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Asplund spaces, the Radon-Nikodym property and optimization

  • Robert R. Phelps
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1364)

Abstract

In this section we are going to look at a duality relationship between Asplund spaces and spaces with the Radon-Nikodym property. Roughly speaking, a Banach space E or, more generally, a closed convex subset C of E, is said to have the Radon-Nikodym property (RNP) if the classical Radon-Nikodym theorem (on the representation of absolutely continuous measures in terms of integrals) is valid for vector-valued measures whose “average range” is contained in C. This property has been characterized in purely geometrical terms (which is the basis of the definition we use below). For the extraordinarily wide range of connections between the RNP and various parts of integration theory, operator theory and convexity, one should read the 1977 survey by Diestel and Uhl [Di-U] and, for more recent results (1983) the comprehensive lecture notes by Bourgin [Bou].

Keywords

Banach Space Nonempty Subset Compact Convex Subset Lower Semicontinuous Function Closed Convex Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Robert R. Phelps
    • 1
  1. 1.Department of Mathematics GN-50University of WashingtonSeattleUSA

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