×

Optimization and differentiation in Banach spaces. (English) Zbl 0633.46042

The author shows that some old ideas of Smulian can be used to give another proof of a theorem of Bourgain which characterizes RNP sets in terms of strongly exposed points. By means of optimization results, he gives a characterization theorem on RNP sets in real Banach spaces with a more concise proof than before. This theorem also shows that a converse of the main result in his earlier paper [Math. Ann. 236, 171-176 (1978; Zbl 0365.49006)] holds. Furthermore, some attempts to extend optimization results obtained in real Banach spaces to the case of complex ones have been made.
Reviewer: M.Matsuda

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
49J27 Existence theories for problems in abstract spaces

Citations:

Zbl 0365.49006
Full Text: DOI

References:

[1] Bourbaki, N., Topologie Générale (1951), Hermann: Hermann Paris · Zbl 0085.37103
[2] Bourgain, J., Strongly exposed points in weakly compact convex sets in Banach spaces, Proc. Amer. Math. Soc., 58, 197-200 (1976) · Zbl 0309.46009
[3] Bourgain, J., On dentability and the Bishop-Phelps property, Israel J. Math., 28, 265-271 (1977) · Zbl 0365.46021
[4] Bourgin, R., Geometric Aspects of Convex Sets with the Radon-Nikodym Property (1983), Springer: Springer Berlin · Zbl 0512.46017
[5] Crandall, M.; Lions, P.-L., Uniqueness of viscosity solutions, J. Funct. Anal., 62, 379-396 (1985), Hamilton-Jacobi equations in infinite dimensions I. Uniqueness of viscosity solutions · Zbl 0627.49013
[6] M. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions II. Existence of bounded viscosity solutions, to appear.; M. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions II. Existence of bounded viscosity solutions, to appear. · Zbl 0639.49021
[7] Leach, E. B.; Whitfield, J. H., Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc., 33, 120-126 (1972) · Zbl 0236.46051
[8] Stegall, C. P., Optimization of functions on certain subsets of Banach spaces, Math. Ann., 236, 171-176 (1978) · Zbl 0365.49006
[9] Stegall, C. P., The Radon-Nikodym property in conjugate Banach spaces II, Trans. Amer. Math. Soc., 264, 2, 507-519 (1981) · Zbl 0475.46016
[10] Stegall, C. P., The duality between Asplund spaces and spaces with the Radon-Nikodym property, Israel J. Math., 29, 408-412 (1978) · Zbl 0374.46015
[11] Stegall, C. P., Lectures on the Radon-Nikodym property in Banach spaces (1982), Vorlesungsreihe der Universität: Vorlesungsreihe der Universität Essen
[12] Stegall, C. P., Applications of Descriptive Topology in Functional Analysis (1985), Linz · Zbl 0621.46003
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.