Let $\sigma$ and $\tau$ be the weak$^*$-topologies of $X^*$ and $Y^*$ respectively, and prove that $S$ is continuous linear mapping of $(Y^*,\tau)$ into $(X^*,\sigma)$ if and only if $S=T^*$ for some $T\in \text B(X,Y).$
I didn't get that how to approach it when i assume some continuous linear mapping $S$ on $(Y^*,\tau)$.
Any type of help will be appreciated. Thanks in advance.
For each $x\in X$, the map $y^*\mapsto \langle x,Sy^*\rangle$ is a $\tau$-continuous linear functional on $Y^*$. Since the dual of $(Y^*,\tau)$ is $Y$, there is some $Tx\in Y$ such that $\langle Tx, y^*\rangle=\langle x,Sy^*\rangle$. This gives us a mapping $T:X\to Y$ given by $x\mapsto Tx$. Linearity of $T$ is easy to check. To show $T$ is continuous, show that it maps norm-bounded sets in $X$ to norm-bounded sets in $Y$.