Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-20T03:32:19.899Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological Completeness of Order Intervals in Riesz Spaces

Published online by Cambridge University Press:  20 November 2018

P. G. Dodds
P. G. Dodds*
Affiliation:
School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, 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.

It is shown that if L is a Dedekind complete Riesz space equipped with a locally solid topology T defined by strongly (A, 0) Riesz pseudonorms, then order intervals of L are T-complete. This is an extension of a well known theorem of Nakano. The second part of the paper gives a necessary and sufficient condition for topological completeness of order intervals in a Dedekind σ-complete Riesz space which has a weak order unit and which is equipped with a locally solid σ-Fatou topology.

Keywords

46A4046A05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 3 , 01 September 1984 , pp. 316 - 323
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Aliprantis, C. D. and Burkinshaw, O., Locally solid Riesz spaces (Academic Press, New York, 1978).Google Scholar
2. Fremlin, D. H., Topological Riesz spaces and measure theory (Cambridge University Press, 1974).Google Scholar
3. Kluvánek, Igor and Knowles, Greg, Vector Measures and Control Systems (North Holland Mathematics Studies 20, Amsterdam, 1975).Google Scholar
4. Luxemburg, W. A. J. and Zaanen, A. C., Notes on Banach function spaces, Note III, Nederl. Akad. Wetensch A66 (1963) 239–250.Google Scholar
5. Luxemburg, W. A. J., Notes on Banach function spaces, Note XVI A, Nederl. Akad. Wetensch. A68 (1965) 646–657.Google Scholar
6. Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I, Amsterdam (North Holland, 1977).Google Scholar