The Hahn-Banach Property and the Axiom of Choice

Mathematical Logic Quarterly 45 (3):299-314 (1999)

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
Keywords Uniformly convex  Axiom of choice  Ekeland's variational principle  Hahn‐Banach theorem  Gâteaux‐differentiable  Axiom of dependent choices
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DOI 10.1002/malq.19990450303
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References found in this work BETA

Definability of Measures and Ultrafilters.David Pincus & Robert M. Solovay - 1977 - Journal of Symbolic Logic 42 (2):179-190.
The Axiom of Choice.Thomas Jech - 1973 - North-Holland.

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