Quasi-interior Points and the Extension of Linear Functionals

R. E. Fullerton and C. C. Braunschweiger

A chapter in Contributions to Functional Analysis, 1966, pp 214-224 from Springer

Abstract: 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 ┴}.

Date: 1966
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DOI: 10.1007/978-3-642-85997-7_11

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