{
  "id": "1605.03538",
  "version": "v1",
  "published": "2016-05-11T18:18:14.000Z",
  "updated": "2016-05-11T18:18:14.000Z",
  "title": "Unbounded Norm Convergence in Banach Lattices",
  "authors": [
    "Y. Deng",
    "M. O'Brien",
    "V. G. Troitsky"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "A net $(x_\\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x \\in X$ if $\\lvert x_\\alpha - x\\rvert \\wedge u$ converges to $0$ in order for all $u\\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\\lVert\\lvert x_\\alpha - x\\rvert \\wedge u\\rVert\\to 0$ for all $u\\in X_+$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-11T18:18:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B42",
      "46A40"
    ],
    "keywords": [
      "banach lattice",
      "unbounded norm convergence",
      "general vector lattices",
      "unbounded order convergent",
      "unbounded norm convergent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}