Arzel\`a-Ascoli theorem in uniform spaces

TL;DR

This paper extends the Arzelà-Ascoli theorem to uniform spaces, introducing the extension property to unify topologies on function spaces and applying it to distribution theory.

Contribution

It introduces the extension property in uniform spaces, enabling a generalized Arzelà-Ascoli theorem for uniform convergence and applications in distribution theory.

Findings

Established relative compactness in spaces of distributions

Unified topologies on function spaces via the extension property

Extended Arzelà-Ascoli theorem to uniform spaces

Abstract

In the paper, we generalize the Arzel\`a-Ascoli theorem in the setting of uniform spaces. At first, we recall well-known facts and theorems coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of extension property which, similarly as equicontinuity, equates different topologies on $C(X,Y)$. This property enables us to prove the Arzel\`a-Ascoli theorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki theorem, we establish relative compactness of subfamily of $C(\mathbb{R},\mathcal{D}'(\mathbb{R}^n))$.