{
  "id": "1604.04005",
  "version": "v1",
  "published": "2016-04-14T00:48:03.000Z",
  "updated": "2016-04-14T00:48:03.000Z",
  "title": "Free topological vector spaces",
  "authors": [
    "Saak S. Gabriyelyan",
    "Sidney A. Morris"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "We define and study the free topological vector space $\\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\\mathbb{V}(X)$ is a $k_\\omega$-space if and only if $X$ is a $k_\\omega$-space. If $X$ is infinite, then $\\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\\mathbb{V}(\\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable. It is proved that $\\mathbb{V}(X)$ is a barrelled topological vector space if and only if $X$ is discrete. This result is applied to free locally convex spaces $L(X)$ over a Tychonoff space $X$ by showing that: (1) $L(X)$ is quasibarrelled if and only if $L(X)$ is barrelled if and only if $X$ is discrete, and (2) $L(X)$ is a Baire space if and only if $X$ is finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-14T00:48:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "54A25",
      "54D50"
    ],
    "keywords": [
      "free topological vector space",
      "tychonoff space",
      "free locally convex spaces",
      "barrelled topological vector space",
      "baire space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160404005G"
    }
  }
}