Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T15:19:50.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Ultrabarrelled groups and the closed graph theorem

Published online by Cambridge University Press:  24 October 2008

S. O. Iyahen
S. O. Iyahen
Affiliation:
University of Ibadan, Ibadan, Nigeria
Get access Rights & Permissions [Opens in a new window]

Extract

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 53 - 58
Copyright
Copyright © Cambridge Philosophical Society 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baker, J. W.Topological groups and the closed graph theorem. J. London Math. Soc. 42 (1967), 217–225.CrossRefGoogle Scholar
(2)Iyahen, S. O.On certain classes of linear topological spaces. Proc. London Math. Soc. 18 (1968), 285–307.CrossRefGoogle Scholar
(3)Kelley, J. L.General topology (Van Nostrand: New York, 1955).Google Scholar
(4)Mahowald, M.Barrelled spaces and the closed graph theorem. J. London Math. Soc. 36 (1961), 108–110.CrossRefGoogle Scholar
(5)Pettis, B. J.On continuity and openness of homomorphisms in topological groups. Ann. of Math. 52 (1950), 293–308CrossRefGoogle Scholar
(6)Robertson, A. P. and Robertson, W.On the closed graph theorem. Proc. Glasgow Math. Assoc. 3 (1956), 9–21.CrossRefGoogle Scholar
(7)Robertson, W.Completions of topological vector spaces. Proc. London Math. Soc. 8 (1958), 242–257.CrossRefGoogle Scholar
(8)Varopoulos, N. Th.Studies in harmonic analysis. Proc. Cambridge Philos. Soc. 60 (1964) 465–516.CrossRefGoogle Scholar