Arzelà-Ascoli's theorem in uniform spaces

Mateusz Krukowski

doi:10.3934/dcdsb.2018020

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2018, Volume 23, Issue 1: 283-294. Doi: 10.3934/dcdsb.2018020

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Arzelà-Ascoli's theorem in uniform spaces

Mateusz Krukowski^1, ,

1.
Łódź University of Technology, Institute of Mathematics, Wólczańska 215, 90-924 Łódź, Poland

* Corresponding author: Mateusz Krukowski
* Corresponding author: Mateusz Krukowski

Received: June 30, 2016

Revised: August 31, 2016

Published: November 2017

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Abstract

In the paper, we generalize the Arzelà-Ascoli's theorem in the setting of uniform spaces. At first, we recall the Arzelà-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 $C(X,Y)$. This property enables us to prove the Arzelà-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 $C({\mathbb{R}},{\mathcal{D}}'({\mathbb{R}}^n))$.

Keywords:

Arzelà-Ascoli's theorem,
uniform spaces,
uniformity of uniform convergence (on compacta).

Mathematics Subject Classification: Primary: 46E10; Secondary: 54E15.

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