Some Applications of BP-Theorem in Approximation Theory (Analysis in Theory and Applications, 2011)
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 $L^1(Ω,G)$ is contained in a maximal proximinal subspace of $L^1(Ω,X)$.
Keywords: Bishop-Phelps theorem; support point; proximinality; $L$-projection.