{
  "id": "1604.07875",
  "version": "v1",
  "published": "2016-04-26T22:21:45.000Z",
  "updated": "2016-04-26T22:21:45.000Z",
  "title": "The Szlenk index of $C(K,X)$",
  "authors": [
    "Ryan M Causey"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Given any Banach space $X$ and any $w^*$-compact subset $K$ of $X^*$, we compute the Szlenk index of the $w^*$-closed, convex hull of $K$ as a function of the Szlenk index of $K$. As a consequence, for any compact, Hausdorff topological space $K$ and any Banach space $X$, we compute the the Szlenk index of $C(K,X)$ as a function of the Szlenk index of $X$ and the Cantor-Bendixson index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product $X\\hat{\\otimes}_\\epsilon Y$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\\ell_1$ or $c_0$ index of a Banach space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-26T22:21:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "szlenk index",
      "banach space",
      "compact subset",
      "convex hull",
      "cantor-bendixson index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407875C"
    }
  }
}