Let $X$ and $Y$ be infinite dimensional normed linear spaces and let $S:Y^*\to X^*$ be a one-one linear operator. I want to show that $S$ can not be weak*-norm continuous.
My idea is to choose a sequence $(y_n^*)$ in $Y^*$ which weak* converges to some $y^*$ in $Y^*$ (for this Banach-Alaoglu theorem will have to be used) such that $S(y^*_n)\not\to S(y^*)$ in norm. But I am not getting how infinite dimension of $X$ and $Y$ (and of course $X^*,Y^*$) will help. Any suggestion will be appreciated.