Ascoli-Arzelà theorem

Let Ω be a boundedPlanetmathPlanetmathPlanetmathPlanetmath subset of n and (fk) a sequence of functions fk:Ωm. If {fk} is equibounded and uniformly equicontinuous then there exists a uniformly convergent subsequence (fkj).

A more abstract (and more general) version is the following.

Let X and Y be totally boundedPlanetmathPlanetmath metric spaces and let F𝒞(X,Y) be an uniformly equicontinuous family of continuous mappings from X to Y. Then F is totally bounded (with respect to the uniform convergence metric induced by 𝒞(X,Y)).

Notice that the first version is a consequence of the second. Recall, in fact, that a subset of a complete metric space is totally bounded if and only if its closurePlanetmathPlanetmath is compactPlanetmathPlanetmath (or sequentially compact). Hence Ω is totally bounded and all the functions fk have image in a totally bounded set. Being F={fk} totally bounded means that F¯ is sequentially compact and hence (fk) has a convergent subsequence.

Title Ascoli-Arzelà theorem
Canonical name AscoliArzelaTheorem
Date of creation 2013-03-22 12:41:00
Last modified on 2013-03-22 12:41:00
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 13
Author paolini (1187)
Entry type Theorem
Classification msc 46E15
Synonym Arzelà-Ascoli theorem
Related topic MontelsTheorem