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Function spaces and the Mosco topology

Published online by Cambridge University Press:  17 April 2009

Gerald Beer  and
Robert Tamaki
Gerald Beer
Affiliation:
Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, United States of America
Robert Tamaki
Affiliation:
Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, United States of America
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Abstract

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Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 1 , August 1990 , pp. 7 - 19
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Attouch, H., Variational Convergence for Functions and Operators (Pitman Publishers, Boston, 1984).Google Scholar
[2]Beer, G., ‘Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance’, Proc. Amer. Math. Soc. 95 (1985), 653–658.CrossRefGoogle Scholar
[3]Beer, G., ‘On Mosco convergence of convex sets’, Bull. Austral. Math. Soc. 38 (1988), 239–253.CrossRefGoogle Scholar
[4]Beer, G., ‘On the Young-Fenchel transform for convex functions’, Proc. Amer. Math. Soc. 104 (1988), 1115–1123.CrossRefGoogle Scholar
[5]Beer, G., ‘Convergence of continuous linear functionals and their level sets‘, Archiv. Math. (Basel) 52 (1989), 482–491.CrossRefGoogle Scholar
[6]Beer, G. and Browein, J., ‘Mosco convergence and reflexivity’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[7]Beer, G. and Pai, D., ‘On convergence of convex sets and relative Chebyshev centers’, J. Approx. Theory (to appear).Google Scholar
[8]Beer, G. and Pai, D., ‘The Prox map’, J. Math. Anal. Appl. (to appear).Google Scholar
[9]Borwein, J. and Fitzpatrick, S., ‘Mosco convergence and the Kadec property’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[10]Dolecki, S., Salinetti, S. and Wets, R., ‘Convergence of functions: equi-semicontinuity’, Trans. Amer. Math. Soc. 276 (1983), 409–429.CrossRefGoogle Scholar
[11]Dugundji, J., Topology (Allyn and Bacon, Boston, 1966).Google Scholar
[12]Dunford, N. and Schwartz, J., Linear Operators, Part 1 (Wiley-Interscience, New York, 1957).Google Scholar
[13]Effros, E., ‘Convergence of closed subsets in a topological space’, Proc. Amer. Math. Soc. 16 (1963), 929–931.CrossRefGoogle Scholar
[14]Fell, J., ‘A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space‘, Proc. Amer. Math. Soc. 13 (1962), 472–476.CrossRefGoogle Scholar
[15]Hola, L., ‘Most of the optimization problems have unique solution’, C.R. Acad. Bulgare des Science (to appear).Google Scholar
[16]Klein, E. and Thompson, A., Theory of Correspondences (Wiley-Interscience, New York, 1984).Google Scholar
[17]Mosco, U., ‘Convergence of convex sets and solutions of variational inequalities’, Adv. in Math. 3 (1969), 510–585.CrossRefGoogle Scholar
[18]Mosco, U., ‘On the continuity of the Young-Fenchel transform’, J. Math. Anal. Appl. 35 (1971), 518–535.CrossRefGoogle Scholar
[19]Mrowka, S., ‘On the convergence of nets of sets’, Fund. Math. 45 (1958), 416–428.CrossRefGoogle Scholar
[20]McCoy, R. and Ntantu, I., ‘Topological Properties of Spaces of Continuous Functions’: Lecture Notes in Mathematics 1315 (Springer-Verlag, Berlin, Heidelberg, New York 1988).Google Scholar
[21]Naimpally, S., ‘Graph topology for function spaces’, Trans. Amer. Math. Soc. 123 (1966), 267–272.CrossRefGoogle Scholar
[22]Poppe, H., ‘Über Graphentopologien für Abbildungsraüme I’, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 15 (1967), 71–80.Google Scholar
[23]Poppe, H., ‘Einige Bemerkungen über den Raum der abgeschlossenen Mengen’, Fund. Math. 59 (1966), 159–169.CrossRefGoogle Scholar
[24]Rockafellar, R.T. and Wets, R., ‘Variational systems, an introduction‘, in Multifunction and integrands: Lecture notes in mathematics 1091, Editor Salinetti, G. (Springer-Verlag, Berlin, Heidelberg, New York, 1984).Google Scholar
[25]Sonntag, Y., ‘Convergence au sens de Mosco; théorie et applications à l'approximation des solutions d'inéquations’, (Thése d'Etat. Université de Provence, Marseille, 1982).Google Scholar
[26]Willard, S., General Topology (Addison-Wesley, Reading, Mass., 1968).Google Scholar