Let $X$ be a separable Banach space such that $X$ and its dual $X^\*$ have Radon-Nikodym property. Let $C$ be a convex, closed and bounded subset of $X$.


Can we say that $C$ is **weakly compact** or **weakly locally compact**?


An idea please



$C$ is **weakly compact** or **weakly locally compact**?
The closed unit ball of a Banach space $X$ is weakly compact if and only if it is reflexive. So if $X$ is a non-reflexive space such that $X$ and its dual have RNP, take $C$ to be the unit ball to see that the answer to your first question is negative.


For instance, takes $X$ to be the classical James space; its dual is separable, hence has RNP; its bidual is also separable, hence has RNP; and $X$ is a closed subspace of $X^{\*\*}$ hence also has RNP.


I suspect that the same counterexample also works for "locally weakly compact" but I have not checked the details.


