The Radon-Nikodým Property
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In this chapter we introduce the Radon-Nikodým property for Banach spaces. We begin with a study of vector measures, that is, measures with values in a Banach space. Those spaces for which the classical Radon—Nikodým Theorem extends to vector valued measures are said to have the Radon—Nikodým property. The identification of injective and projective tensor products of spaces of scalar measures in terms of spaces of vector measures sheds some light on this property. We then examine the representability of various types of operators on C(K) and L 1(μ) spaces and we uncover some classes of Banach spaces, such as the reflexive spaces and the separable dual spaces, that possess the Radon-;Nikodým property. We also relate the possession of this property to the coincidence of the integral and nuclear operators Finally, we give some applications of the Radon-Nikodým property, including the Principle of Local Reflexivity.
KeywordsBanach Space Positive Measure Vector Measure Nuclear Norm Nuclear Operator
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