## You are here

HomeAscoli-Arzel\`a theorem

## Primary tabs

# Ascoli-Arzelà theorem

Let $\Omega$ be a bounded subset of $\mathbb{R}^{n}$ and $(f_{k})$ a sequence of functions $f_{k}\colon\Omega\to\mathbb{R}^{m}$. If $\{f_{k}\}$ is equibounded and uniformly equicontinuous then there exists a uniformly convergent subsequence $(f_{{k_{j}}})$.

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

Let $X$ and $Y$ be totally bounded metric spaces and let $F\subset\mathcal{C}(X,Y)$ 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 $\mathcal{C}(X,Y)$).

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 $\Omega$ is totally bounded and all the functions $f_{k}$ have image in a totally bounded set. Being $F=\{f_{k}\}$ totally bounded means that $\overline{F}$ is sequentially compact and hence $(f_{k})$ has a convergent subsequence.

## Mathematics Subject Classification

46E15*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz