# A condition for a bounded $T:X^* \rightarrow Y^*$ being a dual operator

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.

• Is it by any chance assumed that the Banach spaces are reflexive? – pitariver Mar 8 at 18:42
• Don't need reflexivity. The question is correct. – Idonknow Mar 9 at 14:43
• One direction is easy. Another direction is tricky. – Idonknow Mar 9 at 14:47
• 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 at 9:57