The Hahn-Banach Property and the Axiom of Choice

✍ Scribed by Juliette Dodu; Marianne Morillon

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 1021 KB
• Volume: 45
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19990450303

✦ Synopsis

Abstract

We work in set theory ZF without axiom of choice. Though the Hahn‐Banach theorem cannot be proved in ZF, we prove that every Gateaux‐differentiable uniformly convex Banach space E satisfies the following continuous Hahn‐Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ℝ is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ℝ such that g extends f and g ≤ p. We also prove that the continuous Hahn‐Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn‐Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn‐Banach property on Gateaux‐differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn‐Banach property, they do not satisfy the whole Hahn‐Banach property in ZF+DC.