# Ascoli-Arzelà theorem

Let $\mathrm{\Omega}$ be a bounded^{} subset of ${\mathbb{R}}^{n}$ and $({f}_{k})$ a sequence of functions ${f}_{k}:\mathrm{\Omega}\to {\mathbb{R}}^{m}$. 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 $\mathrm{\Omega}$ 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.

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 |