Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-21T00:58:20.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Polar locally convex topologies and Attouch-Wets convergence

Published online by Cambridge University Press:  17 April 2009

Gerald Beer
Gerald Beer
Affiliation:
Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, United States of America
Rights & Permissions [Opens in a new window]

Abstract

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.

Let X be a Hausdorff locally convex space. We show that convergence of a net of continuous linear functionals on X with respect to a given polar topology on its continuous dual X′ can be explained in terms of the convergence of the corresponding net of its graphs in X × R, and the corresponding net of level sets at a fixed height in X, with respect to a natural generalisation of Attouch-Wets convergence in normable spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 33 - 46
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Attouch, H. and Wets, R., ‘Quantitative stability of variational systems: I.’, The epigraph-ical distance, Trans. Amer. Math. Soc. 328 (1991), 695–730.Google Scholar
[2]Attouch, H., Lucchetti, R. and Wets, R., ‘The topology of the ρ−Hausdorff distance’, Ann. Mat. Pura Appl. (to appear).Google Scholar
[3]Azé, D. and Penot, J.-P., ‘Operations on convergent families of sets and functions’, Optimization 21 (1990), 521–534.CrossRefGoogle Scholar
[4]Beer, G., ‘Convergence of continuous linear functionals and their level sets’, Arch. Math. 52 (1989), 482–491.CrossRefGoogle Scholar
[5]Beer, G., ‘Conjugate convex functions and the epi-distance topology’, Proc. Amer. Math. Soc. 108 (1990), 117–126.CrossRefGoogle Scholar
[6]Beer, G., ‘A Second look at set convergence and linear analysis’, Sem. Rend. Math. Fis. Milano (to appear).Google Scholar
[7]Beer, G. and Di Concilio, A., ‘Uniform continuity on bounded sets and the Attouch-Wets topology’, Proc. Amer. Math. Soc. 112 (1991), 235–243.CrossRefGoogle Scholar
[8]Beer, G. and Lucchetti, R., ‘Convex optimization and the epi-distance topology’, Trans. Amer. Math. Soc. 327 (1991), 795–813.Google Scholar
[9]Castaing, C. and Valadier, M., ‘Convex analysis and measurable multifunctions’, in Lecture notes in mathematics 580 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[10]Hola, L., ‘The Attouch-Wets topology and a characterisation of normable spaces’, Bull. Austral. Math. Soc. 44 (1991), 11–18.Google Scholar
[11]Kato, T., Perturbation theory for linear operators (Springer-Verlag, Berlin, Heidelberg, New York, 1966).Google Scholar
[12]Klein, E. and Thompson, A., Theory of correspondences (Wiley, New York, 1984).Google Scholar
[13]Page, W., Topological uniform structures (Dover, New York, 1988).Google Scholar
[14]Penot, J.-P., ‘The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity’, Proc. Amer. Math. Soc. 387 (1991), 275–286.Google Scholar
[15]Robertson, A. and Robertson, W., Topological vector spaces (Cambridge University Press, Cambridge, 1973).Google Scholar