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The Attouch-Wets topology and a characterisation of normable linear spaces

Published online by Cambridge University Press:  17 April 2009

Ľubica Holá
Affiliation:
Department of Probability and Mathematical Statistics, MFF UK, 842 15 Bratislava, Czechoslovakia
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Abstract

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Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Attouch, H. and Wets, R., Quantiiative stability of variational systems: I, The epigraphical distance. Working paper, II ASA (Laxenburg, Austria, 1988).Google Scholar
[2]Azé, D. and Penot, J.P., Operations on convergent families of sets and functions, AVA-MAC report (Perpignan, 1987).Google Scholar
[3]Azé, D. and Penot, J.P., ‘Recent quantitative results about the convergence of convex sets and functions’, in Functional analysis and approximation, Editor Papini, P.L. (Pitagora Editrice, Bologna, 1989).Google Scholar
[4]Beer, G. and Lucchetti, R., ‘Convex optimization and the epi-distance topology’, (preprint).Google Scholar
[5]Beer, G., ‘A second look at set convergence and linear analysis’, Rend. Sem. Mat. Fis. Milano (to appear).Google Scholar
[6]Beer, G., ‘Conjugate convex functions and the epi-distance topology’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[7]Beer, G., ‘On Mosco convergence of convex sets’, Bull. Austral. Math. Soc. 38 (1988), 239253.Google Scholar
[8]Beer, G. and Di Concilio, A., ‘Uniform continuity on bounded sets and the Attouch-Wets topology’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[9]Beer, G., ‘On convergence of closed sets in a metric space and distance functions’, Bull. Austral. Math. Soc. 31 (1985), 421432.CrossRefGoogle Scholar
[10]Engelking, R., General topology (PWN n. 60, Warsaw, 1977).Google Scholar
[11]Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions: Lecture notes in mathematics 580 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[12]Hamlett, T. and Harringhton, L., The closed graph and p-closed graph properties in general topology: AMS Contemporary Series 3 (American Mathematical Society, Providence, RI, 1981).CrossRefGoogle Scholar
[13]Kuratowski, K., Topology 1 (Academic Press, New York, 1966).Google Scholar
[14]Mosco, U., ‘Convergence of convex sets and of solutions of variational inequalities’, Adv. in Math. 3 (1969), 510585.Google Scholar
[15]Mosco, U., ‘On the continuity of the Young-Fenchel transform’, J. Math. Anal. Appl. 35 (1971), 518535.Google Scholar
[16]Robertson, A. and Robertson, W., Topological vector spaces (Cambridge University Press, 1964).Google Scholar
[17]Wijsman, R., ‘Convergence of sequences of convex sets, cones and functions II’, Trans. Amer. Math. Soc. 123 (1966), 3245.Google Scholar