Ascoli-Arzelà theorem
Let Ω be a bounded subset of ℝn and (fk) a sequence of functions fk:Ω→ℝm. 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 bounded metric spaces and let F⊂𝒞(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 closure is compact
(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 ˉF 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 |