Let $E$ and $F$ be two Banach spaces. Let $T \in \mathscr L(E, F)$ so that $T^* \in \mathscr L(F^*, E^*)$. Prove that $T^*$ is continuous from $F^*$ equipped with $\sigma(F^*, F)$ into $E^*$ equipped with $\sigma(E^*, E)$.
This is my approach: Because of $\sigma(E^*, E)$, we have the following relationship: $$\sigma(F^*, F) \ \ \xrightarrow[]{\text{???}} \ \ \sigma(E^*, E) \ \ \xrightarrow[]{\text{cont.}} \ \ \mathbb R$$
Now, I want to define a mapping from $F^*$ to $\mathbb R$ by: $$f \mapsto \langle T^*f, x \rangle = \langle f, Tx \rangle$$ So, if this is continuous from $\sigma(F^*, F)$ to $\mathbb R$, then $???$ above can be replaced with "cont." also. How do I do that?