On the hereditary proximity to $\ell_1$

Details On the hereditary proximity to $\ell_1$ On the hereditary proximity to $\ell_1$

2009-07-24

arxiv.org/abs/0907.4317v2

Mathematics

Functional Analysis

40 pages, submitted for publication, linguistic corrections

Scientific paper

In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to \ell_1. The second part is devoted to a complete separation of the hereditary local proximity to \ell_1 from the asymptotic one. More precisely for every countable ordinal \xi we construct a separable reflexive space \mathfrak{X}_\xi such that every infinite dimensional subspace of it has Bourgain \ell_1-index greater than \omega^\xi and the space itself has no \ell_1-spreading model. We also present a reflexive HI space admitting no \ell_p as a spreading model.

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