Banach spaces without minimal subspaces

Details Banach spaces without minimal subspaces Banach spaces without minimal subspaces

2007-11-08

arxiv.org/abs/0711.1350v2

Journal of Functional Analysis, Volume 257, Issue 1, 1 July 2009, Pages 149-193

Mathematics

Functional Analysis

Scientific paper

We prove three new dichotomies for Banach spaces \`a la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size $\aleph_1$ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.

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