The Hahn-Banach Property and the Axiom of Choice

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


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



    Upload a copy of this work     Papers currently archived: 92,369

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Some consequences of Rado’s selection lemma.Marianne Morillon - 2012 - Archive for Mathematical Logic 51 (7-8):739-749.
James sequences and Dependent Choices.Marianne Morillon - 2005 - Mathematical Logic Quarterly 51 (2):171-186.
Dependent Choices and Weak Compactness.Christian Delhommé & Marianne Morillon - 1999 - Notre Dame Journal of Formal Logic 40 (4):568-573.
Extending Independent Sets to Bases and the Axiom of Choice.Kyriakos Keremedis - 1998 - Mathematical Logic Quarterly 44 (1):92-98.
Disasters in topology without the axiom of choice.Kyriakos Keremedis - 2001 - Archive for Mathematical Logic 40 (8):569-580.
On generic extensions without the axiom of choice.G. P. Monro - 1983 - Journal of Symbolic Logic 48 (1):39-52.
Borel complexity and computability of the Hahn–Banach Theorem.Vasco Brattka - 2008 - Archive for Mathematical Logic 46 (7-8):547-564.
The axiom of choice.John L. Bell - 2008 - Stanford Encyclopedia of Philosophy.
Direct Proofs of Lindenbaum Conditionals.René Gazzari - 2014 - Logica Universalis 8 (3-4):321-343.


Added to PP

35 (#459,945)

6 months
5 (#648,618)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Three-space type Hahn-Banach properties.Marianne Morillon - 2017 - Mathematical Logic Quarterly 63 (5):320-333.

Add more citations

References found in this work

The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
Definability of measures and ultrafilters.David Pincus & Robert M. Solovay - 1977 - Journal of Symbolic Logic 42 (2):179-190.

Add more references