{
  "id": "2307.09167",
  "version": "v1",
  "published": "2023-07-18T11:43:16.000Z",
  "updated": "2023-07-18T11:43:16.000Z",
  "title": "On linearisation, existence and uniqueness of preduals",
  "authors": [
    "Karsten Kruse"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the problem of existence and uniqueness of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$. Then we turn to the case that $X=\\mathcal{F}(\\Omega)$ is a space of scalar-valued functions on a non-empty set $\\Omega$ and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there is a locally convex Hausdorff space $Y$, a map $\\delta\\colon\\Omega\\to Y$ and a topological isomorphism $T\\colon\\mathcal{F}(\\Omega)\\to Y_{b}'$ such that $T(f)\\circ \\delta= f$ for all $f\\in\\mathcal{F}(\\Omega)$. We also lift such a linearisation to the case of vector-valued functions, covering many previous results on linearisations, and use it to characterise the bornological spaces $\\mathcal{F}(\\Omega)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-18T11:43:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A08",
      "46A20",
      "46A70",
      "46B10",
      "46E10",
      "46E40"
    ],
    "keywords": [
      "uniqueness",
      "bornological locally convex hausdorff space",
      "unique predual",
      "sufficient conditions",
      "non-empty set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}