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.

  • $\begingroup$ Your arguments are correct. $\endgroup$ – Jochen Apr 1 at 6:55

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