Contributions to Functional Analysis pp 214-224 | Cite as

# 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* ┴}.

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