Various notions of best approximation property in spaces of Bochner integrable functions

TL;DR

This paper investigates various approximation properties in spaces of Bochner integrable functions, establishing equivalences and introducing the concept of uniform proximinality, with applications to subspace structures and examples.

Contribution

It characterizes when proximinal properties are preserved in Bochner spaces and introduces the new notion of uniform proximinality with multiple examples.

Findings

Proximinal properties in subspaces are equivalent in Bochner spaces for certain p-values.

Uniform proximinality is a new concept with several examples provided.

Preservation of proximinality in Bochner spaces depends on separability and other conditions.

Abstract

We derive that for a separable proximinal subspace $Y$ of $X$, $Y$ is strongly proximinal (strongly ball proximinal) if and only if for $1\leq p< \infty$, $L_p(I,Y)$ is strongly proximinal (strongly ball proximinal) in $L_p(I,X)$. Case for $p=\infty$ follows from stronger assumption on $Y$ in $X$ (uniform proximinality). It is observed that for a separable proximinal subspace $Y$ in $X$, $Y$ is ball proximinal in $X$ if and only if $L_p(I,Y)$ is ball proximinal in $L_p(I,X)$ for $1\leq p\leq\infty$. Our observations also include the fact that for any (strongly) proximinal subspace $Y$ of $X$, if every separable subspace of $Y$ is ball (strongly) proximinal in $X$ then $L_p(I,Y)$ is ball (strongly) proximinal in $L_p(I,X)$ for $1\leq p<\infty$. We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in \cite{LZ}. Several examples are given having this property, viz. any $U$-subspace of a Banach space, closed unit ball $B_X$ of a space with $3.2.I.P$, closed unit ball of any M-ideal of a space with $3.2.I.P.$ are uniformly proximinal. A new class of examples are given having this property.