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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.

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up vote 4 down vote accepted

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

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