An infinite Ramsey theorem and some Banach-space dichotomies

By William T. Gowers. 2002

135 citations recorded by the source corpus.

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 TomczakJaegermann, this gives a positive answer to BanachXs problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/82aa090b54a6f51ea1f738b15c628f69b5f0aaed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Stevo Todorcevic
2. Natasha Dobrinen

In Geometry and Topology

1. Stevo Todorcevic
2. Spiros A. Argyros
3. Pandelis Dodos
4. Natasha Dobrinen
5. Vassilis Kanellopoulos

In Mathematical Physics

1. Pandelis Dodos
2. Vassilis Kanellopoulos
3. Stevo Todorcevic
4. Spiros A. Argyros
5. Natasha Dobrinen

Citations in the corpus

• Unconditional families in Banach spaces (2008, background): “The class of HI spaces stands in the opposite of the class of spaces with an unconditional basis and Gowers’ dichotomy [ 16 ], a Ramsey theoretic principle for Banach spaces, yields that every Banach space either is HI saturated, or contains an unconditional basic sequence.”
• Tukey types of ultrafilters (2011, methodology): “This will of course be based on Gower’s Theorem for FINk and Lopez-Abad’s canonical Ramsey Theorem for FINk (see [1], [21], [11]).”