I am considering a linear map $T:X^\*\to Y^\*$ where $X^\*$ and $Y^\*$ are dual 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: [1] $T$ is weak* continuous when restricted to the unit ball of $X^\*$; [2] 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.