$Simultaneous proximinality in $L^{\infty}(\mu,X)$$

Simultaneous proximinality in $L^{\infty}(\mu,X)$

Let $X$ be a Banach space and $G$ be a closed subspace of $X$. Let us denote by $L^{\infty}\left( \mu,X\right) $ the Banach space of all $X$-valued essentially bounded functions on a $\sigma$-finite complete measure space $\left( \Omega,\Sigma,\mu\right) .$ In this paper we show that i...

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Bibliographic Details
Main Author:	Eyad Abu-Sirhan
Format:	Article
Language:	English
Published:	Publishing House of the Romanian Academy 2013-08-01
Series:	Journal of Numerical Analysis and Approximation Theory
Subjects:	simultaneous approximation Banach spaces
Online Access:	https://ictp.acad.ro/jnaat/journal/article/view/984

Description
Summary:	Let $X$ be a Banach space and $G$ be a closed subspace of $X$. Let us denote by $L^{\infty}\left( \mu,X\right) $ the Banach space of all $X$-valued essentially bounded functions on a $\sigma$-finite complete measure space $\left( \Omega,\Sigma,\mu\right) .$ In this paper we show that if $G$ is separable, then $L^{\infty}\left( \mu,G\right) $ is simultaneously proximinal in $L^{\infty}\left( \mu,X\right) $ if and only if $G$ is simultaneously proximinal in $X.$
ISSN:	2457-6794 2501-059X

Summary:	Let \(X\) be a Banach space and \(G\) be a closed subspace of \(X\). Let us denote by \(L^{\infty}\left( \mu,X\right) \) the Banach space of all \(X\)-valued essentially bounded functions on a \(\sigma\)-finite complete measure space \(\left( \Omega,\Sigma,\mu\right) .\) In this paper we show that if \(G\) is separable, then \(L^{\infty}\left( \mu,G\right) \) is simultaneously proximinal in \(L^{\infty}\left( \mu,X\right) \) if and only if \(G\) is simultaneously proximinal in \(X.\)
ISSN:	2457-6794 2501-059X

Simultaneous proximinality in \(L^{\infty}(\mu,X)\)

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