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Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator satisfying $x_n'(x) \rightarrow x'(x) \Rightarrow (Tx_n')(x) \rightarrow (Tx')(x)$ for $x_n',x' \in X'$ is continuous in the weak $^*$ topology?

If anything is unclear, please let me know.

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  • 2
    $\begingroup$ If $X$ is not finite-dimensional then $X'$ is not metrizable. In this case, sequential continuity is weaker than continuity. I'm thus expecting the answer to be "no". $\endgroup$ – M.G May 27 '16 at 0:14
  • $\begingroup$ Is $T$ supposed to be linear? $\endgroup$ – gerw May 27 '16 at 17:45
  • $\begingroup$ @gerw yes, it is $\endgroup$ – Tzego May 27 '16 at 19:31

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