Approximative compactness in B\"ochner spaces

TL;DR

This paper investigates the approximation properties of simple functions in Bochner spaces, establishing conditions under which these sets are proximinal and compact in dual Banach spaces with specific topological features.

Contribution

It proves proximinality and compactness of simple functions with limited values in Bochner spaces over dual Banach spaces with certain compactness properties.

Findings

Simple functions with at most k values are proximinal in $L^p(X)$ for dual Banach spaces.

Under additional conditions, these sets are approximatively $w^*$-compact.

Stronger hypotheses lead to approximative norm-compactness.

Abstract

For any $p\in[1,\infty)$, we prove that the set of simple functions taking at most $k$ different values is proximinal in B\"ochner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^*$-sequentially compact unit ball. With additional properties on $X$ and its norm, we show these sets are approximatively $w^*$-compact for $p\in(1,\infty)$ and even approximatively norm-compact under stronger hypothesis.