Notre Dame Journal of Formal Logic

Dependent Choices and Weak Compactness

Christian Delhommé and Marianne Morillon


We work in set theory without the Axiom of Choice ZF. We prove that the Principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact and, in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF and the latter statement does not imply DC. Furthermore, DC does not imply that the closed unit ball of a reflexive space is weakly compact.

Article information

Notre Dame J. Formal Logic, Volume 40, Number 4 (1999), 568-573.

First available in Project Euclid: 30 January 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E35: Consistency and independence results


Delhommé, Christian; Morillon, Marianne. Dependent Choices and Weak Compactness. Notre Dame J. Formal Logic 40 (1999), no. 4, 568--573. doi:10.1305/ndjfl/1012429720.

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