{
  "id": "1812.04029",
  "version": "v1",
  "published": "2018-12-10T19:08:35.000Z",
  "updated": "2018-12-10T19:08:35.000Z",
  "title": "Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert space",
  "authors": [
    "David Buhagiar",
    "Emmanuel Chetcuti",
    "Hans Weber"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair $(S,\\el)$, where $S$ is a pre-Hilbert space and $\\el$ is an orthocomplemented poset of orthogonally closed linear subspaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\\in\\el$ and $F$ is a finite dimensional linear subspace of $S$, then $M+F\\in \\el$). We study the order topology $\\tau_o(\\el)$ on $\\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological space $(\\el,\\tau_o(\\el))$. It will be seen that the remarkable lack of a proper probability-theory on pre-Hilbert space logics -- for an incomplete $S$ -- comes out elementarily from this topological characterization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-10T19:08:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "order topology",
      "orthocomplemented poset",
      "pre-hilbert space logic",
      "finite dimensional linear subspace",
      "finite dimensional perturbations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}