Some remarks on proximinality in higher dual spaces

Abstract

In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}_{{α∈ Γ}} we show that any finite co-dimensional proximinal subspace of X=c_0⊕Xα is strongly proximinal in all the duals of even order of X.