# Q: Exercise 3.10 in Brezis' Functional Analysis

I'm trying to solve exercise 3.10 (page 81) from H. Brezis' "Functional Analysis, Sobolev Spaces and PDE".

Let be $E$ and $F$ Banach space such that $T\in \mathcal{L}(E,F)$ and $T^*\in\mathcal{L}(F^*,E^*)$. Prove that $T^*$ is continuous form $F^*$ equipped with $\sigma(F^*,F)$ into $E^*$ equipped with $\sigma(E^*,E)$.

Can someone, please, give me a hint?

Thank you!

If $V$ is a Banach space, then $\sigma (V^*,V)$ is the weak$^*$ topology in $V^*$.

• What is $\sigma(V^*,V)$? – Fimpellizieri May 11 '16 at 5:03
• The weak* topology. I will add this information in the question. Thanks @Fimpellizieri. – Irddo May 11 '16 at 5:05

It is just about writing the definitions. You have to show that if $\{f_n\}$ is a net in $F^*$ with $f_n\to f$ in $\sigma(F^*,F)$, then $T^*f_n\to T^*f$ in $\sigma(E^*,E)$.
You have, for $e\in E$, $$(T^*f_n)(e)=f_n(Te)\to f(Te)=(T^*f)(e).$$