Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-27T07:26:58.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Mosco convergence and weak topologies for convex sets and functions

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

Gerald Beer
Gerald Beer
Affiliation:
Professor G. Beer, Department of Mathematics, California State University, Los Angeles, Los Angeles, California 90032, U.S.A.
Get access

Abstract

Let X be a reflexive Banach space. This article presents a number of new characterizations of the topology of Mosco convergence TM for convex sets and functions in terms of natural geometric operators and functional. In addition, necessary and sufficient conditions are given for TM to agree with the weak topology generated by {d(x, C): x є X}, where each distance functional is viewed as a function of the set argument.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 89 - 104
Copyright
Copyright © University College London 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AT. Attouch, H.. Variational Convergence for Functions and Operators (Pitman publishers, Boston, 1984).Google Scholar
AW. Attouch, H. and Wets., R. Quantitative stability of variational systems: I. The epigraphical distance. To appear in Trans. Amer. Math. Soc.Google Scholar
Be1. Beer, G.. On Mosco convergence of convex sets. Bull. Australian Math. Soc, 38 (1988), 239–253.CrossRefGoogle Scholar
Be2. Beer, G.. On the Young-Fenchel transform for convex functions. Proc. Amer. Math. Soc, 104 (1988), 1115–1123.CrossRefGoogle Scholar
Be3. Beer, G.. Support and distance functional for convex sets. Numer. Fund. Anal. Opt., 10 (1989), 15–36.CrossRefGoogle Scholar
Be4. Beer, G.. Three characterizations of the Mosco topology for convex functions. Archiv derMat., 55 (1990), 285–292.Google Scholar
Be5. Beer, G.. Conjugate convex functions and the epi-distance topology. Proc. Amer. Math. Soc, 108 (1990), 117–126.CrossRefGoogle Scholar
Be6. Beer, G.. A Polish topology for the closed subsets of a Polish space. To appear in Proc. Amer. Math. Soc.Google Scholar
BB. Beer, G. and Borwein, J.. Mosco convergence and reflexivity. Proc. Amer. Math. Soc, 109 (1990), 427–436.CrossRefGoogle Scholar
BLLN. Beer, G., Lechicki, A., Levi, S. and Naimpally, S.. Distance functional and suprema of hyperspace topologies. To appear in Annali Mat. Pura Appl.Google Scholar
Bo. Borwein, J.. A note on ε-subgradients and maximal monotonicity. Pacific J. Math., 103 (1982), 307–314.CrossRefGoogle Scholar
BF. Borwein, J. and Fitzpatrick, S.. Mosco convergence and the Kadec property. Proc. Amer. Math. Soc, 106 (1989), 843–849.CrossRefGoogle Scholar
BR. Bronsted, A. and Rockafellar, R. T.. On the subdifferentiability of convex functions. Proc. Amer. Math. Soc, 16 (1965), 605–611.CrossRefGoogle Scholar
Ch. Christensen, J. P. R.. Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings. Proc. Amer. Math. Soc, 86 (1982), 649–655.CrossRefGoogle Scholar
Co. Cornet, B.. Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la decision #7309 (Universite de Paris Dauphine, 1973).Google Scholar
FLL. Francaviglia, S., Lechicki, A. and Levi, S.. Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions. J. Math. Anal. Appl, 112 (1985), 347–370.CrossRefGoogle Scholar
He. Hess, C.. Contributions à l'étudé, de la mesurabilité, de la loi de probabilité, et de la convergence des multifunctions, thèse d'ètat (Montpellier, 1986).Google Scholar
Ho. Holmes, R.. A course in optimization and best approximation, Lecture notes in mathematics #257 (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
Ke. Kenderov, P.. Most of the optimization problems have unique solution. C. R. Acad. Bulgare des Sci., 37 (1984), 297–300.Google Scholar
KT. Klein, E. and Thompson, A.. Theory of Correspondences (Wiley, Toronto, 1984).Google Scholar
Ku. Kuratowski, K.. Topology, vol. 1 (Academic Press, New York, 1966).Google Scholar
LL. Lechicki, A. and Levi, S.. Wijsman convergence in the hyperspace of a metric space. Bull. Un. Mat. Ital, 5-B (1987), 435–452.Google Scholar
LP. Lucchetti, R. and Patrone, F.. Hadamard and Tyhonov well-posedness of a certain class of convex functions. J. Math. Anal. Appl, 88 (1982), 204–215.CrossRefGoogle Scholar
Mc. McLinden, L.. Successive approximation and linear stability involving convergent sequences of optimization problems. J. Approximation theory, 35 (1982), 311–354.CrossRefGoogle Scholar
Mo1. Mosco, U.. Convergence of convex sets and solutions of variational inequalities. Advances in Math., 3 (1969), 510–585.CrossRefGoogle Scholar
Mo2. Mosco, U.. On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl, 35 (1971), 518–535.CrossRefGoogle Scholar
Ph. Phelps, R.. Convex functions, monotone operators, and differentiability, Lecture notes in mathematics #1364 (Springer-Verlag, Berlin, 1989).Google Scholar
Ro. Rockafellar, R. T.. Convex functions, monotone operators and variational inequalities, in Theory and Applications of Monotone Operators, Ghizetti, A., ed. (Tipografia Oderisi Editrice, Gubbio, Italy (1969), 34-65).Google Scholar
SW. Salinetti, G. and Wets, R.. On the relations between two types of convergence for convex functions. J Math. Anal. Appl, 60 (1977), 211–226.CrossRefGoogle Scholar
So. Sonntag, Y.. Convergence au sens de Mosco, théorie et applications à l'approximation des solutions d'inéquations. Thèse d'Etat (Universite de Provence, Marseille, 1982).Google Scholar
Ts. Tsukada, M.. Convergence of best approximations in a smooth Banach space. J. Approximation Theory, 40 (1984), 301–309.CrossRefGoogle Scholar
Zo. Zolezzi, T.. Variational stability and well-posedness in the optimal control of ordinary differential equations. In Mathematical control theory, Banach center publications, Vol. 14 (PWN, Warsaw, 1985).Google Scholar