Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T04:45:39.562Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm and order properties of Banach lattices

Published online by Cambridge University Press:  09 April 2009

Susan E. Bedingfield  and
Andrew Wirth
Susan E. Bedingfield
Affiliation:
Department of Mathematics University of MelbourneParkvile, Victoria 3052, Australia
Andrew Wirth
Affiliation:
Prahran College of Advanced EducationPrahran, Victoria 3181, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The interrelationships between norm convergence and two forms of convergence defined in terms of order, namely order and relative uniform convergence are considered. The implications between conditions such as uniform convexity, uniform strictness, uniform monotonicity and others are proved. In particular it is shown that a σ-order continuous, σ-order complete Banach lattice is order continuous.

1980 Mathematics subject classification (Amer. Math. Soc.): 46 A 40.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 3 , May 1980 , pp. 331 - 336
Copyright
Copyright © Australian Mathematical Society 1980

References

Berens, H. and Lorentz, G. G. (1973), ‘Korovkin theorems for Banach lattices’, in Approximation theory (editor Lorentz, G. G.) (Academic Press, New York).Google Scholar
Birkhoff, G. (1967), Lattice theory (American Mathematical Society, Providence, Rhode Island), pp. 366–373.Google Scholar
Clarkson, J. A. (1936), ‘Uniform convex spaces’, Trans. Amer. Math. Soc. 40, 396–414.CrossRefGoogle Scholar
Lacey, H. E. (1974), The isometric theory of classical Banach spaces (Springer-Verlag, New York).CrossRefGoogle Scholar
Leader, S. (1970), ‘Sequential convergence in lattice groups’, in Problems in analysis (editor Gunning, R. C.) (University Press, Princeton, New Jersey), pp. 273–290.Google Scholar
Luxemburg, W. A. J. and Zaanen, A. C. (1971), Riesz spaces 1 (North Holland, Amsterdam).Google Scholar
Lotz, H. P. (1974), ‘Minimal and reflexive Banach lattices’, Math. Ann. 209, 117–126.CrossRefGoogle Scholar
Nagel, R. J. (1973), ‘Ordnungsstetigkeit in Banachverbanden’, Manuscripta Mathematica 9, 9–27.CrossRefGoogle Scholar
Vulikh, B. Z. (1967), Introduction to the theory of partially ordered spaces (Wolters-Noordhoff, Groningen), p. 67.Google Scholar
Wirth, A. (1975), ‘Relatively uniform Banach lattices’, Proc. Amer. Math. Soc. 52, 178–180.CrossRefGoogle Scholar