Arzela-Ascoli's Theorem and Applications

Chaoyi Sheng

doi:10.20944/preprints202210.0209.v1

Preprint Essay Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 12 October 2022 / Approved: 14 October 2022 / Online: 14 October 2022 (10:13:19 CEST)

In Functional Analysis as well as Topology, we frequently encounter sets, say $X$, that contain elements close to each other. These sets display a defining "finite-ness" property: for all open cover $\mathcal{O}$ of $X$, there exists a finite subcollection $\mathcal{U}\subseteq\mathcal{O}$ such that $\mathcal{U}$ covers $X$. Such spaces $X$ are called $\textbf{compact}$, and the above "finite-ness" property afford us great convenience, because we can always investigate $X$ by investigating its finite open cover $\mathcal{U}$Arzela-

Arzela-Ascoli's theorem; functional analysis; topology

Computer Science and Mathematics, Mathematics

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Arzela-Ascoli's Theorem and Applications

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