Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Continuous selections and reflexive Banach spaces

Authors: Valentin Gutev and Stoyan Nedev
Journal: Proc. Amer. Math. Soc. 129 (2001), 1853-1860
MSC (2000): Primary 54C65, 54C60, 46A25; Secondary 54B20, 46B10, 26B25
Published electronically: November 3, 2000
MathSciNet review: 1814119
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Every l.s.c. mapping $\Phi$ from a space $X$ into the non-empty closed convex subsets of a reflexive Banach space $Y$ admits a continuous selection provided it satisfies a ``weak'' u.s.c. condition. This result partially generalizes some known selection theorems. Also, it is successful in solving a problem concerning the set of proper lower semi-continuous convex functions on a reflexive Banach space.

References [Enhancements On Off] (What's this?)

  • 1. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, System & Control, Foundation and Applications, vol. 2, Birkhäuser, Boston, 1990. MR 91d:49001
  • 2. G. Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), 239-253. MR 90a:46026
  • 3. -, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), 1115-1123. MR 89i:49006
  • 4. -, Support and distance functionals for convex sets, Numer. Funct. Anal. Optim. 10 (1989), 15-36. MR 89m:46031
  • 5. -, Topologies on closed convex sets and the Effros measurability of set valued functions, Sém. d'Anal. Convexe Montpellier 21 (1991), exposé No 2. MR 93d:46020
  • 6. -, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces, Sém. d'Anal. Convexe Montpellier 21 (1991), exposé No 3; Nonlinear Anal. 19 (1992), 271-290. MR 93c:52001; MR 93h:49025
  • 7. -, Topologies on closed and closed convex sets, Mathematics and its applications, vol. 268, Kluwer Academic Publishers, The Netherlands, 1993. MR 95k:49001
  • 8. G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pure Appl. 162 (1992), 367-381. MR 94c:54016
  • 9. J. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843-852. MR 90i:46025
  • 10. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Longman, Harlow, United Kingdom, 1993. MR 94d:46012
  • 11. J. Diestel, Geometry of Banach spaces, LNM # 485, Springer-Verlag, Berlin, 1975. MR 57:1079
  • 12. S. Francaviglia, A. Lechicki, and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. MR 87e:54025
  • 13. V. Gutev, Weak factorizations of continuous set-valued mappings, Topology Appl. 102 (2000), 33-51.
  • 14. C. Hess, Contributions à l'ètude de la mesurabilité, de la loi de probabilité, et de la convergence des multifunctions, Thèse d'état, Université Montpellier II, 1986.
  • 15. A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Boll. Un. Mat. Ital. B(7) 1 (1987), 435-452. MR 88e:54007
  • 16. E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-382. MR 17:990e
  • 17. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 45:7560
  • 18. -, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 44:817
  • 19. S. Nedev, A selection example, C. R. Acad. Bulgare Sci. 40 (1987), no. 11, 13-14. MR 89e:54034
  • 20. Y. Sonntag and C. Zalinescu, Set convergences: An attempt of classification, Proc. Intr. Conf. on Diff. Equations and Control Theory, Iasi, Romania, August 1990. MR 93a:49008
  • 21. S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1971), 173-180. MR 46:5995

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C65, 54C60, 46A25, 54B20, 46B10, 26B25

Retrieve articles in all journals with MSC (2000): 54C65, 54C60, 46A25, 54B20, 46B10, 26B25

Additional Information

Valentin Gutev
Affiliation: School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa

Stoyan Nedev
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., bl. 8, 1113 Sofia, Bulgaria

Keywords: Set-valued mapping, selection, lower semi-continuous, weakly continuous, hyperspace topology, convex function
Received by editor(s): November 18, 1995
Received by editor(s) in revised form: September 27, 1999
Published electronically: November 3, 2000
Communicated by: James E. West
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society