Hostname: page-component-6bf8c574d5-zc66z Total loading time: 0 Render date: 2025-03-04T14:49:09.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Barrelled spaces and dense vector subspaces

Published online by Cambridge University Press:  17 April 2009

W.J. Robertson ,
S.A. Saxon  and
A.P. Robertson
W.J. Robertson
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, Western Australia 6009
S.A. Saxon
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, U.S.A.
A.P. Robertson
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6150, 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.

This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 3 , June 1988 , pp. 383 - 388
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Eberhardt, V. und Roelcke, W., ‘Über einen Grapliensatz für Abbildungen mit metrisierbarein Zeilraum’, Manuscripta Math 13 (1974), 53–68.CrossRefGoogle Scholar
[2]Horvéth, J., Topological Vector Spaces and Distributions, 1 (Addison-Wesley, 1966).Google Scholar
[3]Komura, Y., ‘On linear topological spaces’, Kumamoto J. Sci. Ser. A 5 (1962), 148–157.Google Scholar
[4]Köthe, C., Topological Vector Spaces 1 (Springer-Verlag, 1969).Google Scholar
[5]Robertson, A.P. and Robertson, W.J., Topological Vector Spaces (Cambridge University Press, 1964).Google Scholar
[6]Robertson, W.J., Twedclle, I. and Yeomans, F.E., ‘On the stability of barrelled topologies III’, Bull. Austrul. Math. Soc. 22 (1980), 99–112.CrossRefGoogle Scholar
[7]Robertson, W.J. and Yeomans, F.E., ‘On the stability of barrelled topologies I’, Bull. Austral. Math. Soc. 20 (1979), 385–395.CrossRefGoogle Scholar
[8]Saxon, S. and Levin, M., ‘Every countable-codimensional subspace of a barrelled space is barrelled’, Proc. Amer. Math. Soc. 29 (1971), 91–96.CrossRefGoogle Scholar
[9]Tsirulnikov, B., ‘On conservation of barrelledness properties in locally convex spaces’, Bull. Soc. Roy. Sci. Liège 49 (1980), 5–25.Google Scholar