Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-19T11:14:02.536Z Has data issue: false hasContentIssue false
  • English
  • Français

Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Marián Fabian  and
Boris S. Mordukhovich
Marián Fabian
Affiliation:
Mathematical Institute, Czech Republic Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
Boris S. Mordukhovich
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
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 develop a method of separable reduction for Fréchet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

Keywords

49J5258C2046B20nonsmooth analysisBanach spacesseparable reductionFréchet-like normals and subdifferentialssupporting propertiesAsplund spaces
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 1 , 01 February 1999 , pp. 26 - 48
Copyright
Copyright © Canadian Mathematical Society 1999

References

[BP] Borwein, J. M. and Preiss, D., A smooth variational principle with applications to subdifferentiability and differentiability of convex functions. Trans. Amer. Math. Soc. 303 (1987), 517–527.Google Scholar
[BS] Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach spaces. Part I: Theory. Canad. J. Math. 38 (1986), 431–452. Part II: Applications. Canad. J. Math. 39 (1987), 517–527.Google Scholar
[D] Diestel, J., Geometry of Banach Spaces. Lecture Notes in Math. 485 , Springer, 1975.Google Scholar
[EL] Ekeland, I. and Lebourg, G., Generic Fréchet differentiability and perturbed optimization problems in Banach spaces. Trans. Amer. Math. Soc. 224 (1976), 193–216.Google Scholar
[F] Fabian, M., Subdifferentiability and trustworthiness in the light of the smooth variational principle of Borwein and Preiss. Acta Univ. Carolin. Math. Phys. 30 (1989), 51–56.Google Scholar
[FM] Fabian, M. and Mordukhovich, B. S., Nonsmooth characterizations of Asplund spaces and smooth variational principles. Department of Mathematics, Wayne State University, Research Report 38 , 1997; Set- Valued Anal., to appear.Google Scholar
[FZ] Fabian, M. and Zhivkov, N. V., A characterization of Asplund spaces with help of local ε supports of Ekeland and Lebourg. C. R. Acad. Bulgare Sci. 38 (1985), 671–674.Google Scholar
[KM] Kruger, A. Y. and Mordukhovich, B. S., Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24 (1980), 684–687.Google Scholar
[L] Loewen, P. D., A mean value theorem for Fréchet subgradients. Nonlinear Anal. 23 (1994), 1365–1381.Google Scholar
[MS1] Mordukhovich, B. S. and Shao, Y., Extremal characterizations of Asplund spaces. Proc. Amer. Math. Soc. 124 (1996), 197–205.Google Scholar
[MS2] Mordukhovich, B. S. and Shao, Y., Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348 (1996), 1235– 1280.Google Scholar
[P] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability. 2nd edn, Lecture Notes in Math. 1364 , Springer, 1993.Google Scholar