2
$\begingroup$

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 $V^*$.

$\endgroup$
  • $\begingroup$ What is $\sigma(V^*,V)$? $\endgroup$ – Fimpellizieri May 11 '16 at 5:03
  • $\begingroup$ The weak* topology. I will add this information in the question. Thanks @Fimpellizieri. $\endgroup$ – Irddo May 11 '16 at 5:05
2
$\begingroup$

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 $e\in E$, $$ (T^*f_n)(e)=f_n(Te)\to f(Te)=(T^*f)(e). $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.