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| {{unreferenced|date=January 2010}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| The purpose of this article is to serve as an [[Annotation|annotated]] [[Index (publishing)|index]] of various modes of convergence and their logical relationships. For an expository article, see [[Modes of convergence]]. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick reference, and indepth descriptions and discussions are reserved for their respective articles.
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| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
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| ''Guide to this index.'' To avoid excessive verbiage, note that each of the following types of objects is a special case of types preceding it: [[set (mathematics)|set]]s, [[topological space]]s, [[uniform space]]s, [[topological abelian group]]s (TAG), [[normed vector space]]s, [[Euclidean space]]s, and the [[Real number|real]]/[[Complex number|complex]] numbers. Also note that any [[metric space]] is a uniform space. Finally, '''subheadings will always indicate special cases of their superheadings.'''
| | Registered users will be able to choose between the following three rendering modes: |
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| The following is a list of modes of convergence for:
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==A sequence of elements {''a<sub>n</sub>''} in a topological space (''Y'')==
| | <!--'''PNG''' (currently default in production) |
| * '''[[Limit of a sequence|Convergence]]''', or "topological convergence" for emphasis (i.e. the existence of a limit).
| | :<math forcemathmode="png">E=mc^2</math> |
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| ===...in a uniform space (''U'')===
| | '''source''' |
| * '''[[Cauchy sequence|Cauchy-convergence]]'''
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| Implications:
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| - Convergence <math>\Rightarrow</math> Cauchy-convergence
| | ==Demos== |
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| - Cauchy-convergence and convergence of a subsequence together <math>\Rightarrow</math> convergence.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| - ''U'' is called "complete" if Cauchy-convergence (for nets) <math>\Rightarrow</math> convergence.
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| Note: A sequence exhibiting Cauchy-convergence is called a ''cauchy sequence'' to emphasize that it may not be convergent.
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| ==A series of elements Σ''b<sub>k</sub>'' in a TAG (''G'')== | | ==Test pages == |
| * '''[[Limit of a sequence|Convergence]]''' (of partial sum sequence)
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| * '''[[Cauchy sequence|Cauchy-convergence]]''' (of partial sum sequence)
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| * '''[[Unconditional convergence]]'''
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| Implications:
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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| - Unconditional convergence <math>\Rightarrow</math> convergence (by definition).
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| ===...in a normed space (''N'')===
| | ==Bug reporting== |
| * '''[[Absolute convergence|Absolute-convergence]]''' (convergence of <math>\sum |b_k|</math>) | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
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| Implications:
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| - Absolute-convergence <math>\Rightarrow</math> Cauchy-convergence <math>\Rightarrow</math> absolute-convergence of some grouping<sup>1</sup>.
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| - Therefore: ''N'' is [[Banach space|Banach]] (complete) if absolute-convergence <math>\Rightarrow</math> convergence.
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| - Absolute-convergence and convergence together <math>\Rightarrow</math> unconditional convergence.
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| - Unconditional convergence <math>\not\Rightarrow</math> absolute-convergence, even if ''N'' is Banach.
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| - If ''N'' is a Euclidean space, then unconditional convergence <math>\equiv</math> absolute-convergence.
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| <sup>1</sup> Note: "grouping" refers to a series obtained by grouping (but not reordering) terms of the original series. A grouping of a series thus corresponds to a subsequence of its partial sums.
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| ==A sequence of functions {''f<sub>n</sub>''} from a set (''S'') to a topological space (''Y'')==
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| * '''[[Pointwise convergence]]''' | |
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| ===...from a set (''S'') to a uniform space (''U'')===
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| * '''[[Uniform convergence]]'''
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| * '''Pointwise Cauchy-convergence'''
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| * '''[[Uniformly Cauchy sequence|Uniform Cauchy-convergence]]
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| Implications are cases of earlier ones, except:
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| - Uniform convergence <math>\Rightarrow</math> both pointwise convergence and uniform Cauchy-convergence.
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| - Uniform Cauchy-convergence and pointwise convergence of a subsequence <math>\Rightarrow</math> uniform convergence.
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| ====...from a topological space (''X'') to a uniform space (''U'')==== | |
| For many "global" modes of convergence, there are corresponding notions of ''a'') "local" and ''b'') "compact" convergence, which are given by requiring convergence to occur ''a'') on some neighborhood of each point, or ''b'') on all compact subsets of ''X''. Examples:
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| * '''[[Local uniform convergence]]''' (i.e. uniform convergence on a neighborhood of each point)
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| * '''[[Compact convergence|Compact (uniform) convergence]]''' (i.e. uniform convergence on all compact subsets)
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| * further instances of this pattern below.
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| Implications:
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| - "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.:
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| Uniform convergence <math>\Rightarrow</math> both local uniform convergence and compact (uniform) convergence.
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| - "Local" modes of convergence tend to imply "compact" modes of convergence. E.g.,
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| Local uniform convergence <math>\Rightarrow</math> compact (uniform) convergence.
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| - If <math>X</math> is locally compact, the converses to such tend to hold:
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| Local uniform convergence <math>\equiv</math> compact (uniform) convergence.
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| ==== ...from a measure space (S,μ) to the complex numbers (C) ====
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| * '''[[Pointwise_convergence#In_measure_theory|Almost everywhere convergence]]'''
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| * '''[[Uniform_convergence#Almost_uniform_convergence|Almost uniform convergence]]'''
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| * '''[[Lp_space#Lp_spaces|L<sup>p</sup> convergence]]'''
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| * '''[[Convergence in measure]]'''
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| * '''[[Convergence in distribution]]'''
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| Implications:
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| - Pointwise convergence <math>\Rightarrow</math> almost everywhere convergence.
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| - Uniform convergence <math>\Rightarrow</math> almost uniform convergence. | |
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| - Almost everywhere convergence <math>\Rightarrow</math> convergence in measure. (In a finite measure space)
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| - Almost uniform convergence <math>\Rightarrow</math> convergence in measure.
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| - L<sup>p</sup> convergence <math>\Rightarrow</math> convergence in measure.
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| - Convergence in measure <math>\Rightarrow</math> convergence in distribution if μ is a probability measure and the functions are integrable.
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| ==A series of functions Σ''g<sub>k</sub>'' from a set (''S'') to a TAG (''G'')== | |
| * '''[[Pointwise convergence]]''' (of partial sum sequence)
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| * '''[[Uniform convergence]]''' (of partial sum sequence)
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| * '''Pointwise Cauchy-convergence''' (of partial sum sequence)
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| * '''[[Uniformly Cauchy sequence|Uniform Cauchy-convergence]]''' (of partial sum sequence)
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| * '''Unconditional pointwise convergence'''
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| * '''Unconditional uniform convergence'''
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| Implications are all cases of earlier ones.
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| ===...from a set (''S'') to a normed space (''N'')===
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| Generally, replacing "convergence" by "absolute-convergence" means one is referring to convergence of the series of nonnegative functions <math>\Sigma|g_k|</math> in place of <math>\Sigma g_k</math>.
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| * '''Pointwise absolute-convergence''' (pointwise convergence of <math>\Sigma|g_k|</math>)
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| * '''[[Uniform absolute-convergence]]''' (uniform convergence of <math>\Sigma|g_k|</math>)
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| * '''[[Normal convergence]]'''<sup>[http://eom.springer.de/N/n067430.htm]</sup> (convergence of the series of [[uniform norm]]s <math>\Sigma||g_k||_u</math>)
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| Implications are cases of earlier ones, except:
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| - Normal convergence <math>\Rightarrow</math> uniform absolute-convergence
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| ===...from a topological space (''X'') to a TAG (''G'')===
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| * '''[[Local uniform convergence]]''' (of partial sum sequence)
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| * '''[[Compact convergence|Compact (uniform) convergence]]''' (of partial sum sequence)
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| Implications are all cases of earlier ones.
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| ====...from a topological space (''X'') to a normed space (''N'')====
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| * '''[[Uniform_absolute-convergence#Generalizations|Local uniform absolute-convergence]]'''
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| * '''[[Uniform_absolute-convergence#Generalizations|Compact (uniform) absolute-convergence]]'''
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| * '''[[Normal_convergence#Local_normal_convergence|Local normal convergence]]'''
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| * '''[[Normal_convergence#Compact_normal_convergence|Compact normal convergence]]
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| Implications (mostly cases of earlier ones):
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| - Uniform absolute-convergence <math>\Rightarrow</math> both local uniform absolute-convergence and compact (uniform) absolute-convergence.
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| Normal convergence <math>\Rightarrow</math> both local normal convergence and compact normal convergence.
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| - Local normal convergence <math>\Rightarrow</math> local uniform absolute-convergence.
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| Compact normal convergence <math>\Rightarrow</math> compact (uniform) absolute-convergence.
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| - Local uniform absolute-convergence <math>\Rightarrow</math> compact (uniform) absolute-convergence.
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| Local normal convergence <math>\Rightarrow</math> compact normal convergence
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| - If ''X'' is locally compact:
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| Local uniform absolute-convergence <math>\equiv</math> compact (uniform) absolute-convergence.
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| Local normal convergence <math>\equiv</math> compact normal convergence
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| == See also ==
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| * [[Limit of a sequence]]
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| [[Category:Convergence (mathematics)]]
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