Convex w∗-closures versus convex norm-closures in dual Banach spaces

A subset Y of a dual Banach space X∗ is said to have the property (P) if for every weak*-compact subset H of Y. The purpose of this paper is to give a characterization of the property (P) for subsets of a dual Banach space X∗, and to study the behavior of the property (P) with respect to additions, unions, products, whether the closed linear hull has the property (P) when Y does, etc. We show that the property (P) is stable under all these operations in the class of weak* K-analytic subsets of X∗.