Simultaneous proximinality in \(L^{∞}(μ,X)\) - Abu-Sirhan, Eyad (Rev. Anal. Numér. Théor. Approx., 2013)
Let \(X\) be a Banach space and \(G\) be a closed subspace of \(X\). Let us denote by \(L^{∞}\left( μ,X\right) \) the Banach space of all \(X\)-valued essentially bounded functions on a \(σ\)-finite complete measure space \(\left( Ω,Σ,μ\right) .\) In this paper we show that if \(G\) is separable, then \(L^{∞}\left( μ,G\right) \) is simultaneously proximinal in \(L^{∞}\left( μ,X\right) \) if and only if \(G\) is simultaneously proximinal in \(X.\)