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UG-differentiability entails Hahn-Banach
Abstract
Denoting by ACN the countable axiom of choice, we show in ZF+ACN 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
Keywords: Compactness; gauge space; uniformly Gâteaux differentiable normed space; weak* topology; axiom of choice
Quaestiones Mathematicae 33(2010), 131–146