Compactness in Lipschitz spaces and around

TL;DR

This paper characterizes (pre)compactness in Lipschitz and Hölder spaces, extending existing criteria with new concepts of equicontinuity and equinormed sets, providing a deeper understanding of function space compactness.

Contribution

It introduces novel criteria for compactness in Lipschitz/Hölder spaces using a new concept of equinormed sets and extends classical compactness results.

Findings

Established new compactness criteria for Lipschitz/Hölder spaces.

Introduced a concept of equinormed sets related to compactness.

Extended classical results with generalized conditions.

Abstract

The aim of the paper is to characterize (pre)compactness in the spaces of Lipschitz/H\"older continuous mappings acting from a compact metric space to a normed space. To this end some extensions and generalizations of already existing compactness criteria for the spaces of bounded and continuous mappings with values in normed spaces needed to be established. Those auxiliary results, which are interesting on their own since they use a concept of equicontinuity not seen in the literature, are based on an abstract compactness criterion related to the recently introduced notion of an equinormed set.