Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-01T01:02:28.506Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of normed M-spaces

Published online by Cambridge University Press:  18 May 2009

Garfield C. Schmidt
Garfield C. Schmidt
Affiliation:
Mathemetics Department, University of Lowell Lowell, MA 01854, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

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.

Linear spaces on which both an order and a topology are defined and related in various ways have been studied for some time now. Given an order on a linear space it is sometimes possible to define a useful topology using the order and linear structure. In this note we focus on a special type of space called a linear lattice and determine those lattice properties which are both necessary and sufficient for the existence of a classical norm, called an M-norm, for the lattice. This result is a small step in a program to determine which intrinsic order properties of an ordered linear space are necessary and sufficient for the existence of various given types of topologies for the space. This study parallels, in a certain sense, the study of purely topological spaces to determine intrinsic properties of a topology which make it metrizable and the study of the relation between order and topology on spaces which have no algebraic structure, or. algebraic structures other than a linear one.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 89 - 92
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Birkhoff, G., Lattice theory, (Amer. Math. Soc. Colloquium Publication, Vol. 25, 1967).Google Scholar
2.Day, M. M., Normed linear spaces, 2nd ed., (Springer-Verlag, 1962).CrossRefGoogle Scholar
3.Jameson, G. J. O., Ordered linear spaces, (Springer-Verlag, 1970).CrossRefGoogle Scholar
4.Kakutani, S., Concrete representations of abstract (M)-spaces, Ann. of Math., 42 (1941), 994–1024.CrossRefGoogle Scholar
5.Kelley, J. and Namioka, I., Linear topological spaces (D. Van Nostrand Co., 1963).CrossRefGoogle Scholar