Stability of unique Hahn–Banach extensions and associated linear projections

Abstract

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in and when this association forms a linear operator of norm-1 from to . It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if has property-U (SU) in . We prove that when has the Radon–Nikodým Property for 1 < p < ∞, L p (μ, Y) has property-U (property-SU) in L p (μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c 0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c 0 which have property-U (SU).