weak star convergent net is uniformly convergent on weakly compact sets?

Question

Let $X$ be a Banach space and $X^\ast$ its (topological)dual space.

Let $x_\alpha^\ast,x^\ast\in B_{X^\ast}$, where $B_{X^\ast}$ denotes the closed unit ball of $X^\ast$, such that $x_\alpha^\ast\to x^\ast$ in the weak star topology, and $A\subset X$ is weakly compact. Does $x_\alpha^\ast$ converge to $x^\ast$ uniformly on $A$?

Under what condition(s) could this statement be true?

Answer 1

The answer is no in general. If $X$ is reflexive, then $B_X$ is weakly compact and clearly $x_\alpha^*\to x^\star$ uniformly on $A=B_X$ iff the convergence is in norm.