Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T07:28:07.406Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Pták homomorphism theorem

Part of: Topological linear spaces and related structures General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

B. Rodrigues
B. Rodrigues
Affiliation:
Department of Mathematical Sciences, Loyola UniversityNew Orleans, Louisiana 70118, U.S.A.
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.

In this note, a brief and accessible proof is given of an extension of the Pták homomorphism theorem to a larger class of spaces—spaces that are not necessarily assumed to be locally convex. This is done by first proving a counterpart of the Bourbaki-Grothendieck homomophism theorem for the non-locally-convex case. Our presentation utilizes the simplifying properties of seminorms.

Keywords

semi-barrelled spacebarrelled spacesemi-B-completeB-completequotient seminormadequate mapsmall adjointsemi-openweakly opennearly semi-opennearly semi-continuous.

MSC classification

Secondary: 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness) 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 322 - 333
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bourbaki, N., Topological vector spaces, (Springer-Verlag, New York-Heidelberg-Berlin, 1987).CrossRefGoogle Scholar
[2]Edwards, R. E., Functional analysis, theory and applications, (Holt Rinehart and Winston, New York, 1965).Google Scholar
[3]Horváth, J., Locally convex spaces, (Lecture notes in mathematics, 331, Ed. Waelbroeck, L., Springer-Verlag, New York-Heidelberg-Berlin, 1973).Google Scholar
[4]Jarchow, H., Locally convex spaces, (B. G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[5]Köthe, G., ‘General linear transformations of locally convex spaces’, Math. Ann. 159 (1965), 309–328.CrossRefGoogle Scholar
[6]Köthe, G., Topological vector spaces I, (Springer-Verlag, Berlin-Heidelberg-New York, 1969).Google Scholar
[7]Köthe, G., Topological vector spaces II, (Springer-Verlag, New York-Heidelberg-Berlin, 1979).CrossRefGoogle Scholar
[8]Pták, V., ‘Completeness and the open mapping theorem’, Bull. Soc. Math. France 86 (1958), 41–74.CrossRefGoogle Scholar
[9]Robertson, W., ‘Completions of topological vector spaces’, Proc. London Math. Soc. 8 (1958), 242–257.CrossRefGoogle Scholar
[10]Schaefer, H. H., Topological vector spaces, (Springer-Verlag, New York-Heidelberg-Berlin, 1986).Google Scholar
[11]Wilansky, A., Modern methods in topological vector spaces, (McGraw-Hill, New York, 1978).Google Scholar