UNIFORM SMOOTHNESS ENTAILS HAHN-BANACH

Edmond Albius ERMIT, Département de Mathématiques et Informatique, Université de La Réunion, 15 avenue René Cassin—BP 7151-97715 Saint-Denis Messag., Cedex 9, France

Marianne Morillon ERMIT, Département de Mathématiques et Informatique, Université de La Réunion, 15 avenue René Cassin—BP 7151-97715 Saint-Denis Messag., Cedex 9, France E-mail: [email protected]

Abstract

We show in set theory ZF (without the Axiom of Choice), that uniformly smooth normed spaces satisfy an effective and geometric form of the Hahn-Banach property. We also compare in ZF the two notions of Gâteaux differentiability and smoothness of a norm, and we obtain a new equivalent of the Hahn-Banach axiom.

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UNIFORM SMOOTHNESS ENTAILS HAHN-BANACH

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