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Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak.

  1. I can see that $T$ being continuous strong-strong implies that $T$ is continuous strong-weak. How to see weak-weak, please?
  2. I have no idea on the other direction. Could anyone help me, please? Thank you!
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  • $\begingroup$ Hint: Assume it's weak-weak but not strong-strong. Then follow a proof by contradiction. $\endgroup$ – Axoren Nov 13 '14 at 11:52
  • $\begingroup$ @Axoren Could add more detail, please? $\endgroup$ – LaTeXFan Nov 13 '14 at 11:54
  • $\begingroup$ You have to use the adjoint of the operator$T$. $\endgroup$ – Neutral Element Nov 13 '14 at 17:09
  • $\begingroup$ For weak-weak implying strong-strong see math.stackexchange.com/questions/5795/… $\endgroup$ – daw Nov 14 '14 at 13:59

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