Trees and branches in Banach spaces Academic Article uri icon

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 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 (Ei) so that all normalized sequences in X which are almost a skipped blocking of (Ei) 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 < ∞ and C < ∞ every weakly null tree T on the sphere of X has a branch C-equivalent to the unit vector basis of lp, then for all ε > 0, there exists a subspace of X having finite codimension which C2 + ε embeds into the lp sum of finite dimensional spaces.

altmetric score

  • 3

author list (cited authors)

  • Odell, E., & Schlumprecht, T. h.

citation count

  • 27

publication date

  • May 2002