Quasi-interior points and the extension of linear functionals

Abstract

Let X be a real normed space, M a linear subspace of X, and X’ and M’ the spaces conjugate to X and M, respectively. It is known from the Hahn-Banach theorem [2] that to each f in M’ there corresponds at least one x’ in X’ with the same norm such that x’ (x) = f(x) for each x in M. The question of when this norm preserving extension is unique has received considerable attention. Taylor [15] and Foguel [4] have shown that this extension is unique for every subspace M of X and for each f in M’ if and only if the unit ball in X’ is strictly convex. Phelps [12] has shown that each f in M’ has a unique norm preserving extension in X’ if and only if the annihilator M┴ of M has the Haar property in X’; that is, if and only if to each x’ in X’ corresponds a unique y’ in M┴ such that ||x’, − y’|| = inf {||x’ − z’||: z’ ϵ M ┴}.