Asplund spaces, the RNP and perturbed optimization
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In this section we are going to look at a duality relationship between Asplund spaces and spaces with the Radon-Nikodým property (RNP). Roughly speaking, a Banach space E or, more generally, a closed convex subset C of E, is said to have the Radon-Nikodým property if the classical Radon-Nikodým 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].
KeywordsBanach Space Nonempty Subset Compact Convex Subset Lower Semicontinuous Function Closed Convex Hull
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