{
  "id": "1108.0169",
  "version": "v3",
  "published": "2011-07-31T11:57:09.000Z",
  "updated": "2011-12-21T03:05:50.000Z",
  "title": "Continuity of bilinear maps on direct sums of topological vector spaces",
  "authors": [
    "Helge Glockner"
  ],
  "comment": "22 pages, LaTeX; v3: update of references",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on R^n to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T(E) as the locally convex direct sum of the projective tensor powers T^j(E) of E. We show that T(E) is a topological algebra if and only if every sequence of continuous seminorms on E has an upper bound. In particular, if E is metrizable, then T(E) is a topological algebra if and only if E is normable. Also, T(E) is a topological algebra whenever E is a DFS-space, or a hemicompact k-space.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-12-21T03:05:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46M05",
      "42A85",
      "44A35",
      "46A13",
      "46A11",
      "46A16",
      "46E25",
      "46F05",
      "46M40"
    ],
    "keywords": [
      "topological vector spaces",
      "bilinear maps",
      "topological algebra",
      "continuity",
      "locally convex direct sum"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.0169G"
    }
  }
}