It is shown that if the unit
ball BX∗∗ of X∗∗ is Eberlein compact in the weak∗ topology, or if x∗ is
isomorphic to a subspace of a weakly compactly generated Banach space
then x∗ possesses the Radon-Nikodým property (RNP). This extends the
classical theorem of N. Dunford and B. J. Pettis. If X is a Banach space with
X∗∗∕X separable then both x∗ and x ∗∗ (and hence X) have the RNP. It
is also shown that if a conjugate space X∗ possesses the RNP and X is
weak∗ sequentially dense in X∗∗ then BX∗∗ is weak∗ sequentially compact.
Thus, in particular, if x ∗∗∕X is separable then BX∗∗∗ is weak∗ sequentially
compact.
