Is the set $\overline{\text{conv}}^{w^*} C$ weakly* compact?

Exercise :

Let $$X$$ be a Banach space and $$C \subseteq X^*$$ be $$w^*-$$compact. Is the set $$\overline{\text{conv}}^{w^*} C$$ $$w^*-$$compact ?

Thoughts :

I (think) that I know that $$w^*-$$compact sets are norm bounded. A proof of that statement can be found here.

Would that mean that since $$C$$ is $$w^*-$$compact, then it is $$C \subseteq n B_1^{X^*}$$ for some $$n \in \mathbb N$$ ? Note that I denote $$B_1^{X^*}$$ the unit ball in $$X^*$$.

If that's the case, that would mean that $$\overline{\text{conv}}^{w^*} C \subseteq n B_1^{X^*}$$ and since the unit ball of $$X^*$$ is $$w^*-$$compact by the Banach-Alaoglu Theorem, then $$\overline{\text{conv}}^{w^*} C$$ would be $$w^*-$$compact as well ?

I haven't had big experience with regards to $$w^*$$ topology so I apologise if any part of my intuition is nonsense.

I would very much appreciate any hints or thorough elaborations.

• Your arguments are correct. – Jochen Apr 1 at 6:55