Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T07:39:16.319Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Minimal Usco Maps

Published online by Cambridge University Press:  20 November 2018

Andrei Verona  and
Maria Elena Verona
Andrei Verona
Affiliation:
Department of Mathematics, California State University, Los Angeles, CA, USA 90032
Maria Elena Verona
Affiliation:
Department of Mathematics, University of California, Riverside, CA, USA 92521
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.

We prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.

Keywords

54C6046B22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 3 , 01 September 1991 , pp. 412 - 416
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Asplund, E., Fréchet differentiability of convex functions , Acta Math. 121(1968), 31–47.Google Scholar
2. Borwein, J. M., Fitzpatrick, S. P. and Kenderov, P. S., Minimal convex uscos and monotone operators on small sets. Preprint, 1989.Google Scholar
3. Drewnowski, L. and Labuda, I., On minimal upper semicontinuous, compact-valued maps, Rocky Mountain J. of Math. 20 (1990), 1–16.Google Scholar
4. Drewnowski, L. and Labuda, I., On minimal convex usco and maximal monotone maps, Real Analysis Exchange 15(1989/1990), 729–742.Google Scholar
5. Fitzpatrick, S. P., Continuity of non-linear monotone operators, Proc. Amer. Math. Soc. 62 (1977), 111–116.Google Scholar
6. Kenderov, P. S., The set-valued monotone mappings are almost everywhere sing le-valued, C. R. Acad. Bulgare Sci. 27 (1974), 1173–1175.Google Scholar
7. Phelps, R. R., Convex Functions, Monotone Operators and Differentiability , Lecture Notes in Mathematics 1364, Springer-Verlag, Berlin, 1989.Google Scholar
8. Stegall, C., Gäteaux differentiation of functions on a certain class ofBanach spaces , Functional Analysis: Surveys and Recent Results III, North-Holland, 1984, 35–46.Google Scholar
9. Verona, M. E., Generic differentiability of convex functions and monotone operators. Ph.D. Thesis, California Institute of Technology, Pasadena, 1989.Google Scholar
10. Verona, A. and Verona, M. E., Locally efficient monotone operators, Proc. Amer. Math. Soc. 109 (1990), 195–204.Google Scholar