Is the following true?
Let $X$ and $Y$ be separable Banach spaces and consider their dual spaces $X^*$ and $Y^*$ equipped with weak* topology. Suppose that a linear map $T:X^*\to Y^*$ is sequentially continuous. Then, is it true that $T$ is continuous?
I suspect this may be true using the following two facts:
(i) Any closed ball in $X^*$ (and in $Y^*$) is metrizable as $X$ and $Y$ are separable.
(ii) A convex set in $X^*$ (and in $Y^*$) is weak* closed iff it is sequentially closed (by Krein-Smulian theorem).
From (i), we have that $T$, when restricted to a closed ball, is continuous. But I am not able to go further from this.
Any help or hint would be appreciated. Thank you.