The Radon-Nikodym Property in Conjugate Banach Spaces

Abstract

ABSTRACT. We characterize conjugate Banach spaces X* having the Radon-Nikodym Property as those spaces such that any separable subspace of X has a separable conjugate. Several applications are given.Introduction. There are several equivalent formulations of the Radon-Nikodym Property (RNP) in Banach spaces; we give perhaps the earliest definition: a Banach space X has RNP if given any finite measure space (S, 2, ju) and any X valued measure m on E, with m having finite total variation and being absolutely continuous with respect to fi, then m is the indefinite integral with respect to ß of an X valued Bochner integrable function on S. The first study of this property

Unconditional families in Banach spaces

“…We also state and prove some of its consequences. As we have mentioned, the proof is based on the following fundamental construction due to Stegall [36]. A variation of Stegall's construction has been presented by Godefroy and Talagrand [15] in the more general context of representable Banach spaces (see also [14]).…”

“…Although the above statement is not explicitly isolated in [36], it is the precise content of the proof.…”

“…Assume that (i) does not hold. It follows that there exists a separable subspace Z of Y such that Z * is non-separable (see [36]). By Theorem 13, we get that Z * * contains an unconditional basic sequence.…”

“…In particular, we use results concerning definable partitions of certain classes of antichains of the dyadic tree, which we call increasing and decreasing, as well as, definable partitions of finite products of perfect sets. Theorem 12, extracted from Stegall's fundamental construction [36] for separable Banach spaces with non-separable dual, also plays a key role. More precisely, using the Ramsey properties of increasing and decreasing antichains, proved in [3], we obtain the following extension of Stern's theorem [37] (see Sect.…”

On interpolation of Asplund operators

“…Asplund operators have been studied widely. See, for example, the papers by Edgar [12], Stegall [24] and the book by Diestel, Jarchow and Tonge [10]; see also the book by Pietsch [22] and the paper by Heinrich [15] where they are referred to as decomposing operators. Asplund operators A form a closed injective and surjective operator ideal in the sense of Pietsch [22].…”