Concerning the Bourgain $el{l}_1$ index of a Banach space

Abstract

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 $\alpha$.

References in corpus (2)

• Complexity of weakly null sequences
• Banach Spaces Of The Type Of Tsirelson