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Trees and Branches in Banach Spaces

Edward Odell·Thomas Schlumprecht
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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 $1 0$, there exists a finite codimensional subspace of X which $C^2+ε$ embeds into the $\ell_p$ sum of finite dimensional spaces.

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Edward Odell, Thomas Schlumprecht. 2000-02-25. Trees and Branches in Banach Spaces. https://arxiv.org/abs/math/0002219

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