Marianne Morillon

The Hahn-Banach Property and the Axiom of Choice

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 Ekelands 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.

Three generators for minimal writing-space computations

Wen construct, for each integer n ,n three functions from {0,1} n to {0,1} such that any boolean mapping fromn {0,1} n to {0,1} n can be computed with a finite sequence of assignationsn only using the n input variables and those three functions.

Sequential computation of linear Boolean mappings

We prove that any linear Boolean mapping of dimension n can be computed with a double sequence of linear assignments of the n variables. The proof is effective and gives a decomposition of Boolean matrices and directed graphs.

Quadratic Sequential Computations of Boolean Mappings

AbstractnThis paper proposes a constructive proof thatnany mapping on n boolean variables can bencomputed by a straight-line program made up of n2nassignments of the n input variables.nn

Some consequences of Rado's selection lemma

We prove in set theory without the Axiom of Choice, that Rado’s selection lemma (

Uniform Smoothness Entails Hahn-Banach

{mathbf{RL}}

Helly Spaces and Radon Measures on Complete Lines

) implies the Hahn-Banach axiom. We also prove that