| Abstrakt: |
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 Y ∗ and when this association forms a linear operator of norm-1 from Y ∗ to X ∗ . 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 X ⊗ ε ∨ Y has property-U (SU) in X ⊗ ε ∨ Z. We prove that when X ∗ has the Radon–Nikodým Property for 1 < p < ∞, Lp(μ, Y) has property-U (property-SU) in Lp(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 c0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c0 which have property-U (SU). [ABSTRACT FROM AUTHOR] |
