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The Hahn‐Banach Property and the Axiom of Choice
Author(s) -
Dodu Juliette,
Morillon Marianne
Publication year - 1999
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19990450303
Subject(s) - mathematics , banach space , differentiable function , separable space , banach manifold , pure mathematics , infinite dimensional vector function , axiom of choice , discrete mathematics , lp space , mathematical analysis , set (abstract data type) , set theory , computer science , programming language
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.