{
  "id": "1504.04202",
  "version": "v1",
  "published": "2015-04-16T12:13:39.000Z",
  "updated": "2015-04-16T12:13:39.000Z",
  "title": "The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces",
  "authors": [
    "S. Gabriyelyan",
    "J. Kakol",
    "G. Plebanek"
  ],
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\\mathbb{R}$-space, hence any $k$-space, is Ascoli. Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_\\mathbb{R}$-space iff $X$ is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of $\\ell_1$, we show that the following assertions are equivalent for a Banach space $E$: (i) $E$ does not contain isomorphic copy of $\\ell_1$, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_\\mathbb{R}$-space, (iv) $B_{w}$ is an Ascoli space. We prove also that a Fr\\'{e}chet lcs $F$ does not contain isomorphic copy of $\\ell_1$ iff each closed and convex bounded subset of $F$ is Ascoli in the weak topology. However we show that a Banach space $E$ in the weak topology is Ascoli iff $E$ is finite-dimensional. We supplement the last result by showing that a Fr\\'{e}chet lcs $F$ which is a quojection is Ascoli in the weak topology iff either $F$ is finite dimensional or $F$ is isomorphic to the product $\\mathbb{K}^{\\mathbb{N}}$, where $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{C}\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-16T12:13:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A04",
      "46B03",
      "54C30"
    ],
    "keywords": [
      "weak topology",
      "function spaces",
      "ascoli property",
      "fréchet spaces",
      "contain isomorphic copy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150404202G"
    }
  }
}