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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.

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    $\begingroup$ 1 and 2 are trivially equivalent from linearity of $T$. The interesting part is whether they are equivalent to $T$ is w* continuous on all of $X^*$. $\endgroup$ Mar 31, 2022 at 13:40

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

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