Best simultaneous approximation in L^{p}(S,X)

As a counterpart to the best approximation in normed linear spaces, the best simultaneous approximation was introduced. In this paper, we shall consider relation between simultaneous proximinality W in X and L^p(S,W) in L^p(S,X) for ۱\leq p\leq\infty. Also, we consider the relation between w-simultaneous proximinality W in X and L^p(S,W) in L^p(S,X) for ۱\leq p\leq\infty.