Simultaneously proximinality in $L_\infty(\mu, X)$

TL;DR

This paper demonstrates that for any weakly compact set in a Banach space, the associated set of essentially bounded functions is N-simultaneously proximinal in the space of bounded functions, for any monotonic norm.

Contribution

It establishes the N-simultaneous proximinality of $L_(,)$ sets in $L_(,X)$ for weakly compact sets in Banach spaces, generalizing previous results.

Findings

$L_(,W)$ is N-simultaneously proximinal in $L_(,X)$

Applicable for arbitrary monotonous norms in $^n$

Results extend known proximinality properties in Banach space function spaces

Abstract

It is shown that for any W weakly compact set of a real Banach space X, the set $L_\infty(\mu ,W)$ is N-simultaneously proximinal in $L_\infty(\mu ,X)$ for arbitrary monotonous norm N in $\mathbb{R}^n$.