Unbounded norm topology in Banach lattices

Creative Commons License

Kandic M., Marabeh M. A. A., Troitsky V. G.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.451, no.1, pp.259-279, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 451 Issue: 1
  • Publication Date: 2017
  • Doi Number: 10.1016/j.jmaa.2017.01.041
  • Journal Name: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.259-279
  • Keywords: Banach lattice, Un-convergence, Uo-convergence, Un-topology, ORDER CONVERGENCE
  • Middle East Technical University Affiliated: Yes

Abstract

A net (x(alpha)) in a Banach lattice X is said to un-converge to a vector x if xl A parallel to vertical bar x(alpha) - x vertical bar boolean AND u parallel to -> 0 for every u is an element of X+. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quaRi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B-x is un-complete. For a Banach lattice X, B-x is un-compact if X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence. (C) 2017 Elsevier Inc. All rights reserved.