Trees and branches in Banach spaces

Odell, E.; Schlumprecht, Th.

doi:10.1090/S0002-9947-02-02984-7

Trees and branches in Banach spaces
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by E. Odell and Th. Schlumprecht PDF

Trans. Amer. Math. Soc. 354 (2002), 4085-4108 Request permission

Abstract:

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal {T}$ of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal {T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell _p$, then for all $\varepsilon >0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell _p$ sum of finite dimensional spaces.

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