Uniform Smoothness Entails Hahn-Banach

Abstract

We in Zermelo-Fraenkel set theory ZF (without
the Axiom of Choice), and we denote by ZFC set theory with
the Axiom of Choice. Our paper deals with the role of the Axiom of
Choice in functional
analysis, and more particularly, with the necessity of using the Axiom of Choice
when invoking some consequence of the following Hahn-Banach axiom HB.

Mathematics Subject Classification (2000): Primary 03E25; Secondary
46.
Keywords: axiom of choice, banach space, Hahn-Banach, uniformly smooth,
ZF, functional analysis, Zermelo-Fraenkel, Mazur property, ZFC, HB, Gâteaux
differentiability, Fréchet differentiability, q-Engel series, John Knopfmacher,
partition, identities, study, Santos polynomials, polynomials

Quaestiones Mathematicaes
24 (4) 2001, 425–439