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.