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:


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.



Weak* continuity of linear maps
Yes. From the Krein Smulian theorem (use Google) you get that $T^\*$ maps $Y$ into $X$.


