[Banach] Abstract of a paper by Marianne Morillon

This is an announcement for the paper "Countable choice and compactness" by Marianne Morillon.

Abstract: We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.

Archive classification: math.FA math.GN math.LO

Mathematics Subject Classification: 03E25, 46B26, 54D30

The source file(s), figure.tex: 548 bytes final.bbl: 2612 bytes final.tex: 55144 bytes icone-ermit.eps: 24310 bytes The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0803.3131

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http://arXiv.org/abs/0803.3131

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