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| {{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]].
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| ==Definition==
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| Let <math>(X, \mathcal{T})</math> be a [[topological space]] and <math>(Y,d_{Y})</math> be a [[metric space]]. A sequence of functions
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| :<math>f_{n} : X \to Y</math>, <math>n \in \mathbb{N},</math>
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| 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>,
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| :<math>(f_{n})|_{K} \to f|_{K}</math>
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| [[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>,
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| :<math>\lim_{n \to \infty} \sup_{x \in K} d_{Y} \left( f_{n} (x), f(x) \right) = 0.</math>
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| ==Examples==
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| * 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.
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| * 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.
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| * 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.
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| ==Properties==
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| * If <math>f_{n} \to f</math> uniformly, then <math>f_{n} \to f</math> compactly.
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| * 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.
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| * 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.
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| * 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.
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| ==See also==
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| *[[Modes of convergence (annotated index)]]
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| *[[Montel's theorem]]
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| ==References==
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| *R. Remmert ''Theory of complex functions'' (1991 Springer) p. 95
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| {{DEFAULTSORT:Compact Convergence}}
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| [[Category:Functional analysis]]
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| [[Category:Convergence (mathematics)]]
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| [[Category:Topology of function spaces]]
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| [[Category:Topological spaces]]
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Mechanical Engineering Draftsperson Elden Earlywine from North Portal, has hobbies and interests for instance fencing, como ganhar dinheiro na internet and fitness. Last month just traveled to Djoudj National Bird Sanctuary.