An infinite Ramsey theorem and some Banach-space dichotomies

• 成果类型: Article
• 署名作者: Gowers, WT
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3597282
• 发表日期: 2002
• 页码: 797-833
• 关键词: sets; Operators; subspaces

摘要

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.