Some Applications of BP-Theorem in Approximation Theory

Abstract

In this paper we apply Bishop-Phelps property to show that if $X$ is a Banach space and $G \subseteq X$ is the maximal subspace so that $G^\bot = \{x^* \in X^*|x^*(y) = 0; \forall y \in G\}$ is an $L$-summand in $X^*$, then $L^1(\Omega,G)$ is contained in a maximal proximinal subspace of $L^1(\Omega,X)$.