Weak compactness in dual spaces (Q1599478)

A variety of results describing types of compact subsets of the dual are established by elementary means without using measure theory arguments. These relate to and generalize results of H. H. Schaefer and X. D. Zhang. Given a \(\sigma\)-Dedekind complete M-space \(E\) with order unit and a bounded subset \(A\) of \(E^*\), the \(\sigma( E^*,E)\)-sequentially precompactness of \(A\) is characterized in various ways, for example, \(A\) is relatively \(\sigma(E^*,E^{**})\)-compact. Consequences for spaces of operators from \(E\) to a Banach space are also considered. A number of these theorems can be viewed as generalizations of a Vitali-Hahn-Saks Theorem for vector measures.