An infinite Ramsey theorem and some Banach-space dichotomies

W. T. Gowers

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An infinite Ramsey theorem and some Banach-space dichotomies

W. T. Gowers

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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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W. T. Gowers. 2005-01-07. An infinite Ramsey theorem and some Banach-space dichotomies. https://arxiv.org/abs/math/0501105

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