Arzelà-Ascoli's theorem in uniform spaces

Abstract

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