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 the author characterizes, 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 characterization, he shows 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. He also shows that a Banach space \(X\) has the Radon-Nikodým 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 \(X^*\).
scientific article; zbMATH DE number 720404