The Relationship Between Goldstine′s Theorem and the Convex Point of Continuity Property

Goldstine′s Theorem says that the natural embedding of the closed unit ball B(X) of a Banach space X is weak* dense in the second dual ball B(X**). In this paper we characterise, in terms of the geometry of B(X), when the natural embedding of B(X) into B(X**) is not only weak* dense, but also residual. Using this characterisation, we show that a Banach space X has the convex point of continuity property, if and only if, for each equivalent norm ball B(X), the natural embedding of B(X) into B(X**) is residual with respect to the weak* topology. We also show that a Banach space X has the Radon-Nikodym property if and only if, for each equivalent norm ball B(X), the set of linear functionals in X* which attain their norm on B(X) is residual in X*.