Hadamard code - Revision history

en>TheSawTooth: Reverted good faith edits by 101.98.195.26 (talk). (TW)

2014-11-09T00:55:41Z

Reverted good faith edits by 101.98.195.26 (talk). (TW)

← Older revision		Revision as of 02:55, 9 November 2014
Line 1:		Line 1:
	~~{{refimprove\|date=January 2010}}~~		Mechanical Engineering Draftsperson Elden Earlywine from North Portal, has hobbies and interests for instance fencing, como [http://comoganhardinheiropelainternet.comoganhardinheiro101.com/ ganhar dinheiro] na internet and fitness. Last month just traveled to Djoudj National Bird Sanctuary.
	In [[mathematics]] '''compact convergence''' (or '''uniform convergence on compact sets''') is a type of [[limit of a sequence\|convergence]] which generalizes the idea of [[uniform convergence]]. It is associated with the [[compact-open topology]].

	~~==Definition==~~
	~~Let <math>(X~~, ~~\mathcal{T})</math> be a [[topological space]]~~ and ~~<math>(Y,d_{Y})</math> be a [[metric space]]. A sequence of functions~~

	~~:<math>f_{n} : X \to Y</math>, <math>n \in \mathbb{N},</math>~~

	~~is said to '''converge compactly''' as <math>n \to \infty</math> to some function <math>f : X \to Y</math> if,~~ for ~~every [[compact set]] <math>K \subseteq X</math>~~,

	~~:<math>(f_{n})\|_{K} \to f\|_{K}</math>~~

	[[~~uniform convergence\|converges uniformly]] on <math>K</math> as <math>n \to \infty</math>. This means that for all compact <math>K \subseteq X</math>,~~

	~~:<math>\lim_{n \to \infty} \sup_{x \in K} d_{Y} \left( f_{n} (x), f(x) \right) = 0.</math>~~

	~~==Examples==~~
	* If <math>X = (0, 1) \subset \mathbb{R}</math> and <math>Y = \mathbb{R}</math> with their usual topologies, with <math>f_{n} (x) :~~= x^{n}<~~/~~math>, then <math>f_{n}<~~/~~math> converges compactly to the constant function with value 0, but not uniformly~~.

	* If <math>X=(0,1]</math>, <math>Y=\R</math> and <math>f_n(x)=x^n</math>, then <math>f_n</math> converges [[pointwise convergence\|pointwise]] to the function that is zero on <math>(0,1)</math> and one at <math>1</math>, but the sequence does not converge compactly.

	* A very powerful tool for showing compact convergence is the [[Arzelà–Ascoli theorem]]. There are several versions of this theorem, roughly speaking it states that every sequence of [[equicontinuous]] and [[uniformly bounded]] maps has a subsequence which converges compactly to some continuous map.

	~~==Properties==~~
	* If <math>f_{n} \to f</math> uniformly, then <math>f_{n} \to f</math> compactly.
	* If <math>(X, \mathcal{T})</~~math> is a [[compact space]~~] and ~~<math>f_{n} \to f</math> compactly, then <math>f_{n} \to f</math> uniformly~~.
	* If <math>(X, \mathcal{T})</math> is [[locally compact]], then <math>f_{n} \to f</math> compactly if and only if <math>f_{n} \to ~~f</math> locally uniformly~~.
	* If <math>(X, \mathcal{T})</math> is a [[compactly generated space]], <math>f_n\to f</math> compactly, and each <math>f_n</math> is [[continuous function\|continuous]], then <math>f</math> is continuous.

	~~==See also==~~
	*[[Modes of convergence (annotated index)]]
	*[[Montel's theorem]]

	~~==References==~~
	*R. Remmert ''Theory of complex functions'' (1991 Springer) p. 95


	~~{{DEFAULTSORT:Compact Convergence}}~~
	~~[[Category:Functional analysis]]~~
	~~[[Category:Convergence (mathematics)]]~~
	~~[[Category:Topology of function spaces]]~~
	~~[[Category:Topological spaces]]~~

en>Awaterl: Add periods at the end of two sentences.

2013-08-06T05:13:14Z

Add periods at the end of two sentences.

← Older revision		Revision as of 07:13, 6 August 2013
Line 1:		Line 1:
	~~Marvella~~ is ~~what you can call her but it's not~~ the ~~most feminine name out there~~. ~~Hiring has been my occupation for~~ some ~~time but I've already applied~~ for ~~another 1. His wife doesn't like it the way he does but~~ [~~http~~://~~www~~.~~revleft~~.~~com~~/vb/~~member.php?u~~=~~160656 at home std testing] what he really at [http~~://~~203~~.~~250.78.160~~/~~zbxe~~/~~?document_srl~~=~~817810&mid=gallery home std test~~] ~~likes doing~~ is ~~to do aerobics~~ and ~~he's been doing~~ it ~~for fairly~~ a ~~whilst~~. ~~Years in the past we moved~~ to ~~North Dakota and I adore every [http:~~//~~Www~~.~~Aahealth.org~~/~~programs~~/~~comm-diseases~~/~~hiv-aids~~/~~hotlines day living~~] ~~right here~~.<br><br>~~Here~~ is ~~my web site; std home test (~~[~~http:~~//~~www~~.~~fuguporn~~.~~com/user/RMitchell Discover More~~])		{{refimprove\|date=January 2010}}
			In [[mathematics]] '''compact convergence''' (or '''uniform convergence on compact sets''') is a type of [[limit of a sequence\|convergence]] which generalizes the idea of [[uniform convergence]]. It is associated with the [[compact-open topology]].

			==Definition==
			Let <math>(X, \mathcal{T})</math> be a [[topological space]] and <math>(Y,d_{Y})</math> be a [[metric space]]. A sequence of functions

			:<math>f_{n} : X \to Y</math>, <math>n \in \mathbb{N},</math>

			is said to '''converge compactly''' as <math>n \to \infty</math> to some function <math>f : X \to Y</math> if, for every [[compact set]] <math>K \subseteq X</math>,

			:<math>(f_{n})\|_{K} \to f\|_{K}</math>

			[[uniform convergence\|converges uniformly]] on <math>K</math> as <math>n \to \infty</math>. This means that for all compact <math>K \subseteq X</math>,

			:<math>\lim_{n \to \infty} \sup_{x \in K} d_{Y} \left( f_{n} (x), f(x) \right) = 0.</math>

			==Examples==
			* If <math>X = (0, 1) \subset \mathbb{R}</math> and <math>Y = \mathbb{R}</math> with their usual topologies, with <math>f_{n} (x) := x^{n}</math>, then <math>f_{n}</math> converges compactly to the constant function with value 0, but not uniformly.

			* If <math>X=(0,1]</math>, <math>Y=\R</math> and <math>f_n(x)=x^n</math>, then <math>f_n</math> converges [[pointwise convergence\|pointwise]] to the function that is zero on <math>(0,1)</math> and one at <math>1</math>, but the sequence does not converge compactly.

			* A very powerful tool for showing compact convergence is the [[Arzelà–Ascoli theorem]]. There are several versions of this theorem, roughly speaking it states that every sequence of [[equicontinuous]] and [[uniformly bounded]] maps has a subsequence which converges compactly to some continuous map.

			==Properties==
			* If <math>f_{n} \to f</math> uniformly, then <math>f_{n} \to f</math> compactly.
			* If <math>(X, \mathcal{T})</math> is a [[compact space]] and <math>f_{n} \to f</math> compactly, then <math>f_{n} \to f</math> uniformly.
			* If <math>(X, \mathcal{T})</math> is [[locally compact]], then <math>f_{n} \to f</math> compactly if and only if <math>f_{n} \to f</math> locally uniformly.
			* If <math>(X, \mathcal{T})</math> is a [[compactly generated space]], <math>f_n\to f</math> compactly, and each <math>f_n</math> is [[continuous function\|continuous]], then <math>f</math> is continuous.

			==See also==
			*[[Modes of convergence (annotated index)]]
			*[[Montel's theorem]]

			==References==
			*R. Remmert ''Theory of complex functions'' (1991 Springer) p. 95


			{{DEFAULTSORT:Compact Convergence}}
			[[Category:Functional analysis]]
			[[Category:Convergence (mathematics)]]
			[[Category:Topology of function spaces]]
			[[Category:Topological spaces]]

en>Phil Boswell: convert dodgy URL to ID, replaced: url = http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.154.2879&rep=rep1&type=pdf → id = {{citeseerx|10.1.1.154.2879}} using AWB

2012-07-17T08:18:42Z

convert dodgy URL to ID, replaced: url = http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.154.2879&rep=rep1&type=pdf → id = {{citeseerx|10.1.1.154.2879}} using AWB

New page

Marvella is what you can call her but it's not the most feminine name out there. Hiring has been my occupation for some time but I've already applied for another 1. His wife doesn't like it the way he does but [http://www.revleft.com/vb/member.php?u=160656 at home std testing] what he really at [http://203.250.78.160/zbxe/?document_srl=817810&mid=gallery home std test] likes doing is to do aerobics and he's been doing it for fairly a whilst. Years in the past we moved to North Dakota and I adore every [http://Www.Aahealth.org/programs/comm-diseases/hiv-aids/hotlines day living] right here.<br><br>Here is my web site; std home test ([http://www.fuguporn.com/user/RMitchell Discover More])