Dunford–Pettis sets

Abstract

Abstract. Bibasic sequences are used to study relative weak compactness and relative norm compactness of Dunford-Pettis sets.A Banach space X has the Dunford-Pettis property provided that every weakly compact operator with domain X and range an arbitrary Banach space Y maps weakly compact sets in X into norm compact sets in Y . Localizing this notion, a bounded subset A of X is said to be a Dunford-Pettis subset of X if T (A) is relatively norm compact in Y whenever T : X → Y is a weakly compact operator. Consequently, a Banach space X has the Dunford-Pettis property if and only if each of its weakly compact sets is a Dunford-Pettis set. The survey article by Diestel [5] is an excellent source of information about classical results in Banach spaces which relate to the Dunford-Pettis property.Kevin Andrews utilized Dunford-Pettis sets in a study of the Bochner integral in E. Bator showed in [2] that a dual space has the weak Radon-Nikodym property if and only if each Dunford-Pettis subset of X is relatively compact. In addition to reproducing Bator's result, Emmanuele [8] established several other structure properties for Banach spaces in which all Dunford-Pettis sets are relatively compact. Since every bounded subset of a Banach space X whose dual space X * has the Schur property is a Dunford-Pettis subset of X, it is clear that there are Dunford-Pettis sets which are not relatively weakly compact. However, we note that Odell [13, p. 377] showed that every sequence in a Dunford-Pettis set has a weakly Cauchy subsequence. In this paper we study Dunford-Pettis sets which fail to be relatively norm or weakly compact.

Tensor products and Dunford–Pettis sets

“…A Banach space X is said to have the Dunford-Pettis property (DP P ) if every weakly compact operator T from X to a Banach space Y is completely continuous (=Dunford-Pettis operator), and X is said to have the hereditary Dunford-Pettis property if every closed linear subspace of X has the DP P . See [1,7,9,11,17,20] for a guide to the extensive literature dealing with the DPP. Note that the classical spaces c 0 and 1 have the hereditary DPP [7, pp.…”