Compactness in the Lebesgue-Bochner spaces L^p(\mu;X)

TL;DR

This paper provides an elementary proof of the Diaz–Mayoral theorem, characterizing relative compactness in Lebesgue-Bochner spaces through uniform p-integrability, tightness, and scalar relative compactness.

Contribution

It offers a simplified proof of a key theorem linking compactness criteria in Lebesgue-Bochner spaces, enhancing understanding and accessibility.

Findings

Relatively compact subsets are characterized by uniform p-integrability.

Uniform tightness is essential for compactness in L^p(;X).

Scalar relative compactness is a necessary condition.

Abstract

Let (\Omega,\mu) be a finite measure space, X a Banach space, and let 1\le p<\infty. The aim of this paper is to give an elementary proof of the Diaz--Mayoral theorem that a subset V of L^p(\mu;X) is relatively compact if and only if it is uniformly p-integrable, uniformly tight, and scalarly relatively compact.