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.
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Is it by any chance assumed that the Banach spaces are reflexive? – pitariver Mar 08 '19 at 18:42
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Don't need reflexivity. The question is correct. – Idonknow Mar 09 '19 at 14:43
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One direction is easy. Another direction is tricky. – Idonknow Mar 09 '19 at 14:47
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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. – Jochen Mar 10 '19 at 09:57