Compactness in normed spaces: a unified approach through semi-norms

TL;DR

This paper introduces a unified framework for compactness in normed spaces using semi-norms, leading to new criteria that encompass classical results like Arzelà-Ascoli and extend to spaces of functions of bounded variation.

Contribution

It develops a general approach to compactness via equinormed sets and semi-norms, providing new criteria and applications to specific function spaces.

Findings

Established new abstract compactness criteria in normed spaces.

Unified classical compactness theorems within a semi-norm framework.

Characterized linear operators between variation spaces using kernel conditions.

Abstract

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,\mathbb{R})$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known compactness criteria (including the Arzel\`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation and provide a full characterization of linear integral operators acting from the space of functions of bounded Jordan variation to the space of functions of bounded Schramm variation in therms of their generating kernels.