{
  "id": "math/9709217",
  "version": "v1",
  "published": "1997-09-18T00:00:00.000Z",
  "updated": "1997-09-18T00:00:00.000Z",
  "title": "On asymptotic properties of Banach spaces under renormings",
  "authors": [
    "Edward Odell",
    "Thomas Schlumprecht"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "It is shown that a separable Banach space $X$ can be given an equivalent norm $|\\!|\\!|\\cdot |\\!|\\!|$ with the following properties:\\quad If $(x_n)\\subseteq X$ is relatively weakly compact and $\\lim_{m\\to\\infty} \\lim_{n\\to\\infty}\\break |\\!|\\!| x_m + x_n |\\!|\\!| = 2\\lim_{m\\to\\infty} |\\!|\\!| x_m|\\!|\\!|$ then $(x_n)$ converges in norm. This yields a characterization of reflexivity once proposed by V.D.~Milman. In addition it is shown that some spreading model of a sequence in $(X, |\\!|\\!|\\cdot |\\!|\\!|)$ is 1-equivalent to the unit vector basis of $\\ell_1$ (respectively, $c_0$) implies that $X$ contains an isomorph of $\\ell_1$ (respectively, $c_0$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "1997-09-18T00:00:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B45"
    ],
    "keywords": [
      "asymptotic properties",
      "renormings",
      "unit vector basis",
      "equivalent norm",
      "separable banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1997math......9217O"
    }
  }
}