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 Gâteaux-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 Gâteaux—differentiable normed spaces satisfy the (effective) continuous Hahn-Banach property in ZF. This enhances a result previously obtained in [1] for uniformly Fréchet differentiable Banach spaces.