{
  "id": "1702.07867",
  "version": "v1",
  "published": "2017-02-25T10:00:39.000Z",
  "updated": "2017-02-25T10:00:39.000Z",
  "title": "Topological properties of strict $(LF)$-spaces and strong duals of Montel strict $(LF)$-spaces",
  "authors": [
    "Saak Gabriyelyan"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1611.02994",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fr\\'{e}chet space or $E=\\phi$. We prove that the strong dual $E'_\\beta$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fr\\'{e}chet--Montel space, so $E'_\\beta$ is a sequential non-Fr\\'{e}chet--Urysohn space, or (ii) $E=\\phi$, so $E'_\\beta= \\mathbb{R}^\\omega$. Consequently, the space $\\mathcal{D}(\\Omega)$ of test functions and the space of distributions $\\mathcal{D}'(\\Omega)$ are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-25T10:00:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A13",
      "46A11",
      "22A05"
    ],
    "keywords": [
      "montel strict",
      "strong dual",
      "topological properties",
      "compact subset",
      "tychonoff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}