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Extending Edgar's ordering to locally convex spaces

Published online by Cambridge University Press:  18 May 2009

Neill Robertson
Neill Robertson
Affiliation:
Department of Mathematics, University of Cape Town, Rondebosch 7700, South Africa
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By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 2 , May 1992 , pp. 175 - 188
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Bourgin, R. D., Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics 993 (Springer-Verlag, 1983).CrossRefGoogle Scholar
2.Buchwalter, H., Espaces localement convex semi-faibles, Ileme Coll. Anal. Fonct.Bordeaux 1973, Pub. Dep. Math. Lyon 10 (1973).Google Scholar
3.Dierolf, P., Theorems of the Orlicz-Pettis type for locally convex spaces, Manuscripta Math. 20 (1977), 73–94.CrossRefGoogle Scholar
4.Dierolf, P. and Dierolf, S., Topological properties of the dual pair <ℬ(Ω)′ ℬ(Ω8)″), Pacific J. Math. 108 (1983, 51–82).CrossRefGoogle Scholar
5.Diestel, J., Sequences and series in Banach spaces, Graduate Texts in Mathematics 92 (Springer-Verlag, 1984).CrossRefGoogle Scholar
6.Drewnowski, L., Another note on Kalton's theorem, Studia Math. 52 (1975), 233–237.CrossRefGoogle Scholar
7.Edgar, G. A., Measurability in a Banach space II, Indiana Univ. Math. J. 28 (1979), 559–579.CrossRefGoogle Scholar
8.Edgar, G. A., An ordering for the Banach spaces, Pacific J. Math. 108 (1983), 83–98.CrossRefGoogle Scholar
9.Faires, B. T., Varieties and vector measures, Math. Nachr. 85 (1978), 303–314.CrossRefGoogle Scholar
10.Floret, K., Weakly compact sets, Lecture Notes in Mathematics 801 (Springer-Verlag, 1980).CrossRefGoogle Scholar
11.Freniche, F. J., Grothendieck locally convex spaces of continuous vector valued functions, Pacific J. Math. 120 (1985), 345–355.CrossRefGoogle Scholar
12.Freniche, F. J., Some remarks on the average range of a vector measure, Proc. Amer. Math. Soc. 107 (1989), 119–124.CrossRefGoogle Scholar
13.Garnir, H. G., De Wilde, M. and Schmets, J., Analyse Fonctionelle Tome I, (Birkhäuser-Verlag, 1968).Google Scholar
14.Geitz, R. F., Geometry and the Pettis integral, Trans. Amer. Math. Soc. (2) 269 (1982), 535–548.CrossRefGoogle Scholar
15.Godefroy, G. and Talagrand, M., Classes d'espaces de Banach à prédual unique. C.R. Acad. Sci. Paris 292 (1981), 323–325.Google Scholar
16.Hunter, R. J. and Lloyd, J., Weakly compactly generated locally convex spaces, Math. Proc. Cambr. Philos. Soc. 82 (1977), 85–98.CrossRefGoogle Scholar
17.Jarchow, H., Locally convex spaces, (Teubner, 1981).CrossRefGoogle Scholar
18.Kappeler, T., Banach spaces with the condition of Mazur, Math. Z. 191 (1986), 623–631.CrossRefGoogle Scholar
19.Kalton, N. and Wilansky, A., Tauberian operators on Banach spaces, Proc. Amer. Math.Soc. 57 (1976), 251–255.CrossRefGoogle Scholar
20.Lindström, M., A characterization of Schwartz spaces, Math. Z. 198 (1988), 423–430.CrossRefGoogle Scholar
21.Neidinger, R. D., Concepts in the real interpolation of Banach spaces, Functional Analysis: Proceedings of the Seminar at the University of Texas at Austin, 1986–1987 (Odell, E. and Rosenthal, H., eds), 43–53, Lecture Notes in Mathematics 1332 (Springer-Verlag, 1988).CrossRefGoogle Scholar
22.Pfister, H., Bemerkungen zum Satz über die Separabilität der Fréchet-Montel-Räume, Arch. Math. (Basel) 27 (1976), 86–92.CrossRefGoogle Scholar
23.Sentilles, F. D. and Wheeler, R. F., Pettis integration via the Stonian transform, Pacific J. Math. 107 (1983), 473–496.CrossRefGoogle Scholar
24.Talagrand, M., Espaces de Banach faiblement ℌ-analytiques, Ann. Math. 110 (1979), 407–438.CrossRefGoogle Scholar
25.Wilansky, A., Mazur spaces, Intern. J. Math. Math. Sci. 4 (1981), 39–55.CrossRefGoogle Scholar