Concerning the Bourgain $ell_1$ index of a Banach space

Details Concerning the Bourgain $ell_1$ index of a Banach space Concerning the Bourgain $ell_1$ index of a Banach space

1996-06-05

arxiv.org/abs/math/9606209v1

Mathematics

Functional Analysis

Scientific paper

A well known argument of James yields that if a Banach space $X$ contains $\ell_1^n$'s uniformly then $X$ contains $\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\ell_1$-index of certain Banach spaces. Furthermore we show that the $\ell_1$-index of a separable Banach space not containing $\ell_1$ must be of the form $\omega^{\alpha}$ for some countable ordinal $\alpha$.

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