Structure of strongly proximinal subspaces

Abstract

In this paper we extended some results of G. Godefroy, originally proved for c0 using its special geometry, to the class of complex Banach spaces X whose dual is isometric to L1(μ). We exhibit classes of complex Banach spaces in which every strongly proximinal finite codimensional subspace is a finite dimensional perturbation of a M-ideal. We also show for this class, when the unit ball has an extreme point, if every proximinal subspace of codimension one is strongly proximinal, then the space is isometric to the k-dimensional space with the maximum norm, ℓ∞(k).