On nonlinear simultaneous approximation problems

In this paper we prove that if Y is a reflexive subspace of a Banach space X, then L (a)(mu, Y) is simultaneously proximinal in L (a)(mu, X). Furthermore if X is reflexive and mu (0) is the restriction of mu to a sub-sigma-algebra, then L (a)(mu (0), X) is simultaneously proximinal in L (a)(mu, X).