Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-28T17:06:56.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results involving weak compactness in C(X), C(υX) and C(X)′

Published online by Cambridge University Press:  20 January 2009

I. Tweddle
I. Tweddle
Affiliation:
University of Stirling
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.

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 221 - 229
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Buchwalter, H., Sur le théorème de Nachbin-Shirota, J. Math, pures et appl. 51 (1972), 399–418.Google Scholar
(2) Buchwalter, H. and Schmets, J., Sur quelques propriétés de l'espace C3(T), J. Math, pures et appl. 52 (1973), 337–352.Google Scholar
(3) De Wilde, M. and Schmets, J., Caractérisation des espaces C(X) ultrabornologiques, Bull. Soc. Roy. Sc. Liège, 40e année, 3–4 (1971), 119–121.Google Scholar
(4) Gillman, L. and Jerison, M., Rings of Continuous Functions (Van Nostrand Reinhold Company, New York, 1960).Google Scholar
(5) Kalton, N. J., Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc. 70 (1971), 401–408.CrossRefGoogle Scholar
(6) Kirk, R. B., Topologies on spaces of Baire measures, Bull. Amer. Math. Soc. 79 (1973), 542–545.CrossRefGoogle Scholar
(7) KÖthe, G., Topological Vector Spaces I (Springer-Verlag, Berlin, 1969).Google Scholar
(8) Nachbin, L., Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 471–474.Google Scholar
(9) Pryce, J. D., A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. (3) 23 (1971), 532–546.Google Scholar
(10) Robertson, A. P. and Robertson, W. J., Topological Vector Spaces (Cambridge University Press, Cambridge, 1963).Google Scholar
(11) Shirota, T., On locally convex vector spaces of continuous functions, Proc. Japan Acad. 30 (1954), 294–298.Google Scholar
(12) Tweddle, I., Weak compactness in locally convex spaces, Glasgow Math. J. 9(1968), 123–127.CrossRefGoogle Scholar
(13) Tweddle, I., Vector-valued measures, Proc. London Math. Soc. (3) 20 (1970), 469–489.CrossRefGoogle Scholar
(14) Tweddle, I., Unconditional convergence and vector-valued measures, J. London Math. Soc. (2) 2 (1970), 603–610.Google Scholar
(15) Warner, S., The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265–282.Google Scholar
(16) Webb, J. H., Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341–364.CrossRefGoogle Scholar
(17) Constanttnescu, C., Smulian-Eberlein spaces, Comm. Math. Helv. 48 (1973), 254–317.CrossRefGoogle Scholar
(18) Kirk, R. B., Complete topologies on spaces of Baire measure, Trans. Amer. Math. Soc. 184 (1973), 1–29.CrossRefGoogle Scholar