Equicontinuous

Equicontinuous

Preliminaries

Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018

If the common domain of a set of functions S is compact, then equicontinuity on the domain follows from equicontinuity at each point in the domain. This observation often makes it easier to verify equicontinuity on a domain. A set of functions S in C[a,b] is Functionequicontinuous familyEquicontinuousequicontinuous atx0 in [a,b] if given any ε>0 there is a δ > 0 depending on ɛ and on x0 such that for all f in S, |f(x)−f(x0)|<ε for x in [a,b] with |x−x0|<δ.

Normed Linear Spaces

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Published in James K. Peterson, Basic Analysis III, 2020

James K. Peterson

So what does a compact subset of an infinite dimensional set look like? We can give you one nice example using the idea of an equicontinuous family of continuous functions and use the Arzela - Ascoli Theorem to prove such a family is a compact set.

Analysis

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Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018

Dan Zwillinger

Ascoli-Arzela theorem Let K be a compact set in R. If {fn(x)} is uniformly bounded and equicontinuous on K, then {fn(x)} contains a uniformly convergent sub‐sequence on K.

Global behaviours of an in-host viral model with general incidence terms

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Journal Information

Published in Applicable Analysis, 2018

Hong Yang, Junjie Wei

This implies that is uniformly bounded. In view of the Arezla–Ascoli theorem, it suffice to show that is equicontinuous in for all . By the similar arguments as in [11, Lemma 4.1], we prove that , and are all equicontinuous in for all . We first show that is equicontinuous in for all . It is easy to see that

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Preliminaries

Normed Linear Spaces

Analysis

Global behaviours of an in-host viral model with general incidence terms