The Hahn-Banach Property and the Axiom of Choice

We work in the set theory without the axiom of choice: ZF. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gâteauxdifferentiable 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 Gâteaux-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).