Proximinality in \(L_p(\mu,X)\)
Let \( X \) be a Banach space, let \( p\in [1,\infty] \) and let \( L_p(\mu,X) \) be the Banach space of \(X\)-valued Bochner \(p\)-integrable functions defined on a positive complete \(\sigma\)-finite measure space \( (\Omega,\Sigma,\mu)\). The author gives an example of a pair of Banach spaces \( (X,Y), Y\subseteq X\), with \( Y \) proximinal in \( X \) such that \( L_p(\mu,Y) \) is not proximinal in \( L_p(\mu,X)\). The pair \( (X,Y) \) was also used in other context by \textit{R. Holmes} and \textit{B. Kripke} [Mich. Math. J. 15, 225-248 (1968; Zbl 0177.16201)]. On the other hand the author shows that for separable proximinal subspaces \( Y \) of \( X, L_p(\mu,Y) \) is proximinal in \( L_p(\mu,X)\). A number of known results are derived from this last proposition.
scientific article; zbMATH DE number 1197085