UG-differentiability entails Hahn-Banach

Abstract

Denoting by AC_N the countable axiom of choice, we show in ZF+AC_N that the dual ball of a uniformly G^ateaux-differentiable normed space is compact in the weak* topology. In ZF, we prove that this dual ball is (closely) convex-compact in the weak* topology. We deduce that uniformly G^ateaux–differentiable normed spaces satisfy the (effective) continuous Hahn-Banach property in ZF. This enhances a result previously obtained in [1] for uniformly Fr´echet differentiable Banach spaces.

Keywords: Compactness; gauge space; uniformly Gâteaux differentiable normed space; weak* topology; axiom of choice

Quaestiones Mathematicae 33(2010), 131–146