lt;br&gt; [[w:Mitsubishi Galant|Galant]] (1994-2012),&lt;br&gt; [[w:Mitsubishi Endeavor|Endeavor]] (2004-2011),&lt;br&gt; [[w:Mitsubishi ASX#First generation (GA; 2010)|Outlander Sport/RVR]] (2013-15)
| Originally established as Diamond-Star Motors, a 50/50 joint venture between Chrysler and Mitsubishi which assembled both Chrysler and Mitsubishi vehicles. Production began in September 1988. In October 1991, Chrysler sold its 50% stake in the plant to Mitsubishi but production for Chrysler continued under contract. On May 24, 1993, production of the Mitsubishi Galant began at the Diamond-Star plant. In July 1995, the plant was renamed Mitsubishi Motors Manufacturing America. In January 2002, the plant was renamed Mitsubishi Motors North America Manufacturing Division. In January 2003, production of the Mitsubishi Endeavor began, the first SUV built at the Mitsubishi plant. In February 2005, production of vehicles for Chrysler ended. All further production was only of Mitsubishi-branded vehicles. In mid-2012, the plant began producing the Mitsubishi Outlander Sport. The Outlander Sport is sold as the RVR in Canada. On November 30, 2015, vehicle production ended. The Mitsubishi plant produced 3,283,549 vehicles. The plant continued to produce replacement parts until May 2016, when the plant closed completely. In June 2016, the plant was sold to liquidation firm Maynards Industries. In January 2017, EV startup [[w:Rivian|Rivian Automotive]] bought the former Mitsubishi plant. Rivian began production at the Normal, IL plant in September 2021 with the [[w:Rivian R1T|R1T]] electric pickup.
|-
| U
| [[w:Eurostar Automobilwerk|Eurostar]] - Chrysler Graz Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1991
| 2002
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (1992-2002), [[w:Chrysler PT Cruiser|Chrysler PT Cruiser]] (2002)
| Originally, a 50/50 joint venture between Chrysler and Steyr-Daimler-Puch founded in 1990. The Eurostar plant is next to the Steyr Fahrzeugtechnik plant solely owned by Steyr-Daimler-Puch. Production of the Chrysler Voyager and Grand Voyager minivans began in October 1991. This was the 2nd generation Chrysler minivan. The 3rd generation began production on September 25, 1995. In 1996, production of right-hand drive minivans began. The 4th generation began production in January 2001. Production of the PT Cruiser began in July 2001 on the same line as the Voyager minivans. In 1999, DaimlerChrysler bought the 50% stake in Eurostar held by Steyr-Daimler-Puch Fahrzeugtechnik, now majority owned by Magna International, making Eurostar a subsidiary of DaimlerChrysler. DaimlerChrysler sold 100% of Eurostar to Magna Steyr in July 2002. PT Cruiser production in Austria ended and was consolidated in Toluca, Mexico. Production of the Chrysler Voyager and Grand Voyager minivans moved next door to the main Magna Steyr plant for 2003. Magna International had acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. 
|-
| 3 (1959),&lt;br&gt; E (1958)
| Evansville Assembly
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1919, 1928 (became part of Chrysler)
| 1959 
| Graham Brothers trucks (through 1932),&lt;br&gt; Dodge Brothers trucks &lt;br&gt; (through 1932),&lt;br&gt; Plymouth cars (1936-1958),&lt;br&gt; Dodge cars (1937-1938),&lt;br&gt; [[w:Plymouth Savoy|Plymouth Savoy]] (1959), [[w:Plymouth Belvedere|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1959)
| Located at 1625 N. Garvin St. Built in 1919 by Graham Brothers Truck Company to build trucks. In 1925, Dodge Brothers bought a controlling 51% stake in Graham Brothers and then bought the rest in 1926, completely merging the 2 companies. This gave Dodge Brothers plants in Evansville, IN and Stockton, CA. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. At that point, trucks with a Dodge Brothers nameplate were rated as a half-ton; larger rated trucks were sold under the Graham Brothers name. On January 1, 1929, the Graham Brothers brand was eliminated, and all trucks produced became Dodge trucks. In 1932, Chrysler closed the Evansville plant due to the Great Depression. In 1935, as the economy improved, Chrysler reopened the Evansville plant and renovated and expanded it. It began building Plymouth cars for 1936. Dodge cars were also built for 1937 and 1938. During World War II, the plant became the Evansville Ordinance Plant, which produced more than 3.26 billion ammunition  cartridges - about 96% of all the .45 automatic ammunition produced for all the armed forces. The Evansville Ordinance Plant also rebuilt 1,600 Sherman tanks and 4,000 military trucks. After the war ended, Plymouth car production resumed. However, in the early 1950s, during the Korean War, the Evansville plant retooled and dedicated about a third of its space and manpower to building 60-foot aluminum hulls for Grumman  UF-1 Albatross air-sea rescue planes for the Navy and Coast Guard. Evansville built its 1 millionth Plymouth in March 1953. The plant was closed in 1959 and was replaced by the larger, more modern St. Louis plant in Fenton, MO, which had access to more railroad lines for shipping than Evansville did. 
|-
| 
| Evansville Stamping
| [[w:Evansville, Indiana|Evansville, Indiana]]
| [[w:United States|United States]]
| 1935, 1953 (became part of Chrysler)
| 1959 
| Body panel stampings
| Opened by Briggs Manufacturing Company when Chrysler reopened Evansville Assembly in 1935 to build Plymouth cars. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Made body panels for the nearby Evansville Assembly plant. Closed when Evansville Assembly closed in 1959.
|-
| B (1968-1980),&lt;br&gt; 2 (1960-1967),&lt;br&gt; 2 (1959)
| [[w:Dodge Main|Hamtramck Assembly (Dodge Main)]]
| [[w:Hamtramck, Michigan|Hamtramck, Michigan]]
| [[w:United States|United States]]
| 1911,&lt;br&gt; 1928 (became part of Chrysler)
| 1980 
| Dodge (1914-1958),&lt;br&gt; [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1959),&lt;br&gt; [[w:Dodge Royal#Third generation (1957–1959)|Dodge Royal]] (1959),&lt;br&gt; [[w:Dodge Custom Royal|Dodge Custom Royal]] (1959), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959),&lt;br&gt; [[w:Dodge Matador|Dodge Matador]] (1960),&lt;br&gt; [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962),&lt;br&gt; [[w:Dodge Polara|Dodge Polara]] (1960-1964),&lt;br&gt; [[w:Dodge 330|Dodge 330]] (1963-1964),&lt;br&gt; [[w:Dodge 440|Dodge 440]] (1963-1964),&lt;br&gt; [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1975), [[w:Plymouth Duster|Plymouth Duster]] (1970-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1969, 1972-1975), [[w:Dodge_Dart#1971|Dodge Dart Demon]] (1971-1972),&lt;br&gt; [[w:Dodge Charger (1966)|Dodge Charger]] (1967-1969), [[w:Dodge Charger Daytona#First generation (1969)|Dodge Charger Daytona]] ('69), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-74), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970-1974), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volaré|Plymouth Volaré]] (1976-1980),&lt;br&gt; Engines
| Located at 7900 Joseph Campau Ave. This plant predated Dodge being part of Chrysler Corp. On November 4, 1914, the first Dodge Brothers passenger car was produced at the Hamtramck plant. Prior to that, Dodge Brothers made components for other automakers, primarily Ford. The Hamtramck plant was fully vertically integrated, capable of building almost every part needed to build a complete car. Dodge Brothers was bought by the Chrysler Corporation on July 31, 1928. Even after the Chrysler takeover, Hamtramck remained Dodge's home plant. From the early 1950s, various operations were automated or moved to other plants and Hamtramck became more of an assembly plant by the early 1960s. Closed January 4, 1980. Last vehicle built was a Silver Metallic 1980 Dodge Aspen R/T 2-door. 13,943,221 vehicles were produced at the plant. Demolished in 1981. Replaced by the General Motors Detroit/Hamtramck Assembly Plant (Factory Zero), which opened in 1985. 
|-
| 
| [[w:Highland Park Chrysler Plant|Highland Park Plant]]
| [[w:Highland Park, Michigan|Highland Park, Michigan]]
| [[w:United States|United States]]
| 1909,&lt;br&gt; 1925 (became part of Chrysler)
| 1960s (end of manufacturing)
| Maxwell cars (1910-1925), Chrysler Series 50 (1926-27), Chrysler Series 52 (1928), Plymouth Model Q (1929), Chrysler Series 60 (1927), Chrysler Series 62 (1928), DeSoto Series K (1929-1930), DeSoto Series CK (1930), DeSoto Series CF (1930), Fargo Trucks (1928-1930) &lt;br&gt; Parts including fluid coupling and torque converter for [[w:Fluid Drive|Fluid Drive]]
| Located at at 12000 Chrysler Service Drive (formerly Chrysler Drive). Originally, this was [[w:Maxwell Motor Company|Maxwell Motor Company]]'s main plant. However, before Maxwell Motor Co.'s formation, parts of the site was used by several car and truck manufacturers: Grabowsky Power Wagon Company used 1 building, Brush Runabout Co. used another building, and Gray Motor Co. owned another building. Gray only used the western 1/3 of the building so they leased the middle third to Alden- Sampson Truck Co. and the eastern third to Maxwell-Briscoe Motor Co. All these companies, along with several others, combined under the [[w:United States Motor Company|United States Motor Company]] in 1910. In 1913, United States Motor Company collapsed and Maxwell was the only surviving part. The U.S. Motor Co. assets were purchased by Walter Flanders, who reorganized the company as the Maxwell Motor Co. Maxwell hired Walter P. Chrysler to turn the company around in 1921 after its finances deteriorated in the post-World War I recession in 1920. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including its Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924 at the Jefferson Ave. plant. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925 at Highland Park. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 built at Highland Park and was then reworked again into the first Plymouth in 1928, also built at Highland Park. Plymouth production was moved to the new Lynch Road Assembly plant in 1929 and DeSoto production also moved to the new Lynch Road Assembly plant in 1931. Highland Park still built parts but it no longer built vehicles. Highland Park was used more for design,  engineering, and management and served as Chrysler Corporation's headquarters through 1996. During the 1990's, Chrysler moved to its current headquarters at the Chrysler Technology Center in Auburn Hills. Much of the site has been demolished though Chrysler still has a small presence at the site with the FCA Detroit Office Warehouse. Other parts of the site are now occupied by several automotive suppliers including Magna, Valeo, Mobis, Avancez, and Yanfeng.
|-
|
| [[w:Indiana Transmission#Indiana Transmission Plant II|Indiana Transmission Plant II]]
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 2003 (as Indiana Transmission Plant II)
| 2019 (as Indiana Transmission Plant II)
| [[w:W5A580|Mercedes W5A580 (A580) &lt;br&gt; 5-speed auto. trans.]], Transmission components
| Located at 3360 North U.S. Highway 931. Plant was originally known as Indiana Transmission Plant II, which built automatic transmissions and transmission components from 2003-2019, when it was idled. Production began in November 2003. 5-speed auto. trans. production ended in August 2018 while production of components for the 8-speed auto. transmission ended in the fall of 2019. Starting in 2020, the plant was converted to engine production and is now known as Kokomo Engine Plant. Engine production began in late February 2022. 
|-
|
| Indianapolis Electrical Plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1953
| 1988
| Transmission plant: [[w:Chrysler PowerFlite transmission|Chrysler PowerFlite 2-speed auto. transmission]] &lt;br&gt; Electrical Plant: Alternators, distributors, starters, power steering units, voltage regulators, windshield wiper motors, and other electrical parts for cars
| Located at 2900 Shadeland Ave. Began making transmissions in 1953. In January 1959, Chrysler housed its new Electrical Division at the Shadeland Avenue plant, replacing transmission production. Became part of Chrysler's Acustar components subsidiary in 1987. Production ended on November 30, 1988 but shipping and other activities continued until March 1989 when the factory closed. Certain portions have been demolished and improvements were made to the remainder, which is now the Shadeland Business Center. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Naomi Street plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1946 (became part of Chrysler)
| 1970's
| Engine blocks
| Located at 1535 Naomi Street. Purchased from American Foundry Company in 1946. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Kept operating even after the Tibbs Ave. plant was launched. This location is now Wilco Gutter Supply. 
|-
|
| [[w:Indianapolis Foundry|Indianapolis Foundry]] - Tibbs Avenue plant
| [[w:Indianapolis|Indianapolis]], [[w:Indiana|Indiana]]
| [[w:United States|United States]]
| 1950 
| 2005
| Engine heads and blocks and other components
| Located at 1100 S. Tibbs Avenue. Operated as a subsidiary of Chrysler Corp. called American Foundry Co. until 1959, when it was merged into Chrysler Corp. Production ended on September 30, 2005 and the facility closed. Demolished in 2006. 
|-
| C (1968-1990),&lt;br&gt; 3 (1960-1967),&lt;br&gt; 1 (1959)
| [[w:Detroit Assembly Complex – Jefferson#Jefferson Avenue Assembly|Jefferson Avenue Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1908,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| [[w:Chrysler Six|Chrysler Series 70 (B-70)]] (1924-1925), [[w:Chrysler Six|Chrysler Series 70 (G-70)]] (1926-1927), [[w:Chrysler Six|Chrysler Series 72]] (1928), [[w:Chrysler Royal|Chrysler Royal]] (1933, 1937-1950), DeSoto SD (1933), Chrysler Airflow (1934-1937), Chrysler Airstream (1935-36), [[w:Chrysler (brand)|Chrysler brand]] (1937-1958), Desoto Airflow (1934-1936), DeSoto Airstream (1935-1936), [[w:Chrysler Imperial|Chrysler Imperial]] (1926-1942, 1946-1954), [[w:Imperial (automobile)|Imperial]] (1955-1958, 1962-1975), [[w:Chrysler 300 letter series|Chrysler 300 letter series]] (1959-1965), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-1978), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1961), [[w:Chrysler Newport|Chrysler Newport]] (1961-1978), [[w:Chrysler 300 non-letter series|Chrysler 300 Sport Series]] (1962-1971), [[w:Chrysler Town &amp; Country|Chrysler Town &amp; Country]] (1968-1972), [[w:DeSoto Firedome|DeSoto Firedome]] (1959), [[w:DeSoto Fireflite|DeSoto Fireflite]] (1959-1960), [[w:DeSoto Adventurer|DeSoto Adventurer]] (1959-1960), [[w:DeSoto (automobile)#1961|DeSoto]] (1961), [[w:Dodge Matador|Dodge Matador]] (1960), [[w:Dodge Polara|Dodge Polara]] (1965-1966), [[w:Dodge D series|Dodge D/W series]] (1979-1980), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1979-1980), [[w:Plymouth Trailduster|Plymouth Trailduster]] (1979-1980), [[w:Dodge Aries|Dodge Aries]] (1981-1989), [[w:Plymouth Reliant|Plymouth Reliant]] (1981-1989), [[w:Dodge 400|Dodge 400]] 4-d (1982-1983), [[w:Chrysler LeBaron|Chrysler LeBaron]] 4-d (1982-1984), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker (E-body)]] (1983-1987), [[w:Chrysler New Yorker#1983–1988|Chrysler New Yorker Turbo (E-body)]] (1988), [[w:Dodge 600|Dodge 600]] 4-d (1983-1988), [[w:Chrysler E-Class|Chrysler E-Class]] (1983-1984), [[w:Plymouth Caravelle|Plymouth Caravelle]] (US: 1985-1988), [[w:Plymouth Caravelle|Plymouth Caravelle]] 4-d (Canada: 1983-1988), [[w:Dodge Omni|Dodge Omni]] (1989-1990), [[w:Plymouth Horizon|Plymouth Horizon]] (1989-1990),&lt;br&gt; Engines
| Located at 12200 East Jefferson Ave. Plant was originally opened by [[w:Chalmers Automobile|Chalmers Motor Co.]]. After falling on hard times, Chalmers agreed in 1917 to build cars for [[w:Maxwell Motor Company|Maxwell Motor Co.]] at the Jefferson Ave. plant in Detroit. In exchange, Chalmers cars would be sold through Maxwell dealers. After having its own financial problems, Maxwell stopped producing cars at the Chalmers plant in 1921. Maxwell hired Walter P. Chrysler to turn the company around in 1921. In early 1921, Maxwell Motor Co. was liquidated and replaced by Maxwell Motor Corp. with Walter P. Chrysler as Chairman. On December 7, 1922, Maxwell took over the bankrupt Chalmers including the Jefferson Ave. plant. Chrysler brand cars began to be produced in 1924. Chalmers was discontinued in late 1923 with the last cars being 1924 models. Maxwell production ended in May 1925. Maxwell Motor Corp. was reorganized into the Chrysler Corporation on June 6, 1925. The 1925 Maxwell was reworked into an entry-level, 4-cylinder Chrysler model for 1926-1928 and was then reworked again into the first Plymouth in 1928. The Jefferson Ave. plant was the home plant of the Chrysler brand through 1978. It was also the home plant for the spin-off Imperial brand except for 1959-1961, when Imperial had its own exclusive plant on Warren Ave. in Dearborn. By the time it ended production on February 2, 1990, Jefferson Ave. Assembly had built 8,310,107 vehicles. Demolished in 1991. Replaced by the Jefferson North plant built across Jefferson Ave. from the old plant, where the Kercheval Body Plant used to be. The Jefferson North plant opened in January 1992.
|-
| W (1987-1989 Chrysler M-body) &lt;br&gt;&lt;br&gt; [K for 1981-1983 AMC and 1983-1987 Renault],&lt;br&gt; 0-6 (1966-1980 AMC)
| Kenosha I Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]]&lt;br&gt; (1987-1989),&lt;br&gt; [[w:Dodge Diplomat#Second generation (1980)|Dodge Diplomat]] (1987-1989), [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1987-89), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1987-1989)
| This was the Kenosha Main Plant of [[w:American Motors Corporation|AMC]]. The Kenosha plant was the oldest still operating automobile factory in the world when it ended vehicle production in December 1988. It first built automobiles in 1902 for the Thomas B. Jeffery Company under the Rambler brand. The factory was purchased in 1900 from the Sterling Bicycle Co., which built it in 1895. In 1914, the Thomas B. Jeffery Company rebranded its vehicles under the Jeffery brand. In 1916, the Thomas B. Jeffery Company was bought by Charles Nash and renamed Nash Motors. Kenosha produced Nash vehicles from 1917-1957. Kenosha also produced Nash's entry-level Lafayette brand from 1934-1936. After Nash merged with Hudson to form American Motors in 1954, Kenosha also produced Hudson vehicles from 1955-1957. Kenosha then produced vehicles under the Rambler brand for AMC from 1958-1968 and under the AMC brand from 1966-1983. Kenosha also produced the Alliance for AMC shareholder Renault for 1983-1987 along with the Encore for 1984-1986 and the GTA for 1987. Chrysler signed a deal with AMC in September 1986 to utilize surplus capacity at AMC's Kenosha plant to build Chrysler's trio of rwd M-body sedans beginning in February 1987. Chrysler did not have any capacity left in its own plants to continue building the M-body sedans. The St. Louis North plant that had been building the M-body sedans had been converted to build the extended length minivans. This deal led to Chrysler's acquisition of AMC, announced in March 1987. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 and the M-bodies were discontinued. The site continued building engines until 2010. The plant has since been demolished.
|-
| Y
| Kenosha II Assembly
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 1988
| [[w:Dodge Omni|Dodge Omni]] (1988-1989), [[w:Plymouth Horizon|Plymouth Horizon]] (1988-1989)
| This was the Kenosha Lakefront Plant of [[w:American Motors Corporation|AMC]], located on the shore of Lake Michigan. This property was originally a Simmons mattress manufacturing plant from 1870 to 1960. AMC bought it in 1960 to manufacture and paint auto bodies. Became part of Chrysler in the 1987 buyout of AMC. In September 1987, production of the Dodge Omni and Plymouth Horizon began. Production was moved to Kenosha from Chrysler's Belvidere, IL plant, which was being converted to build Chrysler's C-body sedans (Dynasty/New Yorker). Closed in December 1988. Omni &amp; Horizon production then moved to the Jefferson Ave. plant in Detroit. Demolished in 1990. In 1994, the City of Kenosha purchased the property for $1. The property was subsequently cleaned up and redeveloped into the HarborPark area, which includes a park and open space, a public museum, residential housing, and a marina. 
|-
| 
| [[w:Kenosha Engine|Kenosha Engine Plant]]
| [[w:Kenosha, Wisconsin|Kenosha, Wisconsin]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2010
| [[w:AMC straight-4 engine|AMC straight-4 engine]],&lt;br&gt; [[w:AMC straight-6 engine|AMC straight-6 engine]],&lt;br&gt; [[w:AMC V8 engine|AMC V8 engine]],&lt;br&gt; [[w:Chrysler LH engine|Chrysler 2.7L DOHC V6]], [[w:Chrysler SOHC V6 engine#3.5|Chrysler 3.5L SOHC V6]]
| Located at 5555 30th Avenue. Became part of Chrysler in the 1987 buyout of AMC. Vehicle production ended in December 1988 at the adjacent assembly plant but the site continued building engines until 2010. After the Chrysler buyout, the plant kept building the AMC 5.9L V8 for the SJ Jeep Grand Wagoneer through 1991, the AMC 2.5L I4 through 2002 for Jeeps and the Dodge Dakota, the AMC 4.2L I6 through 1990 for the Jeep Wrangler, and the AMC 4.0L I6 through 2006 for various Jeeps ('06 Wrangler was the last to use the 4.0L). Chrysler started building its own 2.7L V6 at Kenosha in 1997 and its own 3.5L V6 in 2003. Engine production ended in October 2010 and the plant closed. Demolished in 2012-2013.
|-
|
| Kercheval Body Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1920,&lt;br&gt; 1925 (became part of Chrysler)
| 1990
| Automobile Bodies
| Located at 12265 E Jefferson Ave., across Jefferson Ave. from Chrysler's Jefferson Ave. Assembly Plant, which had originally been the Chalmers plant. The assembly plant was on the south side of Jefferson Ave. while the body plant was on the north side. The plant was called Kercheval because the north side of the plant was bordered by Kercheval Ave. The plant was built in 1920 by Wadsworth Manufacturing Co. to replace a previous plant on the same site that burned down in 1919. That fire had also damaged the Chalmers plant across the street. In November 1920, Wadsworth Manufacturing was sold to American Motor Body Co., a division of the American Can Co. On July 1, 1923, American Motor Body Co. became American Motor Body Corp., run by Charles M. Schwab. On September 4, 1925, Chrysler Corp. bought the Detroit plant of the American Motor Body Corp. to quickly increase its manufacturing capacity. In 1955, Kercheval Body Plant was connected to the Jefferson Assembly plant by a bridge crossing over Jefferson Ave. Previously, bodies made at Kercheval had to be transported by truck across Jefferson Ave. to the assembly plant. The plant closed in Feb. 1990, at the same time the Jefferson Ave. Assembly Plant closed. The Kercheval plant was demolished and Chrysler built the new Jefferson North Assembly Plant on the site of the former Kercheval Body Plant, on the north side of Jefferson Ave. 
|-
|
| Kokomo - Home Ave. plant
| [[w:Kokomo, Indiana|Kokomo, Indiana]]
| [[w:United States|United States]]
| 1937
| 1969
| Manual Transmissions (1937-1955), Aluminum die casting (1955-1969)
| Located at 1105 S. Home Ave. Chrysler bought this plant in 1937. This was Chrysler's first plant in Kokomo. It had previously belonged to the [[w:Haynes Automobile Company|Haynes Automobile Co.]], which went out of business in 1925. The plant had been dormant since then. 5,124,211 manual transmissions were built here from 1937-1955. Transmission production then shifted to a new plant about a mile southeast on South Reed Road. The Home Ave. plant then became an aluminum die casting plant until 1969, when that operation shifted to the new Kokomo Casting plant on East Boulevard. Chrysler subsequently sold this plant. The facility was last used by Warren's Auto Parts, an auto salvage yard, which closed in 2020 after nearly 50 years. It is currently empty though still standing as of 2025. 
|-
| M
| [[w:Lago Alberto Assembly|Lago Alberto Assembly]]
| [[w:Nuevo Polanco|Nuevo Polanco district]], [[w:Miguel Hidalgo, Mexico City|Miguel Hidalgo borough]], [[w:Mexico City|Mexico City]]
| [[w:Mexico|Mexico]]
| 1938
| 2002
| &lt;br&gt; Past models: &lt;br&gt;  Mexico only: Dodge Savoy, Dodge Dart, Dodge 330, Dodge 440, Chrysler Valiant (1963-1969), Valiant Barracuda (1965-1969), Dodge Coronet, [[w:Dodge Ram|Dodge Ram pickup]] (1981-02), [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-96), [[w:Dodge Ramcharger#Third generation (1999–2001)|Dodge Ramcharger]] (1999-01) &lt;br&gt; Export to US:&lt;br&gt; [[w:Dodge Ramcharger#Second generation (1981–1993)|Dodge Ramcharger]] (1986-93), [[w:Dodge Ram#First generation (1981; D/W)|Dodge Ram pickup]] (1990-93), [[w:Dodge Ram#Second generation (1994; BR/BE)|Dodge Ram pickup]] (1994-02)
| Located at 320 Lago Alberto Street. Originally part of Fabricas Automex, Chrysler's affiliate in Mexico. In 1968, Fabricas Automex was 45% owned by Chrysler. In December 1971, Chrysler increased its stake to 90.5% and changed the Mexican company's name to Chrysler de Mexico. Chrysler later bought another 8.8% stake, taking its total to 99.3%. Lago Alberto began exporting to the US with the 1986 Dodge Ramcharger, sourced exclusively from Mexico. The Lago Alberto plant was closed in 2002 and Mexican pickup production was consolidated in the newer, more modern Saltillo plant. 
|-
| E (1968-1971),&lt;br&gt; 5 (1960-1967),&lt;br&gt; 4 (1959),&lt;br&gt; L (1958),&lt;br&gt; L (1955-1957 Chrysler brand)
| [[w:Los Angeles (Maywood) Assembly|Los Angeles (Maywood) Assembly]]
| [[w:Commerce, California|City of Commerce, California]]
| [[w:United States|United States]]
| 1932
| 1971
| Plymouth (1946-1958), Dodge (1946-1953, 1955-1958), DeSoto Deluxe/Custom (1948-1952), DeSoto Powermaster Six (1953-1954), DeSoto Firedome (1952-57), DeSoto Fireflite (1955-1957), DeSoto Firesweep (1957-1958), Chrysler Windsor (1948-1958), Chrysler Royal (1949-1950), Chrysler Saratoga (1951-1952, 1957-1958), Chrysler New Yorker (1953-1958), [[w:Chrysler Windsor|Chrysler Windsor]] (1959-1960), [[w:Chrysler Saratoga|Chrysler Saratoga]] (1959-1960), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1959-60), [[w:DeSoto Firesweep|DeSoto Firesweep]] (1959), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1957-1959), [[w:Dodge Custom Royal|Dodge Custom Royal]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Dodge Polara|Dodge Polara]] (1960-1964), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1959), [[w:Plymouth Fury|Plymouth Fury]] (1958-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1959-1961), [[w:Plymouth Savoy|Plymouth Savoy]] (1958-1959, 1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1971), [[w:Plymouth Duster|Plymouth Duster]] (1970-1971), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1971), [[w:Dodge Dart#1971|Dodge Dart Demon]] (1971), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1966, 1970), [[w:Dodge Challenger (1970)|Dodge Challenger]] (1970), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1971), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1971), [[w:Dodge Coronet|Dodge Coronet]] (1965-1971), [[w:Dodge Charger (1966)|Dodge Charger]] (1971), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971)
| Located at 5800 South Eastern Avenue and Slauson Avenue in Maywood, now part of City of Commerce. Across the street from the [[w:Maywood Assembly|Ford Maywood Assembly plant (Los Angeles Assembly plant No. 1)]]. 
|-
| A (1968-1981),&lt;br&gt; 1 (1960-1967),&lt;br&gt; 6 (1959)
| [[w:Lynch Road Assembly|Lynch Road Assembly]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1929
| 1981
| Plymouth (1929-1958), DeSoto (1931-1932), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-64), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (59-61), [[w:Plymouth Fury|Plymouth Fury]] (1959-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (59-61), [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (62-70), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1970, 1973-1974), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1978), [[w:Plymouth GTX|Plymouth GTX]] (1967-1970), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-1970, 1973, 1975), [[w:Plymouth Superbird|Plymouth Road Runner Superbird]] (1970), [[w:Dodge Coronet|Dodge Coronet]] (1965-1976), [[w:Dodge Monaco#Fourth generation (1977–1978)|Dodge Monaco]] (1977-1978), [[w:Dodge Charger (1966)#1966|Dodge Charger]] (1966), [[w:Dodge Charger (1966)#Third generation|Dodge Charger]] (1971-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971),  [[w:Chrysler Newport#1979–1981|Chrysler Newport]] (1979-81), [[w:Chrysler New Yorker#1979–1981|Chrysler New Yorker]] (79-81), [[w:Dodge St. Regis|Dodge St. Regis]] (1979-81), [[w:Plymouth Gran Fury#1980–1981|Plymouth Gran Fury]] (80-81),&lt;br&gt; Engines
| Located at 6334 Lynch Road. This was originally Plymouth's home plant. At the time it opened in 1929, Lynch Road was the largest single story auto plant in the world. DeSoto production was transferred from Highland Park to Lynch Road in 1931. In June 1932, DeSoto production was moved to the Jefferson Ave. plant when the DeSoto brand moved up in the brand hierarchy to between Dodge and Chrysler. Previously, DeSoto was between Plymouth and Dodge. During World War II, Lynch Road made tank transmissions, truck parts, and uranium enrichment diffusers for the Oak Ridge Gaseous Diffusion Plant in Oak Ridge, TN to produce enriched uranium for the atomic bomb. For 1965, Lynch Road began to focus on production of Plymouth and Dodge intermediate models. For 1979, Lynch Road was switched to build Chrysler's R-body full-size cars for all 3 Chrysler car brands. Production ended on April 3, 1981 and the factory closed. Last car produced was a white Plymouth Gran Fury police car. 
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant I]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1998
| 2019
| [[w:Chrysler PowerTech engine#4.7|4.7L PowerTech SOHC V8]],&lt;br&gt; [[w:Chrysler Pentastar engine|3.0L/3.2L/3.6L Pentastar V6]]
| Located at 4000 St. Jean Avenue. Mack Ave. Engine Plant I was built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in 1998. 4.7L V8 engine production ended in April 2013. Pentastar V6 engine production ended in 2019. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
| 
| [[w:Detroit Assembly Complex – Mack|Mack Ave. Engine Plant II]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 2000
| 2012
| [[w:Chrysler PowerTech engine#3.7 EKG|3.7L PowerTech SOHC 90° V6]]
| Located at 4500 St. Jean Avenue. Mack Ave. Engine Plant II was built next to the Mack Ave. Engine Plant I, which had been built on the site of the former New Mack Assembly Plant and the Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. The first engine was built in November 2000. Production ended in September 2012. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021.
|-
|
| Mack Ave. Stamping Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 1979
| Stampings
| Chrysler acquired the Mack Ave. Stamping Plant from Briggs Manufacturing Company in 1953. The plant was originally built in 1916 by the Michigan Stamping Company, which was taken over by Briggs Manufacturing in 1923. The plant was closed in 1979 and the site was basically abandoned. The city of Detroit bought the plant site in 1982 but was unable to find a purchaser or afford environmental remediation for the site and returned it to Chrysler. In 1990, Chrysler began cleanup and demolition of the old plant and built a new factory on the site, which became the New Mack Assembly Plant. The site later became the Mack Ave. Engine Complex and later, the Mack Ave. Assembly Plant, which is now part of the Detroit Assembly Complex.
|-
| 
| McGraw Stamping/McGraw Glass Plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 19?
| 2003
| Oil pans, valve covers, and other small stampings,&lt;br&gt; Automotive Glass (1960-)
| Located at 9400 McGraw Ave. Was around the corner and behind the Wyoming Ave. DeSoto/Export plant. Originally, a stamping plant. In 1960, switched to making automotive glass. Used Safeguard brand. Glass was DOT# 21. Demolished. 
|-
|
| [[w:Mound Road Engine|Mound Road Engine Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1953 (became part of Chrysler)
| 2002
| [[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]],&lt;br&gt; [[w:Chrysler LA engine#239 V6|3.9L 90° V6 engine]],&lt;br&gt; [[w:Chrysler LA engine#Magnum 8.0 L V10|8.0L iron Magnum V10 engine]],&lt;br&gt; [[w:Viper engine|8.0L aluminum Viper V10 engine]] (1992-5/01)
| Located at 20300 Mound Road. One of the plants Chrysler acquired from Briggs Manufacturing Company in 1953. Chrysler used the plant to produce aircraft parts from 1953-1954 and then transferred the plant to Plymouth in 1954 to build its new A-series V8 engine for 1956 model year cars. Converted into an engine plant and enlarged by 71,000 sq. ft., it began building V8 engines for Plymouth in July 1955. Dodge later used the A engine from 1959 in the US in cars and trucks and Chrysler used the A engine from 1960 in the US. The plant was closed in 2002 and demolished in 2003. The land was then paved over and is now used as a storage lot for vehicles produced at the nearby Warren Truck Assembly Plant. Warren Truck Assembly is just to the north of where Mound Road Engine was. Mt. Elliott Tool and Die was located immediately to the south of the Mound Road Engine Plant on Outer Drive East.
|-
|
| [[w:Mount Elliott Tool and Die|Mount Elliott Tool and Die]]/Outer Drive Manufacturing Technology Center/Outer Drive Stamping
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1938, 1953 (became part of Chrysler)
| 2018
| Stamping Dies, Checking Fixtures, Stamping Fixtures 
| Located at 3675 Outer Drive East. Built in 1938 by Briggs Manufacturing Company. One of the plants Chrysler acquired from Briggs Manufacturing in 1953. Chrysler renamed it Outer Drive Stamping. Stamping operations ended in 1983 and operations from the closed Vernor Tool &amp; Die plant were moved here. The plant was then renamed Outer Drive Manufacturing Technology Center. The plant now did tool and die work as well as pilot plant operations and engineering for new stamping technologies. Once the Chrysler Technology Center in Auburn Hills was built, Pilot Operations and Advanced Stamping Manufacturing Engineering moved there and the plant was renamed Mount Elliott Tool and Die. Operations at the plant ended in 2018 and the plant was sold to German automotive supplier Laepple Automotive in 2024. Laepple Automotive is producing stamped body parts at the plant. 
|-
| V
| [[w:Detroit Assembly Complex – Mack|New Mack Assembly Plant]]
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| 1992
| 1995
| [[w:Dodge Viper|Dodge Viper RT/10]] (1992-96)
| Located at 4000 St. Jean Avenue. The New Mack Assembly Plant is built on the site of the former Mack Ave. Stamping Plant that Chrysler acquired from Briggs Manufacturing Company in 1953. Viper production began in May 1992 at the New Mack Assembly Plant. After ending production in 1995, New Mack Assembly was converted into the Mack Ave. Engine Plant I. A new addition was then built to create Mack Ave. Engine Plant II. The 2 engine plants were subsequently converted into a single vehicle assembly plant and a new paint shop was built to create the Mack Ave. Assembly Plant. The Mack Ave. and the Jefferson North Assembly plants have operated as the Detroit Assembly Complex since 2021. 
|-
| F (1968-2009),&lt;br&gt; 6 (1960-1967),&lt;br&gt; 5 (1959),&lt;br&gt; N (1958)
| [[w:Newark Assembly|Newark Assembly]]
| [[w:Newark, Delaware|Newark, Delaware]]
| [[w:United States|United States]]
| 1951,&lt;br&gt; 1957 (Automotive prod.)
| 2008
| Plymouth (1957-1958), Dodge (1958), [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1958-1959, 1961), [[w:Plymouth Fury|Plymouth Fury]] (1959-1974), [[w:Plymouth Savoy|Plymouth Savoy]] (1959-1964), [[w:Dodge Coronet#Fourth generation (1957–1959)|Dodge Coronet]] (1958-1959), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge Polara|Dodge Polara]] (1960-1966), [[w:Dodge Monaco|Dodge Monaco]] (1965-1966, 1968), [[w:Chrysler Newport|Chrysler Newport]] (1965-1970), [[w:Chrysler New Yorker|Chrysler New Yorker]] (1965-70), [[w:Chrysler Town &amp; Country (1941–1988)|Chrysler Town &amp; Country]] (1966-1967), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1964, 1974-1975), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1964, 1974-1975), [[w:Dodge Aspen|Dodge Aspen]] (1976-1980), [[w:Plymouth Volare|Plymouth Volare]] (1976-1980), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1979-1980), [[w:Plymouth Reliant|Plymouth Reliant]] sedan &amp; wagon (1981-1988), [[w:Dodge Aries|Dodge Aries]] sedan &amp; wagon (1981-1988), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (sedan: 1984-1988, wagon: 1982-1988), [[w:Plymouth Acclaim|Plymouth Acclaim]] (1989-1995), [[w:Dodge Spirit|Dodge Spirit]] (1989-1995), [[w:Chrysler LeBaron#Third generation sedan (1990–1994)|Chrysler LeBaron Sedan (A-body)]] (1990, 1993-1994), [[w:Chrysler Saratoga#1989–1995|Chrysler Saratoga]] (For export: 1990-1992), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe (J-body)]] (1992-1993), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron convertible (J-body)]] (1992-1995), [[w:Dodge Intrepid#First generation (1993–1997)|Dodge Intrepid]] (1994-1996), [[w:Chrysler Intrepid#First generation (1993–1997)|Chrysler Intrepid]] (Canada: 1994-1995), [[w:Chrysler Concorde#First generation (1993–1997)|Chrysler Concorde]] (1995-1996), [[w:Dodge Durango#First generation (DN; 1998)|Dodge Durango (DN)]] (1998-2003), [[w:Dodge Durango#Second generation (HB; 2004)|Dodge Durango (HB)]] (2004-2009), [[w:Chrysler Aspen|Chrysler Aspen]] (2007-2009) 
| Chrysler began construction of the Delaware Tank Plant in January 1951 to build [[w:M48 Patton|M48 Patton]] tanks. Production began in April 1952. Production ended in May 1961 and the Tank Plant was closed in October 1961. Low rate initial production of the [[w:M60 tank|M60 tank]] was also done at the Newark plant in 1959 before production was moved to the [[w:Detroit Arsenal (Warren, Michigan)|Detroit Arsenal Tank Plant]] in Warren, MI in 1960. Conversion to automotive production began in 1956. Production of Plymouth and Dodge cars began on April 30, 1957. Production ended on December 19, 2008. Sold to the University of Delaware on October 24, 2009. Most of the plant was demolished in 2010-2011 except for the Administration Building near the front of the complex. The site is now the Science, Technology, and Advanced Research (STAR) campus. The old Chrysler Administration Building has been redesigned and is now being used by the College of Health Sciences. 
|-
| K
| [[w:Pillette Road Truck Assembly|Pillette Road Truck Assembly]]
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1976
| 2003
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1976-1980),&lt;br&gt; [[w:Dodge Ram Van|Dodge Ram Van]] (1981-2003), [[w:Dodge Ram Wagon|Dodge Ram Wagon]] (1981-'02), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1976-1983)
| Located at 2935 Pillette Road. Was originally Windsor Plant 6. The Pillette Road plant was located less than a mile away from Chryslers' main plant complex in Windsor. Production began in January 1976. Production ended on June 12, 2003. 2,309,399 units were built. Demolished in 2004. Part of the site is now the Grand Central Business Park. Another part was a logistics center serving Chrysler's Windsor Assembly Plant and operated by Syncreon. The Syncreon Automotive Windsor site closed in October 2022. The parts sorting and sequencing work done there was now going to be done in-house at Chrysler's Windsor Assembly Plant. 
|-
| 
| [[w:Los Angeles (Maywood) Assembly#San Leandro Assembly|San Leandro Assembly]]
| [[w:San Leandro, California|San Leandro, California]]
| [[w:United States|United States]]
| 1948
| 1954
| Plymouth (1949-1954),&lt;br&gt; Dodge (1949-1954)
| 
|-
| B (1996-2009),&lt;br&gt; G (1968-1991),&lt;br&gt; 7 (1960-1967),&lt;br&gt; 8 (1959)
| [[w:Saint Louis Assembly|St. Louis I Assembly]] - South plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1959
| 2008
| [[w:Plymouth Belvedere#Full-size series|Plymouth Belvedere]] (1960), [[w:Plymouth Fury|Plymouth Fury]] (1960-1964), [[w:Plymouth Savoy|Plymouth Savoy]] (1960-1964), [[w:Plymouth Suburban|Plymouth Suburban wagon]] (1961), [[w:Dodge Dart|Dodge Dart (full-size)]] (1960-1962), [[w:Dodge 330|Dodge 330]] (1963-1964), [[w:Dodge 440|Dodge 440]] (1963-1964), [[w:Plymouth Valiant#First generation (1960–1962)|Valiant]] (1960), [[w:Plymouth Valiant|Plymouth Valiant]] (1961-1965, 1976), [[w:Plymouth Duster|Plymouth Duster]] (1973-1976), [[w:Dodge Lancer#1961–1962: Lancer|Dodge Lancer]] (1961-1962), [[w:Dodge Dart|Dodge Dart (compact)]] (1963-1965, 1973-1976), [[w:Plymouth Barracuda|Plymouth Barracuda]] (1964-1965),&lt;br&gt; [[w:Plymouth Belvedere#Intermediate series|Plymouth Belvedere]] (1965-1970), [[w:Plymouth Satellite|Plymouth Satellite]] (1965-1974), [[w:Plymouth GTX|Plymouth GTX]] (1967-1971), [[w:Plymouth Road Runner|Plymouth Road Runner]] (1968-75), [[w:Plymouth Fury#Seventh generation (1975–1978)|Plymouth Fury]] (1975-1976), [[w:Dodge Coronet|Dodge Coronet]] (1965-1973, 75), [[w:Dodge Charger (1966)|Dodge Charger]] (1968-1974), [[w:Dodge Super Bee|Dodge Super Bee]] (1968-1971), [[w:Dodge Diplomat|Dodge Diplomat]] (1977-1981), [[w:Chrysler LeBaron#First generation (1977–1981)|Chrysler LeBaron (M-body)]] (1977-1981), [[w:Plymouth Caravelle|Plymouth Caravelle (M-body)]] (Canada: 1978-1981), [[w:Plymouth Reliant|Plymouth Reliant]] 2-d (1982-1986), [[w:Dodge Aries|Dodge Aries]] 2-d (1982-1986), [[w:Dodge 400|Dodge 400]] 2-d &amp; convertible (1982-1983), [[w:Dodge 600|Dodge 600]] 2-d &amp; convertible (1984-1986), [[w:Plymouth Caravelle|Plymouth Caravelle]] 2-d (Canada: 1983-1986), [[w:Chrysler LeBaron#Second generation (1982–1988)|Chrysler LeBaron (K-body)]] (2-d &amp; convertible: 1982-1986), [[w:Chrysler Executive|Chrysler Executive]] (1983-1986), [[w:Dodge Daytona|Dodge Daytona]] (1984-1991), [[w:Chrysler Daytona|Chrysler Daytona]] (Canada: 1984-1991), [[w:Dodge Daytona#Chrysler Laser|Chrysler Laser]] (1984-1986), [[w:Chrysler LeBaron#Third generation coupe/convertible (1987–1995)|Chrysler LeBaron coupe/convertible (J-body)]] (1987-1991), [[w:Plymouth Voyager|Plymouth Voyager]] (1996-2000), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1996-2000), [[w:Dodge Caravan|Dodge Caravan]] (1996-2007), [[w:Dodge Caravan|Dodge Grand Caravan]] (1996-2009), [[w:Chrysler Voyager|Chrysler Voyager]] (2001-2003), [[w:Chrysler Voyager|Chrysler Grand Voyager]] (2000), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1996-2001, 2004-2007)
| Located at 1001 N. Hwy Dr. St. Louis South was idled in 1991. The Dodge Daytona was moved to Sterling Heights and the J-body Chrysler LeBaron was moved to Newark, DE. St. Louis South was reopened in 1995 to build minivans, which were moved from the St. Louis North plant. For the 3rd generation, St. Louis South built both SWB and LWB models. For the 4th generation, St. Louis South built all SWB models but also built some LWB models. Closed on October 31, 2008. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1996-2009),&lt;br&gt; X (1973-1995),&lt;br&gt; U (1970-1972),&lt;br&gt; 7 (1967-1969)
| [[w:Saint Louis Assembly|St. Louis II Assembly]] - North plant / Missouri Truck Assembly Plant
| [[w:Fenton, Missouri|Fenton, Missouri]]
| [[w:United States|United States]]
| 1966
| 2009
| [[w:Dodge D series|Dodge D/W series]] (1967-1973), [[w:Dodge Ramcharger|Dodge Ramcharger]] (1974-1977), [[w:Plymouth Trail Duster|Plymouth Trail Duster]] (1974-1976), [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1980), [[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1975-1976, 1980),&lt;br&gt; [[w:Plymouth Gran Fury#1982–1989|Plymouth Gran Fury]] (1984-1987), [[w:Plymouth Caravelle#Canada|Plymouth Caravelle Salon]] (Canada: 1984-1987), [[w:Dodge Diplomat|Dodge Diplomat]] (1984-1987), [[w:Chrysler Fifth Avenue#1982–1989: The M-body years|Chrysler Fifth Avenue]] ('84-'87), [[w:Plymouth Voyager|Plymouth Grand Voyager]] (1987-1995), [[w:Dodge Caravan|Dodge Grand Caravan]] (1987-1995), [[w:Chrysler minivans (S)#Cargo van|Dodge Extended Mini Ram Van]] (1987-1988), [[w:Chrysler Town &amp; Country (minivan)|Chrysler Town &amp; Country]] (1990-1995),&lt;br&gt; [[w:Dodge Ram|Dodge Ram pickup]] (1996-'09)
| Originally opened to build trucks as the Missouri Truck Assembly Plant. In 1980, the plant was idled. Plant was reopened in 1983 to build the rwd, M-body sedans, which were moved from Windsor, ON, Canada so that Windsor could be converted to build minivans. Plant was renamed St. Louis II Assembly. During 1987, the M-body sedans were moved to AMC's plant in Kenosha, WI so that St. Louis North could be converted to build the new LWB minivans. For the first 2 generations of Chrysler minivans, St. Louis North only built the LWB models. For 1996, minivan production moved to St. Louis South and the North plant was converted to build full-size pickups. Closed on July 10, 2009. Demolished in 2011. Site was sold in 2014 and is now the Fenton Logistics Park.
|-
| J (1970-1978),&lt;br&gt; 6 (1968-1969),&lt;br&gt; 9 (1961-1967)
| Tecumseh Road Truck Assembly / Windsor Truck Assembly Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1916,&lt;br&gt; 1925 (became part of Chrysler)
| 1978
| Maxwell (1924-1925),&lt;br&gt; Chrysler (1924-1929),&lt;br&gt; Dodge Trucks (1931-1960), &lt;br&gt; Dodge D-Series Trucks: &lt;br&gt; D100 (1961), D200 (1964), W200 (1965), W100 (1966), D100/W100 (1967), D200 (1970), D500 (1968),&lt;br&gt; D500/D600/D700/D800 &lt;br&gt; medium-duty trucks (1970-1972), D500/D600/W600/D700/D800 medium-duty trucks (1974-1977),&lt;br&gt; D100 (1978),&lt;br&gt; Fargo Trucks (1936-1972)
| Located at 300 Tecumseh Road East. Was originally Windsor Plant 1. Became part of Chrysler upon its founding in 1925. Was previously a Maxwell-Chalmers plant and was originally Maxwell's Canadian plant from 1916. Switched to building trucks in 1931 after Plant 3 opened in 1929. Closed in 1978. Became the Imperial Quality Assurance Centre from 1980-1983, doing extra quality control on the 1981-1983 Imperial built at the Windsor Assembly Plant (the car plant - Plant 3) about 1.5 miles east of Plant 1. The Imperial Quality Assurance Centre closed in 1983 when the Imperial was discontinued. Subsequently demolished. Is now the Plaza 300 shopping mall.
|-
|
| Tipton Transmission Plant
| [[w:Tipton, Indiana|Tipton, Indiana]]
| [[w:United States|United States]]
| 2014
| 2023
| [[w:ZF 9HP transmission|948TE 9-speed auto.]] transmission, SI-EVT transmission
| Located at 5880 W. State Road 28. Originally, the plant was supposed to be a [[w:Getrag|Getrag]] plant focused on supplying Chrysler with dual-clutch transmissions. Chrysler withdrew from the deal in 2008 after a dispute over financing and sued Getrag. The 80% completed facility then sat dormant until Chrysler Group purchased the facility in February 2013 and completed construction. Production began in April 2014 with the ZF-designed 9-speed auto. transmission, built under license from ZF. Production ended in June 2023 and 9-speed production was consolidated into Indiana Transmission Plant I in Kokomo, IN. The SI-EVT transmission for the Pacifica Hybrid was moved to Kokomo Transmission Plant. Sold to IRH Manufacturing LLC in September 2024 to make solar cells.
|-
| L (1989-2001),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Parkway|Toledo Assembly #1 Plant]] - Jeep Parkway plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2001 (ended final assembly), 2006 (ended body assembly)
| [[w:Jeep Cherokee (XJ)|Jeep Cherokee (XJ)]] (1984-01), [[w:Jeep Cherokee (XJ)#Wagoneer|Jeep Wagoneer (XJ)]] (1984-90), [[w:Jeep Comanche|Jeep Comanche]] (1986-1992),&lt;br&gt; Bodies for vehicles made at Stickney Ave. plant &lt;br&gt; 
Models only made before Chrysler takeover: Willys Aero (1952-1955), Kaiser Manhattan (1954-1955), Jeep CJ (1946-1986), Jeep DJ, Jeep Jeepster (1948-1950), Jeep Jeepster Commando (1967-1971), Jeep Commando (1972-1973), Willys Jeep Station Wagon (1946-1964), Willys Jeep Truck (1947-1965), Jeep Gladiator (SJ) (1963-1971), Jeep J-series pickup, Jeep Wagoneer [SJ] (1963-1981), Jeep Cherokee [SJ] (1974-1981), Jeep Forward Control [FC] (1957-1965), Jeep FJ Fleetvan (1961-1975)
| Located at 1000 Jeep Parkway. The John North Willys-owned Overland Automobile Co. purchased the plant in 1909 from Pope-Toledo, another early automaker. Overland Automobile Co. became Willys-Overland in 1912. Began building Jeeps in the 1940s. This was the original Jeep assembly plant. Willys-Overland was bought by Kaiser in 1953. Kaiser then sold its Willow Run plant in Ypsilanti, MI to GM and moved its production to the Willys plant in Toledo, OH. Kaiser and Willys production in the US ended in 1955 and Toledo focused on Jeep production going forward. Kaiser Jeep was sold to AMC in 1970. Became part of Chrysler in the 1987 buyout of AMC. Final assembly ended in 2001 when the XJ Cherokee ended production but painted body production continued until June 30, 2006, when the TJ Wrangler ended production. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant a few miles away.  The Administration Building, used from 1915 through 1974, was imploded on April 14, 1979. A third of the plant was demolished in 2002 after final assembly ended including the Jeep Museum. The remainder was demolished in 2006-2007 after body production ended. One of the three large brick smokestacks was preserved and was dedicated in 2013 to the plant's history and workforce. A bronze plaque was mounted next to the smokestack, which still says &quot;Overland&quot; on it. Over 11 million vehicles were produced at the site, including military Jeeps during World War II. The site was sold to the Toledo-Lucas County Port Authority in 2010. The site has been redeveloped into the Overland Industrial Park. Dana Inc. and Detroit Manufacturing Systems are among the tenants in the Overland Industrial Park and those plants supply the current Jeep plants elsewhere in Toledo. All-Phase Electric Supply Co. is another tenant. 
|-
| P (1989-2006),&lt;br&gt; T (1981-1988)
| [[w:Toledo Complex#Stickney|Toledo Assembly #2 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1987 (became part of Chrysler)
| 2006 
| [[w:Jeep Wagoneer (SJ)#1984: SJ and XJ|Jeep Grand Wagoneer (SJ)]]&lt;br&gt; (1984-1991),&lt;br&gt; [[w:Jeep Wrangler (YJ)|Jeep Wrangler (YJ)]] (1993-95), [[w:Jeep Wrangler (TJ)|Jeep Wrangler (TJ)]] (1997-06)
Models only made before Chrysler takeover:&lt;br&gt; Jeep Wagoneer [SJ] (1981-1983), Jeep Cherokee [SJ] (1981-1983)
| Located at 4000 Stickney Ave. Originally opened in 1942 by the Electric Auto-Lite Co., a maker of spark plugs. Sold to Kaiser-Jeep in 1964, which used it as a machining and engine plant until 1981, when AMC converted it for vehicle production. AMC had taken over Kaiser Jeep in 1970. AMC built the SJ Wagoneer and Cherokee at the Stickney Ave. plant. Body assembly was done at an SJ- or later,  Wrangler-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant. Became part of Chrysler in the 1987 buyout of AMC. Production ended in 2006 with the end of the TJ Wrangler. Production of the replacement JK Wrangler moved to the new Toledo Supplier Park plant built on the site of the old Stickney Ave. plant. 
|-
| W (1994-1996)
| [[w:Toledo Complex#Stickney|Toledo Assembly #3 Plant]] - Jeep Stickney Ave. plant
| [[w:Toledo, Ohio|Toledo, Ohio]]
| [[w:United States|United States]]
| 1993
| 1996 
| [[w:Dodge Dakota#First generation (1987–1996)|Dodge Dakota]] (1994-1996)
| Body assembly was done at a Dakota-specific body shop at the Parkway plant while final assembly was at the Stickney Ave. plant.
|-
|
| Toluca Engine Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2002
| [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Closed in 2002
|-
|
| Toluca Transmission Plant 
| [[w:Toluca|Toluca]], [[w:State of Mexico|State of Mexico]]
| [[w:Mexico|Mexico]]
| ?
| 2001
| Automatic Transmissions for fwd cars
| Closed in 2001
|-
| 
| [[w:Trenton Engine Complex|Trenton Engine North Plant]] 
| [[w:Trenton, Michigan|Trenton, Michigan]]
| [[w:United States|United States]]
| 1952
| 2022
| [[w:Chrysler B engine|Chrysler B V8 engine]],&lt;br&gt; [[w:Chrysler B engine#RB engines|Chrysler RB V8 engine]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]],&lt;br&gt; [[w:Volkswagen EA827 engine#1.7|VW 1.7L EA827 I4 engine]] (adding Chrysler parts to already built VW engines made in W. Germany),&lt;br&gt; [[w:Chrysler 2.2 &amp; 2.5 engine|2.2L/2.5L &quot;K-car&quot; I4 engine]], [[w:Chrysler 1.8, 2.0 &amp; 2.4 engine|1.8L, 2.0L I4 &quot;Neon engine&quot;]], [[w:Chrysler 3.3 &amp; 3.8 engines|3.3L/3.8L OHV V6]], [[w:Chrysler SOHC V6 engine|3.5L/3.2L/4.0L SOHC V6]], [[w:Chrysler Pentastar engine|3.2L/3.6L Pentastar V6 engine]], [[w:World Gasoline Engine#2.4_2|2.4L Tigershark I4]],&lt;br&gt; Engine components,&lt;br&gt; Air raid sirens
| Located at 2000 Van Horn Road. Trenton North began production in fall 1952 and was expanded in 1964, 1967, 1969, 1976, and 1977. At first, Trenton North began by making water pumps and air raid sirens but engines quickly followed. Trenton Engine North was Chrysler’s first dedicated engine factory in the US, separate from the assembly plants. On September 29, 1978, V8 production ended. Trenton North added Chrysler parts such as the intake and exhaust manifolds, water pump, ignition system and other major parts to already built VW 1.7L EA827 I4 engines imported from Salzgitter, W. Germany for use in the Omni/Horizon. Production of the 3.5L V6 moved to Kenosha Engine in 2003. Trenton North was idled in May 2011 when the 3.8 V6 ended production following the end of 3.3 &amp; 4.0 V6 engine production in 2010. Chrysler then announced in June 2011 it would use a fifth of the plant to make components for the Pentastar V6 being made at Trenton South. In January 2012, Trenton North began producing the 3.6L Pentastar V6 engine. Chrysler then installed a flexible production line that could build both the Pentastar V6 and the Tigershark I4. In May 2013, Trenton began producing the 3.2L Pentastar V6. Tigershark I4 production began late in 3rd quarter 2013. By the end of 2022, Pentastar Upgrade engine production moved from Trenton North to Trenton South and the older North plant ended production. Trenton North has been repurposed for warehousing and other non-manufacturing opportunities. 
|-
|
| [[w:Twinsburg Stamping|Twinsburg Stamping]]
| [[w:Twinsburg, Ohio|Twinsburg, Ohio]]
| [[w:United States|United States]]
| 1957
| 2010
| Stampings and assemblies
| Located at 2000 East Aurora Road. Opened in August 1957. Closed July 31, 2010. Sold in 2011. Demolished in 2012-2013. Now the Cornerstone Business Park.
|-
|
| Vernor Tool &amp; Die plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1983
| Tooling &amp; Dies
| Located at 12026 E Vernor Highway. Operations moved to Mount Elliott Tool and Die. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
|
| Vernor Trim plant
| [[w:Detroit, Michigan|Detroit, Michigan]]
| [[w:United States|United States]]
| ?
| 1970's
| Trim
| Located at 12025 E Vernor Highway. The former location seems to have been swallowed up by the Jefferson North Assembly plant.
|-
| 4 (1960-1961),&lt;br&gt; 7 (1959)
| Warren Avenue Plant
| [[w:Dearborn, Michigan|Dearborn, Michigan]]
| [[w:United States|United States]]
| 1927,&lt;br&gt; 1950 (opened as part of Chrysler)
| 1960s
| DeSoto bodies (1950-1958), DeSoto engines &lt;br&gt; (1951-1958),&lt;br&gt; [[w:Imperial (automobile)#Second generation (1957–1966)|Imperial]] (1959-1961)
| Located at 8505 West Warren Avenue. This was previously the factory of Paige and Graham-Paige. Chrysler leased half the plant in 1941 to use for military production. Chrysler produced aircraft components for the [[w:Martin B-26 Marauder|B-26 Marauder]] (nose and center fuselage sections) and the [[w:Boeing B-29 Superfortress|B-29 Superfortress]] (the pressurized nose section, wing leading edges, and engine cowlings). The B-29 had a nose so large that trenches had to be dug in the floor and some of the bracing for the plant’s roof girders had to be removed to accommodate the aircraft. Chrysler bought the plant in 1947. Began building bodies for DeSoto in August 1950. Engine production began in 1951. Production of the Imperial brand moved here from the Jefferson Ave. plant in Detroit for 1959 in an attempt to give the Imperial brand its own, exclusive factory however sales weren't high enough to support its own plant so Imperial production moved back to the Jefferson Ave. plant in Detroit for 1962. Production of small parts followed for a few years as did export operations. The plant was later sold. Most of the plant has seen been demolished but the front building facing on Warren Ave. is still there and is now used as Corporate HQ by Shatila Food Products. The DeSoto logo, featuring a stylized image of Hernando de Soto, can still be seen at the top of the building above the front door. 
|-
| T (1970-), 2 (1966-1969),&lt;br&gt; 
| Warren Truck #2 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1966
| 1975
| Dodge heavy-duty trucks
| Located at 6600 East 9 Mile Road, at the corner of Sherwood Ave and East 9 Mile Road. Closed in 1975 when Dodge exited the heavy-duty truck market. Site now belongs to Sundance Beverage Co., the parent of Everfresh Juice Co.
|-
| V (1971-1979)
| Warren Truck (Compact) #3 Assembly Plant
| [[w:Warren, Michigan|Warren, Michigan]]
| [[w:United States|United States]]
| 1970
| 1979
| [[w:Dodge Sportsman|Dodge Tradesman/Sportsman]] (1971-1979),&lt;br&gt;[[w:Plymouth Voyager#Full-size van (AB; 1974–1983)|Plymouth Voyager]] (1974-1979)
| Located on Hoover Road between 8 Mile Road and 9 Mile Road. 
|-
| 
| Windsor Engine Plant
| [[w:Windsor, Ontario|Windsor]], [[w:Ontario|Ontario]]
| [[w:Canada|Canada]]
| 1938
| 1980
| [[w:Chrysler flathead engine#Straight-6|Chrysler flathead inline 6]],&lt;br&gt; [[w:Chrysler Slant-6 engine|Chrysler Slant-6 engine]], &lt;br&gt;[[w:Chrysler A engine|Chrysler A V8 engine]],&lt;br&gt; [[w:Chrysler LA engine|Chrysler LA V8 engine]]
| Was originally Windsor Plant 2. Located just to the south of Windsor Plant 3, the current minivan factory. Closed in August 1980. Built over 8 million engines. Windsor Assembly Plant (Plant 3) expanded onto the site of the old engine plant when it was being renovated for minivan production. 
|- 
|
| [[w:Detroit Assembly#LaSalle Factory/DeSoto Factory|Wyoming Ave. Assembly (DeSoto Wyoming Ave. plant)]] / Wyoming Export Plant
| [[w:Detroit|Detroit]], [[w:Michigan|Michigan]]
| United States
| 1936
| 1958 (Vehicle prod.),&lt;br&gt; 1980 (export operations)
| [[w:DeSoto (automobile)|DeSoto]] (1937-1958),&lt;br&gt; CKD Export (1960-1980)
| Located at 6000 Wyoming Avenue. Originally built to produce Liberty aircraft engines in World War I, opening in 1917. In 1919, was taken over by Saxon Motor Co., owned by Hugh Chalmers of Chalmers Motor Co. GM bought the plant in 1926 and built the LaSalle there from 1927-1933. GM sold Wyoming Assembly to Chrysler in 1934, which then used it to build its DeSoto brand. Became DeSoto's home plant. During World War II, Chrysler built wing center sections for the [[w:Curtiss SB2C Helldiver|Curtiss SB2C Helldiver]] at the Wyoming Ave. plant. For 1959, DeSoto's entry level model, the Firesweep, was moved to the Dodge plant in Hamtramck and DeSoto's other, higher end models were moved to Chrysler's Jefferson Ave. plant in Detroit. This was done so that bodies and final assembly would be done in either a single facility or a pair of connected facilities. This was part of Chrysler's move to unibody construction for 1960 for all cars except Imperial. After the DeSoto brand was discontinued in late 1960, became Wyoming Export plant which was used to prepare vehicles for export. Plant closed in 1980. Plant was demolished in 1992. Site is now occupied by Comprehensive Logistics Inc.
|}

==Non-Chrysler FCA/Stellantis Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 1
| [[w:Fiat Cassino Plant|Cassino Plant]]
| [[w:Piedimonte San Germano|Piedimonte San Germano]], [[w:Province of Frosinone|Province of Frosinone]]
| [[w:Italy|Italy]]
| 1972
| 
| Past Chrysler Group models: [[w:Lancia Delta#|Chrysler Delta]] (UK/Ireland)
| Fiat plant. 
|-
| 3
| [[w:Alfa Romeo Pomigliano d'Arco plant|Pomigliano d'Arco plant]] (Giambattista Vico plant)
| [[w:Pomigliano d'Arco|Pomigliano d'Arco]], [[w:Metropolitan City of Naples|Metropolitan City of Naples]]
| [[w:Italy|Italy]]
| 1972
|
| Past Chrysler Group models: [[w:Dodge Hornet|Dodge Hornet]] (2023-2025). Related models:&lt;br&gt; [[w:Alfa Romeo Tonale|Alfa Romeo Tonale]] (2023-)
| Originally, an Alfa Romeo plant. Oriiginally owned by Construction Industry Neapolitan Vehicles Alfa Romeo - Alfasud S.p.A., a joint venture between Alfa Romeo (88%), Finmeccanica (10%), and IRI (2%). In 1982, Alfasud S.p.A. was renamed Inca Investments. Alfa Romeo was taken over by Fiat in 1986. Fiat merged with Chrysler to form Fiat Chrysler Automobiles (FCA) in 2014. FCA merged with PSA Group to form Stellantis in 2021. 
|}

==Non-Chrysler Group DaimlerChrysler/Daimler AG Factories Previously Making Chrysler Group Vehicles== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Opened
! style=&quot;width:10px;&quot;|Idled
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 5
| Mercedes-Benz Plant Düsseldorf
| [[w:Düsseldorf|Düsseldorf]], [[w:North Rhine-Westphalia|North Rhine-Westphalia]]
| [[w:Germany|Germany]]
| 1962
|
| [[w:Dodge Sprinter|Dodge Sprinter]] (2003-2009)
| Mercedes-Benz Plant. 
|-
| 9
| Mercedes-Benz Plant Ludwigsfelde
| [[w:Ludwigsfelde|Ludwigsfelde]], [[w:Brandenburg|Brandenburg]]
| [[w:Germany|Germany]]
| 1991 (Mercedes prod. began)
|
| [[w:Dodge Sprinter#Second generation (2006–2018, NCV3)|Dodge Sprinter]] chassis cab (2007-2009)
| Mercedes-Benz Plant. Originally established in 1936 by Daimler-Benz to make airplane engines. The plant was bombed by the US in 1945. After the war ended, what remained of the factory was dismantled and taken to the Soviet Union as reparations. On February 1, 1991, Mercedes-Benz took a 25% stake in the Ludwigsfelde plant, which had previously belonged to East German truckmaker VEB Automobilwerke. It became a 100% owned subsidiary of Mercedes-Benz on January 1, 1994. Sprinter production began in 2006. 
|}

==Former partner factories== 
{| class=&quot;wikitable sortable&quot;
! style=&quot;width:60px;&quot;|VIN
! style=&quot;width:100px;&quot;|Name
! style=&quot;width:80px;&quot;|City/state
! style=&quot;width:80px;&quot;|Country
! style=&quot;width:10px;&quot;|Prod. for Chrysler began
! style=&quot;width:10px;&quot;|Prod. for Chrysler ended
! style=&quot;width:260px;&quot;|Products
! style=&quot;width:370px;&quot; class=&quot;unsortable&quot;|Comments
|-
| 6
| [[w:China Motor Corporation|China Motor Corporation]]
| [[w:Yangmei District|Yangmei District]], [[w:Taoyuan, Taiwan|Taoyuan]]
| [[w:Taiwan|Taiwan]]
| 2006
| 2007
| [[w:Chrysler Town &amp; Country (minivan)#Fourth generation (2001–2007)|Chrysler Town &amp; Country]]&lt;br&gt; (Taiwan: 2006-2007),&lt;br&gt; [[w:Dodge 1000|Dodge 1000]] (Mexico: 2007-'10)
| China Motor Corporation plant. Built for Chrysler under license by China Motor Corporation of Taiwan. Production began April 18, 2006. 
|-
|
| [[w:Carrozzeria Ghia|Carrozzeria Ghia]] 
| [[w:Turin|Turin]]
| [[w:Italy|Italy]]
| 1957
| 1965
| [[w:Imperial (automobile)#Imperial Crown (1955–1965)|Imperial Crown Limousine]] (1957-1965) modified, painted limousine bodies and interiors
| 132 Imperial Crown Limousines were built by Ghia under contract for Chrysler between 1957 and 1965. The 1957-1959 models were based on modified 2-door hardtops with the more rigid chassis from the convertible. The 1960-1965 models were based on 4-door models. Ghia lengthened the frame and modified the bodywork and interiors to create the limousines. After producion ended in 1965, Ghia sold the tooling to Barreiros of Spain, which built another 10 Imperial Crown Limousines. Barreiros had been 35% owned by Chrysler since 1963. That was increased to 77% in 1967 and 100% in 1969.  
|-
| U
| [[w:Hyundai Motor Company|Hyundai Motor Co.]] - [[w:List of Hyundai Motor Company manufacturing facilities#Ulsan Plant|Ulsan plant]]
| [[w:Ulsan|Ulsan]]
| [[w:South Korea|South Korea]]
| 2000
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge Atos|Dodge Atos]] (2001-2012),&lt;br&gt;  [[w:Dodge Verna|Dodge Verna]] (2004-06),&lt;br&gt;  [[w:Dodge Attitude#First generation (MC; 2006)|Dodge Attitude (MC)]] (2007-'11), [[w:Dodge Attitude#Second generation (RB; 2011)|Dodge Attitude (RB)]] (2012-'14), [[w:Dodge H100|Dodge H100 truck]],&lt;br&gt; [[w:Hyundai Starex#Second generation (TQ; 2007)|Dodge H100 Van/Wagon]]
| Rebadged Hyundai models sold as Dodges in Mexico. 
|-
| 
| [[w:Hyundai Motor India|Hyundai Motor India]]
| [[w:Chennai|Chennai]], [[w:Tamil Nadu|Tamil Nadu]]
| [[w:India|India]]
| 2011
| 2014
| Mexico only: &lt;br&gt; [[w:Dodge i10|Dodge i10]] (2012-2014)
| Rebadged Hyundai model sold as a Dodge in Mexico. 
|-
| X
| [[w:Karmann|Karmann Osnabrück Assembly]]
| [[w:Osnabrück|Osnabrück]], [[w:Lower Saxony|Lower Saxony]]
| [[w:Germany|Germany]]
| 2003
| 2007
| [[w:Chrysler Crossfire|Chrysler Crossfire]] (2004-2008)
| Karmann plant. Built under contract for Chrysler. 
|-
| Y
| [[w:Magna Steyr|Magna Steyr]] / Steyr-Daimler-Puch - Chrysler Steyr Assembly
| [[w:Graz|Graz]], [[w:Styria|Styria]]
| [[w:Austria|Austria]]
| 1994
| 2010
| [[w:Jeep Grand Cherokee|Jeep Grand Cherokee]]&lt;br&gt; (1995-2010),&lt;br&gt; [[w:Jeep Commander (XK)|Jeep Commander]] (2006-2010), [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager/Grand Voyager]] (2003-2007),&lt;br&gt; [[w:Chrysler 300#First generation (2005)|Chrysler 300C/300C Touring]] (2005-2010)
| Originally, a Steyr-Daimler-Puch plant. Magna International acquired a majority holding of 66.8% in Steyr-Daimler-Puch in 1998 and acquired the rest by 2002 when it was renamed Magna Steyr. Production of the Chrysler Voyager and Grand Voyager minivans moved from the Eurostar plant  next door to the main Magna Steyr plant for 2003. Chrysler minivan production in Austria ended on November 30, 2007. Built under contract for Chrysler. 
|-
| B
| [[w:Maserati|Maserati]] - [[w:Innocenti|Innocenti]] plant
| [[w:Lambrate|Lambrate district]], [[w:Milan|Milan]]
| [[w:Italy|Italy]]
| 1988
| 1990
| [[w:Chrysler TC by Maserati|Chrysler TC by Maserati]]&lt;br&gt; (1989-1991)
| Developed jointly by Chrysler and Maserati, the TC was built in Italy by Maserati at the Innocenti plant in Milan. Maserati and Innocenti were both owned by DeTomaso at the time. Chrysler bought a 5% stake in Maserati in 1984 and increased its stake to 15.6% in 1986. Production ended in 1990 due to low sales.
|-
| U
| Mitsubishi - Mizushima plant (Line 1)
| [[w:Kurashiki|Kurashiki]], [[w:Okayama Prefecture|Okayama Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| [[w:Plymouth Champ|Plymouth Champ]] (1981-1982), [[w:Plymouth Colt|Plymouth Colt]] (1983-1994), [[w:Dodge Colt|Dodge Colt]] (1981-1994), [[w:Dodge Colt|Dodge/Plymouth Colt]]&lt;br&gt; (Canada only: 1995),&lt;br&gt; [[w:Eagle Summit|Eagle Summit]]&lt;br&gt; (4-d: 1989-1990, 1993-1996,&lt;br&gt; 3-d: 1991-1992, 2-d: 1993-96), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1993-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1993-96)
| Mitsubishi Motors plant.
|-
| Z
| Mitsubishi - Okazaki plant
| [[w:Okazaki, Aichi|Okazaki]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1983
| 1996
| [[w:Plymouth Conquest|Plymouth Conquest]] (1984-86), [[w:Dodge Conquest|Dodge Conquest]] (1984-1986), [[w:Chrysler Conquest|Chrysler Conquest]] (1987-1989), [[w:Dodge Colt Vista#Colt Vista|Dodge Colt Vista]] (1984-1991), [[w:Plymouth Colt Vista#Colt Vista|Plymouth Colt Vista]] (1984-91), [[w:Mitsubishi RVR#North America|Plymouth Colt Vista]] (1992-94), [[w:Mitsubishi RVR#North America|Eagle Summit Wagon]] (1992-96) [[w:Eagle Vista#Vista Wagon|Eagle Vista Wagon]]&lt;br&gt; (Canada: 1989-1991)
| Mitsubishi Motors plant.
|-
| Y (Line 1)&lt;br&gt;/&lt;br /&gt;P (Line 2)
| Mitsubishi - &lt;br&gt; Ooe plant &lt;br&gt; a.k.a. &lt;br&gt; Nagoya #1&lt;br&gt;/&lt;br&gt;Nagoya #2
| Ooe-cho, [[w:Minato-ku, Nagoya|Minato ward]], [[w:Nagoya|Nagoya]], [[w:Aichi Prefecture|Aichi Prefecture]] 
| [[w:Japan|Japan]]
| 1970s
| 1996
| VIN code Y:&lt;br&gt; [[w:Plymouth Sapporo|Plymouth Sapporo]] (1981-1983), [[w:Dodge Challenger#Second generation (1978–1983)|Dodge Challenger]] (1981-1983), [[w:Plymouth Arrow Truck#Chrysler variants|Plymouth Arrow Truck]] ('81-'82), [[w:Dodge Ram 50|Dodge Ram 50]] (1981-1984), [[w:Dodge Stealth|Dodge Stealth]] (1991-1996)
VIN code P:&lt;br&gt; [[w:Dodge Ram 50|Dodge Ram 50]] (1985-1986), [[w:Dodge Ram 50#North America|Dodge Ram 50]] (1987-1993)
| Mitsubishi Motors plant. Closed in 2001. Sold to Mitsubishi Heavy Industries and now used by its Aircraft, Defense &amp; Space Business Area.
|-
| J
| Mitsubishi -  Toyo Koki/Pajero Manufacturing Co., Ltd. plant
| [[w:Sakahogi, Gifu|Sakahogi]], [[w:Gifu Prefecture|Gifu Prefecture]] 
| [[w:Japan|Japan]]
| 1986
| 1989
| [[w:Dodge Raider|Dodge Raider]] (1987-1989)
|  Originally, a Toyo Koki Co. Ltd. plant. Opened in 1976. Built vehicles under contract for Mitsubishi. Mitsubishi Motors owned 35% of Toyo Koki and increased its stake to a majority in March 1995. The plant was then renamed Pajero Manufacturing Co., Ltd. in July 1995. In March 2003, Mitsubishi bought all the remaining shares in Pajero Manufacturing Co., Ltd., making it a wholly owned subsidiary. Closed in 2021. Sold to Daio Paper in 2022.
|-
| H (Attitude), 9 (1200)
| [[w:Mitsubishi Motors (Thailand)|Mitsubishi Motors Thailand]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 2014
| 2024
| [[w:Dodge Attitude#Third generation (A10; 2015)|Dodge Attitude]]&lt;br&gt; (Mexico: 2015-2024),&lt;br&gt; [[w:Mitsubishi Triton#Fifth generation (KJ/KK/KL; 2014)|Ram 1200]]&lt;br&gt; (Middle East: 2017-2019)
| Mitsubishi Motors plant.  
|-
| ?
| [[w:MMC Automotriz|MMC Automotriz]]
| [[w:Barcelona, Venezuela|Barcelona]], [[w:Anzoátegui|Anzoátegui state]]
| [[w:Venezuela|Venezuela]]
| 2002
| 2009
| [[w:Dodge Brisa|Dodge Brisa]]&lt;br&gt; ([[w:Hyundai Accent#First generation (X3; 1994)|2002-2005]]), ([[w:Hyundai Getz|2006-2009]])
| MMC Automotriz plant. Originally, MMC Automotriz was 49% owned by Consorcio Inversionista Fabril S.A. (CIF) of Venezuela and 42% owned by Nissho Iwai Corp. The remaining 9% of the company was owned by the Japan International Development Organization Ltd., a partnership between the government-financed Overseas Economic Cooperation Fund and 98 private companies. Nissho Iwai merged with Nichimen Corp. in 2004 to form Sojitz Corp. Sojitz later increased its stake in MMC Automotriz to 98%, with the other 2% still held by CIF. MMC Automotriz was sold to the the Sylca Group (also known as Yammine Group) in 2015. MMC Automotriz produced Mitsubishi vehicles and from 1996-2012, also produced Hyundai vehicles. The Dodge Brisa was produced for DaimlerChrysler as part of its cooperation with [[w:Hyundai Motor Company|Hyundai]]. MMC Automotriz also produced Mitsubishi Fuso trucks. 
|-
| G
| [[w:Mitsubishi Motors (Thailand)|MMC Sittipol Co., Ltd.]]
| [[w:Laem Chabang|Laem Chabang]], [[w:Chonburi province|Chonburi province]]
| [[w:Thailand|Thailand]]
| 1988
| 1992
| [[w:Plymouth Colt#Fifth generation (1985–1988)|Plymouth Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Dodge Colt#Fifth generation (1985–1988)|Dodge Colt 100]]&lt;br&gt; (Canada: 1988-1992),&lt;br&gt; [[w:Eagle Vista|Eagle Vista]] (Canada: 1988-92)
| Mitsubishi Motors plant. MMC Sittipol is the predecessor company of Mitsubishi Motors (Thailand). These were the first vehicle exports from Thailand. 
|-
| 2
| [[w:Renault|Renault]] - [[w:Maubeuge Construction Automobile|Maubeuge plant]]
| [[w:Maubeuge|Maubeuge]]
| [[w:France|France]]
| 1987
| 1989
| [[w:Renault Medallion|Renault Medallion]] (1988),&lt;br&gt; [[w:Eagle Medallion|Eagle Medallion]] (1989)
| Renault plant. The Medallion was sold through Chrysler's Jeep-Eagle dealer network as a legacy of Chrysler's takeover of AMC from Renault.
|-
| 8,&lt;br&gt; 0
| [[w:Soueast|Soueast]]
| [[w:Fuzhou|Fuzhou]], [[w:Fujian|Fujian province]]
| [[w:China|China]]
| 2008
| 2010
| [[w:Chrysler Voyager#Fourth generation (2001–2007)|Chrysler Voyager]],&lt;br&gt; [[w:Dodge Caravan#Fourth generation (2001–2007)|Dodge Caravan]]
| South East (Fujian) Motor Co., Ltd. plant. Built for Chrysler under license by South East (Fujian) Motor Co., Ltd.
|}</text>
      <sha1>tpyyaj8o2ev8vjircj6gan2mhr39yck</sha1>
    </revision>
  </page>
  <page>
    <title>Nursing Home Social Services Reference/Flow of Care/Hospice</title>
    <ns>0</ns>
    <id>482375</id>
    <revision>
      <id>4631240</id>
      <parentid>4628897</parentid>
      <timestamp>2026-04-18T15:43:10Z</timestamp>
      <contributor>
        <username>Rchaswms01</username>
        <id>3499245</id>
      </contributor>
      <comment>Made some changes to wording for clarity or to add detail.</comment>
      <origin>4631240</origin>
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      <text bytes="4514" sha1="jkczo5usx7t31szzw9oyupdedc1l7th" xml:space="preserve">Residents living in a nursing home have the service of hospice available to them. They can choose any hospice they wish to work with. More typically, nursing homes establish relationships with a few, usually between 2-5 hospices, to provide residents more of a choice. Working closely with a few chosen hospices improves service delivery, enhances communication between the nursing home and hospice, and makes it easier for hospices to deliver care to more than one client. Ultimately though, the resident can choose any hospice they want, and the nursing home is obligated to comply. 

When nursing staff feel a hospice evaluation could be useful to a resident's care, they will bring it up to the resident and/or their decision maker, usually family, guardian, or friend. Nursing home staff are able to provide an overview of hospice services, but it is best left to the hospice to introduce their agency and explain their services. Once a hospice has been chosen to complete an evaluation, the hospice will send an intake nurse who will meet with and assess the resident and review their medical records. 

Medicare is generally the insurer for hospice and the criteria for hospice agency staffing and services are specified in Medicare regulations for eligible hospice services. Hospice can be paid for privately  though this happens rarely. The criteria for getting onto hospice with Medicare is typically a doctors evaluation stating the resident has six months or less to live unless the resident has a diagnosis of cancer in which case the period of hospice care can be one year. Many think, why is hospice needed when they live in a nursing home with nurses on staff 24/7? 

Hospice has a completely different perspective on how to provide care. Hospice aims to ensure a resident is comfortable until death and all curative and/or preventative care is ceased. A nursing home aims to heal for a better life, while hospice aims to comfort for a better death. Hospices may discontinue a resident's medications, change out their durable medical equipment like a hospital bed, and visit the client on a weekly or daily basis in addition to all the services a nursing home provides. The hospice then becomes the primary on all decisions related to the resident's medical care and the nursing home follows the orders and plan of care from the hospice. 

When a resident is getting the care they need, often they can get better and &quot;graduate&quot; from hospice. They may be placed on palliative care where a nurse from the hospice or palliative agency visits them once a month to monitor them for any changes of condition that might make them eligible again for hospice. 

Both nursing home and hospice staff are expected to work together harmoniously with the goal of providing the most benefit to the resident. Key to this goal are ensuring good and complete communication, sharing of updates on resident status, and alerting the hospice when the resident has warning signs of imminent passing are very important. Nursing homes are busy, but they must ensure the hospice agency is kept updated. 

Social services are part of the coordination of hospice services. While nursing provides the medical records for the referral, social services will often send the referral or calls the hospice to discuss the resident prior to them coming out for an evaluation. Social services also ensures that hospice is involved when a resident is due for their care conference or any meetings related to their care. Admissions and marketing may also be involved due to their focus on establishing relationships with community agencies such as hospice. 

Every hospice must have a social worker and chaplain on staff to visit the resident at least once and once per month or more often based on the resident's needs and willingness.  It is important for social services to communicate with the hospice social worker or chaplain.  Welcome any and all feedback and input from outside sources. The goal of social services is to enhance residents' lives, which means being open to outsider input. Sharing your perspective and hearing the perspective of these professionals will help you be a team working to meet the the resident's needs. 

Hospice and nursing homes work together to care for residents who are nearing end of life (EOL). It is the special mission of hospice to prioritize the EOL needs of residents. This relationship between the hospice and the nursing home can be nurtured and enhanced by social services.

{{BookCat}}</text>
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  </page>
  <page>
    <title>Essowe abalo</title>
    <ns>0</ns>
    <id>482663</id>
    <revision>
      <id>4631216</id>
      <timestamp>2026-04-18T12:05:17Z</timestamp>
      <contributor>
        <username>~2026-23920-31</username>
        <id>3577399</id>
      </contributor>
      <comment>add details</comment>
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      <text bytes="1450" sha1="4ephgaoi6ibiuo08v9u5kxxknyrlcs7" xml:space="preserve">
'''Essowe Abalo''' is a '''project management coach, trainer, and founder of Woloyem Consulting''', a firm focused on helping professionals get certified and perform at a high level—especially in '''PMP®, ITIL®, and PRINCE2®'''.

=== Positioning (what actually matters) ===
He’s not positioned as a “trainer.”

He operates as a '''strategic execution coach'''.

=== Core Identity ===

* Founder &amp; Lead Coach at '''Woloyem Consulting'''
* Known as '''“Papa Na Un Coup KO”'''
* Works primarily with:
** Project managers
** IT professionals
** Certification candidates who want results fast

=== What makes him different ===
Most trainers teach content.

He focuses on '''decision-making, structure, and passing the exam efficiently'''.

His method is built on:

* '''Clarity over complexity'''
* '''Execution over theory'''
* '''Results over comfort'''

=== Track record ===

* '''98.7% success rate''' on PMP® candidates
* Official partnerships:
** PMI® ATP (Authorized Training Partner)
** PeopleCert ATO

=== What he actually delivers ===

* Strategic coaching (not just courses)
* Exam simulators + real-case training
* Personalized preparation plans
* High-pressure, outcome-driven support

=== Bottom line ===
Essowe Abalo is positioned as a '''results-driven authority in project management certification''', focused on one thing:&lt;blockquote&gt;'''Turning candidates into certified, confident professionals—fast.'''&lt;/blockquote&gt;</text>
      <sha1>4ephgaoi6ibiuo08v9u5kxxknyrlcs7</sha1>
    </revision>
    <revision>
      <id>4631217</id>
      <parentid>4631216</parentid>
      <timestamp>2026-04-18T12:05:47Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <comment>Marking for speedy deletion: Out of scope</comment>
      <origin>4631217</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1511" sha1="9o5abufup0nqsy8dpai61tk9nmytw83" xml:space="preserve">&lt;noinclude&gt;{{Delete|example=false|Out of scope}}&lt;/noinclude&gt;

'''Essowe Abalo''' is a '''project management coach, trainer, and founder of Woloyem Consulting''', a firm focused on helping professionals get certified and perform at a high level—especially in '''PMP®, ITIL®, and PRINCE2®'''.

=== Positioning (what actually matters) ===
He’s not positioned as a “trainer.”

He operates as a '''strategic execution coach'''.

=== Core Identity ===

* Founder &amp; Lead Coach at '''Woloyem Consulting'''
* Known as '''“Papa Na Un Coup KO”'''
* Works primarily with:
** Project managers
** IT professionals
** Certification candidates who want results fast

=== What makes him different ===
Most trainers teach content.

He focuses on '''decision-making, structure, and passing the exam efficiently'''.

His method is built on:

* '''Clarity over complexity'''
* '''Execution over theory'''
* '''Results over comfort'''

=== Track record ===

* '''98.7% success rate''' on PMP® candidates
* Official partnerships:
** PMI® ATP (Authorized Training Partner)
** PeopleCert ATO

=== What he actually delivers ===

* Strategic coaching (not just courses)
* Exam simulators + real-case training
* Personalized preparation plans
* High-pressure, outcome-driven support

=== Bottom line ===
Essowe Abalo is positioned as a '''results-driven authority in project management certification''', focused on one thing:&lt;blockquote&gt;'''Turning candidates into certified, confident professionals—fast.'''&lt;/blockquote&gt;</text>
      <sha1>9o5abufup0nqsy8dpai61tk9nmytw83</sha1>
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  </page>
  <page>
    <title>User talk:~2026-23920-31</title>
    <ns>3</ns>
    <id>482664</id>
    <revision>
      <id>4631218</id>
      <timestamp>2026-04-18T12:05:47Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <comment>Notifying author of speedy deletion nomination</comment>
      <origin>4631218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="915" sha1="1jhiuxn94c9tuu813x2ag9rgxv04txg" xml:space="preserve">
== I have added a tag to a page you created ==
Hi! I'm MathXplore, and I recently reviewed your page, [[:Essowe abalo]]. I have added a tag to the page, because it &lt;strong&gt;may meet the [[Wikibooks:Deletion policy#Speedy deletions|criteria for speedy deletion]].&lt;/strong&gt; This means that it can be deleted at any time. The reason I provided was: &lt;blockquote&gt;&lt;strong&gt;Out of scope&lt;/strong&gt;&lt;/blockquote&gt; If you believe that your page should not be deleted, please post a message on [[Talk:Essowe abalo|the page's talk page]] explaining why. &lt;strong&gt;If your reasoning is convincing, your page may be saved.&lt;/strong&gt; If you have any questions or concerns, please [[User talk:MathXplore|let me know]]. Thank you! &lt;!-- Substituted from User:JJPMaster/CurateThisPage/authorMsg --&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:05, 18 April 2026 (UTC)</text>
      <sha1>1jhiuxn94c9tuu813x2ag9rgxv04txg</sha1>
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  </page>
  <page>
    <title>User talk:~2026-23858-31</title>
    <ns>3</ns>
    <id>482665</id>
    <revision>
      <id>4631222</id>
      <timestamp>2026-04-18T12:49:09Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <comment>delete1 ([[m:User:ZbVl/VD|Vandoom]])</comment>
      <origin>4631222</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="923" sha1="m5mgzwyug60bj16fn9tq6vxwp2x3jtp" xml:space="preserve">

== 2026-04-18 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-23858-31|your recent contributions]] have been undone because you removed content without adequately explaining why. In the future, it would be helpful to others if you described your changes to &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikibooks&lt;/span&gt; with an accurate [[:m:en:Help:Edit summary|edit summary]]. If this was a mistake, don't worry; the removed content has been restored. If you would like to experiment, please use the sandbox. Thanks. &lt;/div&gt;&lt;!-- Glow-delete1 @ 1776516541897.7s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:49, 18 April 2026 (UTC)</text>
      <sha1>m5mgzwyug60bj16fn9tq6vxwp2x3jtp</sha1>
    </revision>
    <revision>
      <id>4631224</id>
      <parentid>4631222</parentid>
      <timestamp>2026-04-18T12:49:58Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <comment>delete2 ([[m:User:ZbVl/VD|Vandoom]])</comment>
      <origin>4631224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1693" sha1="pb7dl78zzcc56rtx0jvrej5zv1j8r8s" xml:space="preserve">

== 2026-04-18 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-23858-31|your recent contributions]] have been undone because you removed content without adequately explaining why. In the future, it would be helpful to others if you described your changes to &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikibooks&lt;/span&gt; with an accurate [[:m:en:Help:Edit summary|edit summary]]. If this was a mistake, don't worry; the removed content has been restored. If you would like to experiment, please use the sandbox. Thanks. &lt;/div&gt;&lt;!-- Glow-delete1 @ 1776516541897.7s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:49, 18 April 2026 (UTC)

== 2026-04-18 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left; display: inline&quot; lang=&quot;en&quot;&gt;[[File:Information orange.svg|25px|alt=Information icon]] Sorry for writing in English, but please do not remove content or templates from pages on &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikibooks&lt;/span&gt; without giving a valid reason for the removal in the [[:m:en:Help:Edit summary|edit summary]]. Your content removal does not appear to be constructive and has been reverted. If you only meant to make a test edit, please use the sandbox for that. Thank you. &lt;/div&gt;&lt;!-- Glow-delete2 @ 1776516598763.7s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:49, 18 April 2026 (UTC)</text>
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  </page>
  <page>
    <title>Nursing Home Social Services Reference/Self Care/Support Systems</title>
    <ns>0</ns>
    <id>482666</id>
    <revision>
      <id>4631232</id>
      <timestamp>2026-04-18T14:00:27Z</timestamp>
      <contributor>
        <username>Reillylb93</username>
        <id>3572239</id>
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      <comment>Added section</comment>
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      <text bytes="3499" sha1="49ojjzd3edad97bdoxzkdh9t9kingea" xml:space="preserve">Nursing Home social services department works alongside many other departments and community entities in coordinating care for residents. This provides opportunities for collaboration, professional development, and learning. Being in similar industries, it is possible to build support systems for social services workers over time. 

'''Interdisciplinary Collaboration'''

Working alongside other departments at the Nursing Home is a constant and occurs daily. Nursing, business office, therapy, marketing/admissions, activities, to name a few. Typically, Nursing Homes hold a meeting every day where departments can collaborate on resident care and solve facility wide issues. Strongs relationships between departments including nursing staff, therapists, administrators, and any staff who work with residents can reduce role strain and create trust between the two. When social services workers feel respected and included in care planning, their job satisfaction and resilience can improve potentially leading to less burnout and longevity in the role. 

'''Clinical Supervision and Peer Support'''

Residents' emotional well-being and mental health concerns are often overseen by the social services department in collaboration with nursing. If a social services worker has earned their master's in social work, they are eligible to work towards their licensure or LCSW. Typically, Nursing Homes offer supervision if a licensed social work is employed. This individual or individuals can be working in the Nursing Home or oversee multiple Nursing Homes providing guidance and ensuring department compliance. Having a mentor, such as a licensed clinical social worker, provides a space to process complex cases, receive feedback on personal growth and job performance, and address ethical concerns. This kind of support, whether formal or informal, can offer validation and shared understanding. 

A mentor can be anyone who someone feels they benefit from and can trust. This is not a relationship intended to gossip or talk ill of others, rather to have productive conversations to explore their own professional actions and decisions and how to become better moving forward. Working in a Nursing Home is a high-paced, stressful environment, and having someone they can go to for advice and a second opinion can provide additional perspective. 

'''External Resources'''

Nursing Homes often partner with outside entities to provide therapy and psychiatric care to certain residents with higher mental health needs. These entities employ therapists, psychiatrists, psychologists, psychiatric nurses, etc.., who work in a similar capacity as social services workers. These individuals work collaboratively and closely with nursing and social services to provide wraparound care and update on their client's current status. Professional relationships form and these relationships become invaluable insights to residents care so that social services can improve their resident's well-being. Seeking professional opinions as well as relating to the stress and frustrations of the Nursing Home, these individuals can be sources of support. 

Additionally, other entities like Hospices employ social workers and chaplains to provide resource navigation, grief support, advocate, and become the bridge between the hospice, resident, and Nursing Home. Collaborating with hospice workers in a similar setting can help to reduce isolation and fosters shared problem-solving and the sharing of resources.</text>
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    <revision>
      <id>4631233</id>
      <parentid>4631232</parentid>
      <timestamp>2026-04-18T14:04:04Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <minor />
      <comment>Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]]</comment>
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      <text bytes="3512" sha1="7momchgrhg71dlwy14z14mjh9p5ezad" xml:space="preserve">Nursing Home social services department works alongside many other departments and community entities in coordinating care for residents. This provides opportunities for collaboration, professional development, and learning. Being in similar industries, it is possible to build support systems for social services workers over time. 

'''Interdisciplinary Collaboration'''

Working alongside other departments at the Nursing Home is a constant and occurs daily. Nursing, business office, therapy, marketing/admissions, activities, to name a few. Typically, Nursing Homes hold a meeting every day where departments can collaborate on resident care and solve facility wide issues. Strongs relationships between departments including nursing staff, therapists, administrators, and any staff who work with residents can reduce role strain and create trust between the two. When social services workers feel respected and included in care planning, their job satisfaction and resilience can improve potentially leading to less burnout and longevity in the role. 

'''Clinical Supervision and Peer Support'''

Residents' emotional well-being and mental health concerns are often overseen by the social services department in collaboration with nursing. If a social services worker has earned their master's in social work, they are eligible to work towards their licensure or LCSW. Typically, Nursing Homes offer supervision if a licensed social work is employed. This individual or individuals can be working in the Nursing Home or oversee multiple Nursing Homes providing guidance and ensuring department compliance. Having a mentor, such as a licensed clinical social worker, provides a space to process complex cases, receive feedback on personal growth and job performance, and address ethical concerns. This kind of support, whether formal or informal, can offer validation and shared understanding. 

A mentor can be anyone who someone feels they benefit from and can trust. This is not a relationship intended to gossip or talk ill of others, rather to have productive conversations to explore their own professional actions and decisions and how to become better moving forward. Working in a Nursing Home is a high-paced, stressful environment, and having someone they can go to for advice and a second opinion can provide additional perspective. 

'''External Resources'''

Nursing Homes often partner with outside entities to provide therapy and psychiatric care to certain residents with higher mental health needs. These entities employ therapists, psychiatrists, psychologists, psychiatric nurses, etc.., who work in a similar capacity as social services workers. These individuals work collaboratively and closely with nursing and social services to provide wraparound care and update on their client's current status. Professional relationships form and these relationships become invaluable insights to residents care so that social services can improve their resident's well-being. Seeking professional opinions as well as relating to the stress and frustrations of the Nursing Home, these individuals can be sources of support. 

Additionally, other entities like Hospices employ social workers and chaplains to provide resource navigation, grief support, advocate, and become the bridge between the hospice, resident, and Nursing Home. Collaborating with hospice workers in a similar setting can help to reduce isolation and fosters shared problem-solving and the sharing of resources.

{{BookCat}}</text>
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  <page>
    <title>Nursing Home Social Services Reference/Flow of Care/Hospitalization</title>
    <ns>0</ns>
    <id>482667</id>
    <revision>
      <id>4631234</id>
      <timestamp>2026-04-18T14:25:41Z</timestamp>
      <contributor>
        <username>Reillylb93</username>
        <id>3572239</id>
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      <comment>Completed section</comment>
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      <text bytes="4764" sha1="0hoync5rgjpoaj6tqfky87x6dgw6yuy" xml:space="preserve">The social services department is involved in the hospitalization of residents including transfers to/from hospital to skilled rehab and/or long-term care. These hospitalizations are interdisciplinary and involve almost all departments within the Nursing Home, even maintenance who prepares the room and may assist with moving the resident in/out if needed. In this section we will explore all aspects of a resident hospitalization in multiple scenarios. 

'''Transfer In, From Hospital to Skilled Rehab or Long-Term Care'''

Move-ins are coordinated by admissions who collaborate directly with the hospital to ensure the NH has everything needed including medical documentation. Admissions are the primary and social services supports in any way needed. Once a resident arrives, social services meet with the resident and completes a psychosocial evaluation. If the resident is staying long term, the social services worker can answer any questions the resident may have about how the facility works or help to address concerns. Social services are far more involved in the discharge planning that begins the moment a resident transfer into the skilled rehab. Rehab is a short-term stay, and social services works to ensure they have the appropriate services and supports in place once they discharge. 

Entities' involved: 

* Nursing staff and physicians within the facility
* Hospital social workers and discharge planners
* Rehabilitation therapists
* Transportation providers and external services
* Business office to facilitate financials
* Admissions to coordinate between Hospital and NH

This interdisciplinary approach ensures that both medical and psychosocial needs are addressed throughout the transition.

'''Transfer Out, From Nursing Home to Hospital''' 

If a resident's need for hospitalization is more acute or related to mental health, social services may have more of a hands-on involvement. For example, if a resident states they are having suicidal ideation, social services complete the necessary assessments needed for safety. If supports cannot be put in place at the Nursing Home and it is determined that psychiatric hospitalization is needed, social services will coordinate this with emergency dispatchers. 

'''Supporting Residents and Families During Hospitalization'''

If a resident goes to the hospital, nursing and/or social services alert the family member or emergency contacts. Social services can provide the hospital with documentation and coordinate the residents return if needed. More often than not, social services are not needed except to relay their return date to the other departments in the NH to prepare for their return. 

Key responsibilities include:

* Maintaining communication with hospital social workers or case managers
* Providing updates to family members as appropriate
* Assisting families in understanding medical information and care decisions
* Advocating for the resident’s needs, preferences, and baseline functioning

==== Return to the Facility from the Hospital ====
Social services ensure there is open communication about the residents return date/time to the Nursing Home so that all departments can be ready for their return. 

Social workers collaborate with hospital teams to:

* Confirm the resident’s eligibility and readiness to return
* Coordinate transportation and admission timing
* Ensure that necessary equipment, medications, and services are in place
* Communicate any changes in condition or care needs to the Nursing Home team

==== Regulatory and Compliance Considerations ====
Reducing avoidable hospitalizations is key when it comes to quality metrics. Social services contribute by supporting advance care planning, supports are in place, acting as advocates, and helping residents and their involved parties make informed decisions about treatment preferences. 

Hospital transfers and readmissions are also closely monitored. Social services participate to ensure these processes are completed as smoothly as possible. 
Transitions are not always straightforward. Social workers frequently address barriers such as:

* Delays in insurance authorization or coverage issues
* Family disagreements, discharge plans
* Changes in level of care needs that require reassessment
* Limited bed availability or facility readiness

'''Conclusion'''

Social services department plays a vital role in managing and overseeing hospitalizations and care transitions in Nursing Homes. They ensure that communication remains clear, resident preferences are honored, and care remains coordinated. Their involvement not only improves the efficiency of transitions but also preserves the dignity and well-being of residents during some of the most vulnerable moments in their care.</text>
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      <id>4631283</id>
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      <comment>Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]]</comment>
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      <text bytes="4777" sha1="4sg53i1vvpb4l71i5cnaqhma8fow1uz" xml:space="preserve">The social services department is involved in the hospitalization of residents including transfers to/from hospital to skilled rehab and/or long-term care. These hospitalizations are interdisciplinary and involve almost all departments within the Nursing Home, even maintenance who prepares the room and may assist with moving the resident in/out if needed. In this section we will explore all aspects of a resident hospitalization in multiple scenarios. 

'''Transfer In, From Hospital to Skilled Rehab or Long-Term Care'''

Move-ins are coordinated by admissions who collaborate directly with the hospital to ensure the NH has everything needed including medical documentation. Admissions are the primary and social services supports in any way needed. Once a resident arrives, social services meet with the resident and completes a psychosocial evaluation. If the resident is staying long term, the social services worker can answer any questions the resident may have about how the facility works or help to address concerns. Social services are far more involved in the discharge planning that begins the moment a resident transfer into the skilled rehab. Rehab is a short-term stay, and social services works to ensure they have the appropriate services and supports in place once they discharge. 

Entities' involved: 

* Nursing staff and physicians within the facility
* Hospital social workers and discharge planners
* Rehabilitation therapists
* Transportation providers and external services
* Business office to facilitate financials
* Admissions to coordinate between Hospital and NH

This interdisciplinary approach ensures that both medical and psychosocial needs are addressed throughout the transition.

'''Transfer Out, From Nursing Home to Hospital''' 

If a resident's need for hospitalization is more acute or related to mental health, social services may have more of a hands-on involvement. For example, if a resident states they are having suicidal ideation, social services complete the necessary assessments needed for safety. If supports cannot be put in place at the Nursing Home and it is determined that psychiatric hospitalization is needed, social services will coordinate this with emergency dispatchers. 

'''Supporting Residents and Families During Hospitalization'''

If a resident goes to the hospital, nursing and/or social services alert the family member or emergency contacts. Social services can provide the hospital with documentation and coordinate the residents return if needed. More often than not, social services are not needed except to relay their return date to the other departments in the NH to prepare for their return. 

Key responsibilities include:

* Maintaining communication with hospital social workers or case managers
* Providing updates to family members as appropriate
* Assisting families in understanding medical information and care decisions
* Advocating for the resident’s needs, preferences, and baseline functioning

==== Return to the Facility from the Hospital ====
Social services ensure there is open communication about the residents return date/time to the Nursing Home so that all departments can be ready for their return. 

Social workers collaborate with hospital teams to:

* Confirm the resident’s eligibility and readiness to return
* Coordinate transportation and admission timing
* Ensure that necessary equipment, medications, and services are in place
* Communicate any changes in condition or care needs to the Nursing Home team

==== Regulatory and Compliance Considerations ====
Reducing avoidable hospitalizations is key when it comes to quality metrics. Social services contribute by supporting advance care planning, supports are in place, acting as advocates, and helping residents and their involved parties make informed decisions about treatment preferences. 

Hospital transfers and readmissions are also closely monitored. Social services participate to ensure these processes are completed as smoothly as possible. 
Transitions are not always straightforward. Social workers frequently address barriers such as:

* Delays in insurance authorization or coverage issues
* Family disagreements, discharge plans
* Changes in level of care needs that require reassessment
* Limited bed availability or facility readiness

'''Conclusion'''

Social services department plays a vital role in managing and overseeing hospitalizations and care transitions in Nursing Homes. They ensure that communication remains clear, resident preferences are honored, and care remains coordinated. Their involvement not only improves the efficiency of transitions but also preserves the dignity and well-being of residents during some of the most vulnerable moments in their care.

{{BookCat}}</text>
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  </page>
  <page>
    <title>Nursing Home Social Services Reference/Career Paths/License</title>
    <ns>0</ns>
    <id>482668</id>
    <revision>
      <id>4631274</id>
      <timestamp>2026-04-18T22:42:43Z</timestamp>
      <contributor>
        <username>Rchaswms01</username>
        <id>3499245</id>
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      <comment>Added section</comment>
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      <text bytes="1550" sha1="j8r0eg7wkqkm0xbpg4ocddfx3yyrki0" xml:space="preserve">One of a social worker's goals may be to become a licensed socila worker in their state. At this time, licenses for professionals are provide by each state. There is a movement to create 'licensing compacts' to honor the licensing requirements of the various states  and allow those licensed professionals to treat residents of their state without further licensing reviews. 

Nursing home social services work qualifies as social work experience for licensing. To get a license, some number of hours of work experience is required. Each state has different requirements for the number of hours of work experience in order to get a social work license. Most states require some prerequisite for accepting hours of work experience. This prerequisite is usually passing the ACSW MSW test or some registration as a candidate for licensing. Once this is obtained, you may collect supervision hours that are group or individual. Supervision is a chance to demonstrate your competence in various areas of social work. This will be from your experiences working in the nursing home. The assessmetns that you do, the behavioral interventions that you do, the care plansyou design and other skills such as rapport-building will be the needed tools. The person who supervises you will also vouch for your work experience in hours. 

This takes from one and half to two years to achieve your license. Having the license allows you to make diagnoses, in many states to initiate involuntary commitment processes and other privileges will accrue. This may give you</text>
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    <revision>
      <id>4631282</id>
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      <timestamp>2026-04-19T02:51:44Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>3097823</id>
      </contributor>
      <minor />
      <comment>Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]]</comment>
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      <text bytes="1563" sha1="gau9ixhq8oscltv61v7wo7165i0lvh8" xml:space="preserve">One of a social worker's goals may be to become a licensed socila worker in their state. At this time, licenses for professionals are provide by each state. There is a movement to create 'licensing compacts' to honor the licensing requirements of the various states  and allow those licensed professionals to treat residents of their state without further licensing reviews. 

Nursing home social services work qualifies as social work experience for licensing. To get a license, some number of hours of work experience is required. Each state has different requirements for the number of hours of work experience in order to get a social work license. Most states require some prerequisite for accepting hours of work experience. This prerequisite is usually passing the ACSW MSW test or some registration as a candidate for licensing. Once this is obtained, you may collect supervision hours that are group or individual. Supervision is a chance to demonstrate your competence in various areas of social work. This will be from your experiences working in the nursing home. The assessmetns that you do, the behavioral interventions that you do, the care plansyou design and other skills such as rapport-building will be the needed tools. The person who supervises you will also vouch for your work experience in hours. 

This takes from one and half to two years to achieve your license. Having the license allows you to make diagnoses, in many states to initiate involuntary commitment processes and other privileges will accrue. This may give you

{{BookCat}}</text>
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  <page>
    <title>User:JKuroha</title>
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    <revision>
      <id>4631286</id>
      <timestamp>2026-04-19T03:38:26Z</timestamp>
      <contributor>
        <username>JKuroha</username>
        <id>3516066</id>
      </contributor>
      <comment>Created user page</comment>
      <origin>4631286</origin>
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      <text bytes="512" sha1="k08420prk618kx8ea204ymei562c16e" xml:space="preserve">{| style=&quot;float: right; margin-left: 1em; margin-bottom: 0.5em; width: 242px; border: #003366 solid 1px; &quot;
|-
| colspan=&quot;2&quot; style=&quot;text-align:center; background-color:#fff; font-size:120%; color:#003366;&quot; | '''Kuroha'''
|-
| {{#babel: plain=1|yue}}
|-
| {{#babel: plain=1|en}}
|-
| {{#babel: plain=1|cmn-4}}
|-
| {{#babel: plain=1|ja-3}}
|-
| {{#babel: plain=1|lzh-3}}
|-
| {{#babel: plain=1|ryu-1}}
|-
| {{#babel: plain=1|fr-1}}
|-
| {{#babel: plain=1|es-1}}
|-
| {{#babel: plain=1|la-1}}
|}

[[w:User:JKuroha]]</text>
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  <page>
    <title>Category:User ryu</title>
    <ns>14</ns>
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      <timestamp>2026-04-19T03:38:27Z</timestamp>
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      <comment>Automatically creating [[Project:Babel|Babel]] category page</comment>
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      <text bytes="105" sha1="5yv9gya7oetlg3634v06ebn5jgimg9x" xml:space="preserve">Users in this category indicate they have knowledge of language Okinawan.

[[Category:Users by language]]</text>
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  <page>
    <title>Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...Bc5</title>
    <ns>0</ns>
    <id>482672</id>
    <revision>
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      <timestamp>2026-04-19T05:42:32Z</timestamp>
      <contributor>
        <username>OneThreeFiveSevenNine</username>
        <id>3576738</id>
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      <comment>Created page with &quot;{{Chess Opening Theory/Position|= |King's Gambit Accepted: King's Knight Gambit }} == King's Gambit: King's Knight Gambit == === 3...Bc5? === An inaccurate reply against 3. Nf3. Black develops their bishop, however, runs straight into [[Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...Bc5/4. d4|4. d4]], taking the center, gaining tempo on the bishop and opening the queenside bishop to recapture the pawn on f4. Better was 3...g5, 3...Nf6 or 3...d6. Chess Open...&quot;</comment>
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      <text bytes="671" sha1="9nfb3nlsj0l3oei2tvma2nx33vrc0er" xml:space="preserve">{{Chess Opening Theory/Position|=
|King's Gambit Accepted: King's Knight Gambit
}}
== King's Gambit: King's Knight Gambit ==
=== 3...Bc5? ===
An inaccurate reply against 3. Nf3.
Black develops their bishop, however, runs straight into [[Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...Bc5/4. d4|4. d4]], taking the center, gaining tempo on the bishop and opening the queenside bishop to recapture the pawn on f4. Better was 3...g5, 3...Nf6 or 3...d6. [[Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...Bc5/4. Bc4|4. Bc4]] and [[Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...Bc5/4. Nc3|4. Nc3]] are sound options, too.

{{ChessFooter}}</text>
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    <revision>
      <id>4631296</id>
      <timestamp>2026-04-19T09:15:25Z</timestamp>
      <contributor>
        <username>JCrue</username>
        <id>2226064</id>
      </contributor>
      <comment>created</comment>
      <origin>4631296</origin>
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      <text bytes="691" sha1="1o5ull8tljm4raz7v4tftj8weji1v7z" xml:space="preserve">{{Chess Opening Theory/Position
|name=English attack
|eco=[[Chess/ECOB|B90]]
|parent=[[Chess Opening Theory/1. e4/1...c5/2. Nf3/2...d6/3. d4/3...cxd4/4. Nxd4/4...Nf6/5. Nc3/5...a6|Najdorf Sicilian]] → [[../|English attack]]
}}
== 6...b5? ==

6...b5?, apparently a natural move to expand on the queenside as prepared by ...a6, is a mistake on account of '''7. Nd5!'''. 

'''7. Nd5!''' appears to give up White's e-pawn. However, 7...Nxe5?? falls into a common amateur trap. After 8. Nxb5! (threatening Nc6+) axb5 9. Bb6 Black's queen is trapped. If 9...Qd7 10. Bxb5 Qxb5 11. Nc7+ forks the queen. 

{{ChessMid}}

== References ==
{{reflist}}
=== See also ===
{{Chess Opening Theory/Footer}}</text>
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