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

Author:

Abu-Sirhan Eyad

Abstract

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 \(X.\)

Publisher

Academia Romana Filiala Cluj

Reference21 articles.

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5. E. Abu-Sirhan, Best simultaneous approximation in Lp(I,X), Inter. J. Math. Analysis, 3 (2009) no. 24, pp. 1157-1168.

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