X^* closed under pointwise convergence.

Question

The pointwise limit $\psi$ of a sequence $(\psi_n)$ in the topological dual $X^*$ of a Banachspace $X$ is of course linear. How can I show that it is continous, too?

Answer 1

Since X is a banach space you can use the uniform boundedness theorem.

if $\psi_n(x)\to \psi(x)$ then the sequence $\{|\psi_n(x)|\}_{n \in \Bbb N}$ is bounded,thus there is $M>0$

such that $||\psi_n||\leq M$ $\forall n \in \Bbb N$.Now let $x \in X$,

$|\psi(x)|=\lim_{n\to \infty} |\psi_n(x)| \leq \limsup_{n \to \infty} ||\psi_n||||x||\leq M||x||$. So $\psi$ is bounded , hence continuous.