{
  "id": "2408.11944",
  "version": "v1",
  "published": "2024-08-21T18:57:28.000Z",
  "updated": "2024-08-21T18:57:28.000Z",
  "title": "On the completeness of the space $\\mathcal{O}_C$",
  "authors": [
    "Michael Kunzinger",
    "Norbert Ortner"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We give a new proof of the completeness of the space $\\mathcal{O}_C$ by applying a criterion of compact regularity for the isomorphic sequence space $\\lim_{k\\rightarrow} (s\\hat \\otimes (\\ell^\\infty)_{-k})$. Along the way we show that the strong dual of any quasinormable Fr\\'echet space is a compactly regular $\\mathcal{LB}$-space. Finally, we prove that $\\lim_{k\\rightarrow}(E_k\\hat \\otimes_\\iota F) = (\\lim_{k\\rightarrow} E_k) \\hat \\otimes_\\iota F$ if the inductive limit $\\lim_{k \\rightarrow}(E_k \\hat \\otimes_\\iota F)$ is compactly regular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T18:57:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A04",
      "46A08",
      "46A13",
      "46F05",
      "46F10"
    ],
    "keywords": [
      "completeness",
      "isomorphic sequence space",
      "compactly regular",
      "compact regularity",
      "strong dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}