I consider a linear map $T\colon X^*\to Y^*$, where $X^*$ and $Y^*$ are duals of Banach spaces. I would like to know if I can deduce that $T$ is weak* continuous (I consider the weak* topologies on both $X^*$ and $Y^*$) if I know either of the following:
$T$ is weak* continuous when restricted to the unit ball of $X^*$;
for any integer $n$, $T$ is weak* continuous when restricted to the ball $B(0; n)$ in $X^*$.
In my case neither $X$ nor $Y$ is reflexive.