×

Asplund spaces, Stegall variational principle and the RNP. (English) Zbl 1176.46024

Summary: Given a pair of Banach spaces \(X\) and \(Y\) such that one is the dual of the other, we study the relationships between generic Fréchet differentiability of convex continuous functions on \(Y\) (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on \(X\) to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of \(X\) (Radon-Nikodým property).

MSC:

46B22 Radon-Nikodým, Kreĭn-Milman and related properties
49J27 Existence theories for problems in abstract spaces
49J53 Set-valued and variational analysis
54C60 Set-valued maps in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968) · Zbl 0162.17501 · doi:10.1007/BF02391908
[2] Asplund, E., Rockafellar, R.T.: Gradients of convex functions. Trans. Amer. Math. Soc. 139, 433–467 (1969) · Zbl 0181.41901 · doi:10.1090/S0002-9947-1969-0240621-X
[3] Bourgin, R.D.: Geometric aspects of convex sets with the Radon-Nikodým property. In: Lect. Notes in Math., vol. 993. Springer, Berlin (1983) · Zbl 0512.46017
[4] Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16, 605–611 (1965) · Zbl 0141.11801
[5] Christensen, J.P.R., Kenderov, P.S.: Dense strong continuity of mappings and the Radon-Nikodým property. Math. Scand. 54, 70–78 (1984) · Zbl 0557.46016
[6] Čoban, M.M., Kenderov, P.S., Revalski, J.P.: Densely defined selections of multivalued mappings. Trans. Amer. Math. Soc. 344, 533–552 (1994) · Zbl 0847.54020 · doi:10.2307/2154494
[7] Collier, J.: The dual of a space with the Radon-Nikodým property. Pacific. J. Math. 64, 103–106 (1976) · Zbl 0357.46028
[8] Drewnowski, L., Labuda, I.: On minimal convex usco and maximal monotone maps. Real Anal. Exchange 15, 729–742 (1989/90) · Zbl 0716.54015
[9] Fabian, M.: On minimum principles. Acta Polytechnica 20, 109–118 (1983)
[10] Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional analysis and infinite-dimensional geometry. In: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8. Springer, New York (2001) · Zbl 0981.46001
[11] Fitzpatrick, S.P.: Monotone operators and dentability. Bull. Austral. Math. Soc. 18, 77–82 (1978) · Zbl 0374.47027 · doi:10.1017/S0004972700007826
[12] Giles, J.R., Kenderov, P.S., Moors, W.B., Sciffer, S.D.: Generic differentiability of convex functions on the dual of a Banach space. Pacific J. Math. 172, 413–431 (1996) · Zbl 0852.46019
[13] Ghoussoub, N., Maurey, B.: H {\(\delta\)} -embedding in Hilbert space and optimization on G {\(\delta\)} -sets. Mem. Amer. Math. Soc. 62(349), iv+101 (1986) · Zbl 0606.46005
[14] Hansell, R.W., Jayne, J.E., Talagrand, M.: First class selectors for weakly upper semi-continuous multivalued maps in Banach spaces. J. Reine Angew. Math. 361, 201–220 (1985) · Zbl 0573.54012
[15] Jokl, L.: Minimal convex-valued weak* USCO correspondences and the Radon-Nikodým property. Comment. Math. Univ. Carolin. 28, 353–376 (1987) · Zbl 0642.46015
[16] Kenderov, P.S.: Monotone operators in Asplund spaces. C. R. Acad. Bulgare Sci. 30, 963–964 (1977) · Zbl 0377.47036
[17] Lassonde, M., Revalski, J.P.: Fragmentability of sequences of set-valued mappings with application to variational principles. Proc. Amer. Math. Soc. 133, 2637–2646 (2005) · Zbl 1079.49018 · doi:10.1090/S0002-9939-05-07865-2
[18] Mordukhovich, B.S.: Variational analysis and generalized differentiation I & II. In: Fundamental Principles of Mathematical Sciences, vols. 330 & 331. Springer, Berlin (2006)
[19] Moreau, J.-J.: Fonctionnelles convexes, séminaire équations aux dérivées partielles. Collège de France, Paris (1966)
[20] Namioka, I., Phelps, R.R.: Banach spaces which are Asplund spaces. Duke Math. J. 42, 735–750 (1975) · Zbl 0332.46013 · doi:10.1215/S0012-7094-75-04261-1
[21] Phelps, R.R.: Convex functions, monotone operators and differentiability. In: Lect. Notes in Math., vol. 1364. Springer, Berlin (1993) · Zbl 0921.46039
[22] Stegall, C.: Optimization of functions on certain subsets of Banach spaces. Math. Ann. 236, 171–176 (1978) · Zbl 0379.49008 · doi:10.1007/BF01351389
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.