{
  "id": "1908.09115",
  "version": "v1",
  "published": "2019-08-24T09:41:33.000Z",
  "updated": "2019-08-24T09:41:33.000Z",
  "title": "On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence",
  "authors": [
    "Taras Banakh"
  ],
  "comment": "7 pages",
  "journal": "in: General Topology in Banach Spaces (T.Banakh ed.), Nova Sci. Publ., NY, (2001), 171--178",
  "categories": [
    "math.GN",
    "math.FA"
  ],
  "abstract": "In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$, $(Y,weak)$ are sequentially homeomorphic if and only if $\\mathcal W(X)=\\mathcal W(Y)$, where $\\mathcal W(X)$ is the class of topological spaces homeomorphic to closed bounded subsets of $(X,weak)$. Moreover, if $X,Y$ are Banach spaces which are isomorphic to their hyperplanes and have separale duals, then the spaces $(X,weak)$ and $(Y,weak)$ are sequentially homeomorphic if and only of the spaces $(X,c)$ and $(Y,c)$ are homeomorphic. To prove this result, we show that for a normed space $X$ which is isomorphic to its hyperpane and has separable dual, the space $(X,c)$ (resp. $(X,weak)$) is (sequentially) homeomorphic to the product $B\\times\\mathbb R^\\infty$ of the weak unit ball $B$ of $X$ and the linear space $\\mathbb R^\\infty$ with countable Hamel basis and the strongest linear topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-24T09:41:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N17",
      "57N20",
      "46A20",
      "46A19"
    ],
    "keywords": [
      "normed space",
      "weak topology",
      "compact convergence",
      "topological classification",
      "uniform convergence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}