{
  "id": "1810.10250",
  "version": "v1",
  "published": "2018-10-24T09:00:14.000Z",
  "updated": "2018-10-24T09:00:14.000Z",
  "title": "Locally convex properties of free locally convex spaces",
  "authors": [
    "Saak Gabriyelyan"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\\ell_\\infty$-barrelled, (ii) $L(X)$ is $\\ell_\\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\\ell_\\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such that $L(X)$ is a quasi-$(DF)$-space but is not a $c_0$-quasibarrelled space. For each non-metrizable compact space $K$, the space $L(K)$ is a $(df)$-space but is not a quasi-$(DF)$-space. If $X$ is a $\\mu$-space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford--Pettis property for every Tychonoff space $X$. If $X$ is a sequential $\\mu$-space (for example, metrizable), then $L(X)$ has the sequential Dunford--Pettis property iff $X$ is discrete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-24T09:00:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "46A08",
      "54C35"
    ],
    "keywords": [
      "free locally convex space",
      "locally convex properties",
      "sequential dunford-pettis property",
      "compact space",
      "non-discrete metrizable space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}