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On Saab's Characterizations of Weak Radon–Nikodym Sets

1985; Kyoto University; Volume: 21; Issue: 5 Linguagem: Inglês

10.2977/prims/1195178789

1663-4926

Minoru Matsuda, Teiichi KOBAYASHI,

Advanced Harmonic Analysis Research

Throughout this paper X and Y denote real Banach spaces with topological duals X* and F* respectively.The closed unit ball in X is denoted by B x .Recently, in parallel with that of dual Banach spaces with the Radon-Nikodym property (RNP), the study of such spaces with the weak Radon-Nikodym property (WRNP) as well as Banach spaces not containing a copy of ^ has been made by many authors, especially, Pelczynski [14], Rosenthal [18], [19], Odell and Rosenthal [13], Haydon [7], Musial [11], Janicka [9], Riddle and Uhr [17], and Saab and Saab [23].Corresponding to those of dual Banach spaces with the RNP, a number of characterizations of such spaces with the WRNP have been obtained, heavily relying on Rosenthal's signal theorem (Theorem 1 in [18] or Theorem 2.2 in [19]) asserting that the space X contains no copy of /i if and only if every bounded subset of X is weakly pre-compact (For this terminology, see § 3).They are collected below.Theorem A. Each of the following statements characterizes X not containing a copy of /j.(a) (Haydon [7]).Every z**eA r ** is universally weak*-measurable and satisfies the barycentric formula (For this terminology, see § 2) on B x * equipped with the weak*-topology.(b) (Pelczynski [14]).Every bounded linear operator T : L^X* is a Dunford-Pettis operator.(c) (Musial [11] and Janicka [9]).The space X* has the WRNP.(d) (Saab and Saab [23]).The restriction of each x**^X** to each nonempty weak*-compact subset of B x * has a point of weak*-continuity.Succeedingly, as in the case of weak*-compact convex sets with the RNP (for instance, [20]), some attempts to localize the results stated in Theorem A