Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T17:58:08.582Z Has data issue: false hasContentIssue false

In what spaces is every closed normal cone regular?

Published online by Cambridge University Press:  20 January 2009

C. W. McArthur
Affiliation:
The Florida State University, Tallahassee, Florida
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.

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1970

References

REFERENCES

(1) Banach, S., Théorie des opérations linéaires, Warszawa, 1933.Google Scholar
(2) Bessaga, C. and Pelczyński, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151164.Google Scholar
(3) Bessaga, C. and Pelczyński, A., A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165174.Google Scholar
(4) Day, M. M., Normed Linear Spaces (Springer-Verlag, Berlin-Gottingen- Heidelberg, 1962).Google Scholar
(5) Dubinsky, E. and Retherford, J. R., Schauder bases and Köthe sequence spaces, Trans. Amer. Math. Soc. 130 (1968), 265280.Google Scholar
(6) James, R. C., Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518527.Google Scholar
(7) Karijn, S., Bases in Banach spaces, Duke Math. J. 15 (1948), 971985.Google Scholar
(8) Karijn, S., Positive operators, J. Math. Mech. 8 (1959), 907935.Google Scholar
(9) Krasnosel'Skiĭ, M. A., Positive solutions of operator equations (Noordhoff, Groningen, 1964).Google Scholar
(10) Lozanovskiĭ, G. JA., Funkcional. Anal, i Priložen. 1 (1967), no. 3, 92. MR 36 # 3110.Google Scholar
(11) Mcarthur, C. W., Convergence of monotone nets in ordered topological vector spaces, Studia Math., 34 (1970), 116.Google Scholar
(12) Mcarthur, C. W. and Retherford, J. R., Some applications of an inequality in locally convex spaces, Trans. Amer. Math. Soc. 137 (1969), 115123.Google Scholar
(13) Peressini, A. L., Ordered topological vector spaces (Harper & Row, New York and London, 1967).Google Scholar
(14) Pelczyński, A., A connection between weakly unconditional convergence and weakly completeness of Banach spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 6 (1958), 251253.Google Scholar
(15) Tzafriri, L., Reflexivity in Banach lattices and their subspaces, to appear.Google Scholar
(16) Weill, L. J., Unconditional bases in locally convex spaces (Dissertation, Florida State University, 1966).Google Scholar