{
  "id": "1412.1748",
  "version": "v1",
  "published": "2014-12-04T18:02:53.000Z",
  "updated": "2014-12-04T18:02:53.000Z",
  "title": "Networks for the weak topology of Banach and Fréchet spaces",
  "authors": [
    "S. Gabriyelyan",
    "J. Kcakol",
    "W. Kubiś",
    "W. Marciszewski"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We start the systematic study of Fr\\'{e}chet spaces which are $\\aleph$-spaces in the weak topology. A topological space $X$ is an $\\aleph_0$-space or an $\\aleph$-space if $X$ has a countable $k$-network or a $\\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr\\'echet lcs, then $E$ endowed with its weak topology $\\sigma(E,E')$ is an $\\aleph$-space if and only if $(E,\\sigma(E,E'))$ is an $\\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr\\'echet lcs to be an $\\aleph$-space in the weak topology. We prove that a reflexive Fr\\'echet lcs $E$ in the weak topology $\\sigma(E,E')$ is an $\\aleph$-space if and only if $(E,\\sigma(E,E'))$ is an $\\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\\ell_{1}(\\mathbb{R})$ with the weak topology is an $\\aleph$-space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-04T18:02:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "54H11",
      "22A05",
      "54C35"
    ],
    "keywords": [
      "weak topology",
      "fréchet spaces",
      "extend corsons result",
      "nonseparable banach space",
      "reflexive frechet lcs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1748G"
    }
  }
}