{
  "id": "2003.06764",
  "version": "v1",
  "published": "2020-03-15T06:12:23.000Z",
  "updated": "2020-03-15T06:12:23.000Z",
  "title": "The Josefson-Nissenzweig property for locally convex spaces",
  "authors": [
    "{Taras Banakh",
    "Saak Gabriyelyan"
  ],
  "comment": "32 pages",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E',\\sigma(E',E))\\to ( E',\\beta^\\ast(E',E))$ is not sequentially continuous. By the classical Josefson--Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. We show that for a Tychonoff space $X$, the function space $C_p(X)$ has the JNP iff there is a weak$^\\ast$ null-sequence $\\{\\mu_n\\}_{n\\in\\omega}$ of finitely supported sign-measures on $X$ with unit norm. However, for every Tychonoff space $X$, neither the space $B_1(X)$ of Baire-1 functions on $X$ nor the free locally convex space $L(X)$ over $X$ has the JNP. We also define two modifications of the JNP, called the $universal$ $JNP$ and the $JNP$ $everywhere$ (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-15T06:12:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03"
    ],
    "keywords": [
      "josefson-nissenzweig property",
      "tychonoff space",
      "function space",
      "free locally convex space",
      "infinite-dimensional banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}