On the smoothness of the metric projection and its application to proximinality in \(L^ p (S, X)\)
Let \(X\) be a Banach space and let \(G\) be a closed proximinal subspace in \(X\). For \(x\in X\) let \(\pi_G (x)\) denote the set of best approximations to \(x\) in \(G\) and let \(\widehat {G}\) be given by \(\widehat {G}= \{x\in X: d(x,G)= |x|\}\). Finally, the function \(\psi: X\to [0,\infty)\) is defined by \(\psi(x)= d(0, \pi_G(x))\). The author gives sufficient conditions (for \(G\) and \(X\)) in order that \(\psi\) be lower semicontinuous and \(B_G+ \widehat {G}\) be closed, where \(B_G\) is the unit ball of \(G\). For \(p\in (0,\infty ]\) the proximinality of \(L^p (S, G)\) in \(L^p (S, X)\) is derived under similar conditions. Here \(L^p (S, X)= L^p (S, \Sigma, \mu, X)\) is the space of classes of Bochner \(p\)-integrable functions \(f: S\to X\).
scientific article; zbMATH DE number 827933