convergence for the weak-* topology

Question

Let E be a Banach space.

Let $(x^∗_n )$ be a sequence in $E^∗$ verifying $(<x^∗_n , x>)$ converges for any $x ∈ E$.

Prove that $\exists x^∗ ∈ E^∗: (x^∗_n )$ converges vers $x^*$ for the weak-∗ topology.

The solution I have states that it is a corollary of the Banach Steinhaus theorem but I don't see how it is related and I am not aware of such a corollary.

Many thanks for your help.

Answer 1

Okay so here's how you go about it.

Your condition says $ x_n^*(x)$ is convergent and in particular bounded for any $x \in E$

So the uniform Boundedness Principle gives you $$ sup_n ||x_n^*|| <\infty$$ .

Now if you define $$x^* :E \rightarrow \mathbb K $$ $$ x\mapsto lim_n \ x_n^*(x)$$ then this map is bounded since $ ||x^*(x) || \leq sup_n ||x_n^*|| $

Also $x_n^* \rightarrow x^* $ as $n\rightarrow \infty$ by definition of $weak^*$ topology.