So I am trying to learn some functional analsysis, but the weak star stuff really confuses me. I came across this one and am completely lost. Let $X$ and $Y$ be Banach spaces, and let $F:X^*\rightarrow Y^*$ be a bounded linear operator. Show there is a bounded linear operator $G:Y\rightarrow X$ with $G^*=F$ if and only if $F:(X^*,wk*)\rightarrow(Y^*,wk*)$ is continuous. Thanks for any help you can offer.

  • $\begingroup$ Is it by any chance assumed that the Banach spaces are reflexive? $\endgroup$ – pitariver Mar 8 at 18:42
  • $\begingroup$ Don't need reflexivity. The question is correct. $\endgroup$ – Idonknow Mar 9 at 14:43
  • $\begingroup$ One direction is easy. Another direction is tricky. $\endgroup$ – Idonknow Mar 9 at 14:47
  • $\begingroup$ The "tricky" point is that every element of $(X^*, wk*)^*$ is given by an evaluation in $x\in X$, more briefly $(X^*, wk*)^*=X$. Then $G=F^*$ does the job. $\endgroup$ – Jochen Mar 10 at 9:57

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