{
  "id": "1905.12455",
  "version": "v1",
  "published": "2019-05-29T13:44:06.000Z",
  "updated": "2019-05-29T13:44:06.000Z",
  "title": "A $ξ$-weak Grothendieck compactness principle",
  "authors": [
    "Kevin Beanland",
    "R. M. Causey"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For $0\\leqslant \\xi\\leqslant \\omega_1$, we define the notion of $\\xi$-weakly precompact and $\\xi$-weakly compact sets in Banach spaces and prove that a set is $\\xi$-weakly precompact if and only if its weak closure is $\\xi$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\\leqslant \\xi\\leqslant \\omega_1$, we prove that a Banach space $X$ has the $\\xi$-Schur property if and only if every $\\xi$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-29T13:44:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak grothendieck compactness principle",
      "weakly compact set",
      "banach space",
      "weakly precompact",
      "grothendiecks compactness principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}