{
  "id": "math/0002219",
  "version": "v1",
  "published": "2000-02-25T19:17:02.000Z",
  "updated": "2000-02-25T19:17:02.000Z",
  "title": "Trees and Branches in Banach Spaces",
  "authors": [
    "Edward Odell",
    "Thomas Schlumprecht"
  ],
  "comment": "LaTeX, 24pp",
  "categories": [
    "math.FA"
  ],
  "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 $\\epsilon>0$, there exists a finite codimensional subspace of X which $C^2+\\epsilon$ embeds into the $\\ell_p$ sum of finite dimensional spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-02-25T19:17:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B20"
    ],
    "keywords": [
      "banach space",
      "infinite countably branching tree",
      "finite codimensional subspace",
      "finite dimensional spaces",
      "unit vector basis"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......2219O"
    }
  }
}