Trees and Branches in Banach Spaces

Details Trees and Branches in Banach Spaces Trees and Branches in Banach Spaces

2000-02-25

arxiv.org/abs/math/0002219v1

Mathematics

Functional Analysis

LaTeX, 24pp

Scientific paper

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 $10$, there exists a finite codimensional subspace of X which $C^2+\epsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.

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