Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in \(L^ p(\mu,X)\) (Q1335029)

Let \(Y\) be a closed subspace of the Banach space \(X\), \((S,\Sigma,\mu)\) a \(\sigma\)-finite measure space, \(L(S,Y)\) (respectively, \(L(S,X)\)) the space of all strongly measurable functions from \(S\) to \(Y\) (respectively, \(X\)), and \(p\) a positive number. In this paper is proved that \(L(S,Y)\) is pointwise proximinal in \(L(S,X)\) if and only if \(L^ p(\mu,y)\) is proximinal in \(L^ p(\mu,X)\). As an application of this theorem is proved that if \(Y\) is a separable closed subspace of the space \(X\), \(p\) is a positive number, then \(L^ p(\mu,y)\) is proximinal in \(L^ p(\mu,X)\) if and only if \(Y\) is proximinal in \(X\). Some interesting results on pointwise best approximation are also obtained.