An infinite Ramsey theorem and some Banach-space dichotomies

2005/01/07 by W. T. Gowers · 1 citation
Mathematics · #math.FA #msc:46B20 #msc:03E02 #msc:03E15 #msc:05D10 #msc:46B03

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published as Ann. of Math. (2), Vol. 156 (2002), no. 3, 797--833 · 37 pages, published version

arxiv created 2005/01/07 · arxiv updated 2009/12/01

Abstract

A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and Tomczak-Jaegermann, this gives a positive answer to Banach's problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.

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