Compactness in vector-valued Banach function spaces

TL;DR

This paper provides a new proof of a recent characterization of compactness in Lebesgue-Bochner spaces and extends it to vector-valued Banach function spaces with order continuous norms.

Contribution

It introduces a novel proof technique for compactness characterization and extends the results to a broader class of vector-valued Banach function spaces.

Findings

New proof of compactness characterization in $L_X^p$ spaces

Extension of the characterization to $E_X$ spaces with order continuous norm

Broader understanding of compactness in vector-valued Banach function spaces

Abstract

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\le p<\infty$, and extend the result to vector-valued Banach function spaces $E_X$, where $E$ is a Banach function space with order continuous norm.