Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-30T20:22:58.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally convex topologies and the convex compactness property

Published online by Cambridge University Press:  24 October 2008

E. G. Ostling  and
A. Wilansky
E. G. Ostling
Affiliation:
Hofstra, Lehigh, and Reading Universities
A. Wilansky
Affiliation:
Hofstra, Lehigh, and Reading Universities
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 45 - 50
Copyright
Copyright © Cambridge Philosophical Society 1974

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)Hampson, J. K. and Wilansky, A.Sequences in locally convex spaces. Studia Math. 15 (1973), 221–223.CrossRefGoogle Scholar
(2)Kelley, J. L. and Namioka, I.Linear topological spaces (Van Nostrand, 1963).CrossRefGoogle Scholar
(3)Kötas, G.Topological vector spaces (Springer-Verlag, 1969).Google Scholar
(4)Ostling, E. G. Polar topologies and KS spaces (Ph.D. dissertation, Lehigh University, 1972).Google Scholar
(5)Raimi, R. A.Equicontinuity and compactness in locally convex spaces. Michigan Math. J. 5 (1958), 203–211.CrossRefGoogle Scholar
(6)Robertson, A. P. and Robertson, W. J.Topological vector spaces (Cambridge University Press, 1966).Google Scholar
(7)Webb, J. H.Sequential convergence in locally convex spaces. Proc. Cambridge Philos. Soc. 64 (1968), 341–364.CrossRefGoogle Scholar