Closed unit ball of $X^*$ has extreme points.

Question

Let $X$ be a Banach space. I have to show that the closed unit ball of the dual space $X^*$ has extreme points.

I just used the Banach-Alaoglu and Krein-Milman theorems to prove this but I'm not sure if this is correct because we haven't specified a topology on $X^*$. Because this argument obviously only holds when we consider the weak* topology so if anyone could clarify this that would be greatly appreciated.

Answer 1

The concept of extreme points doesn't depend on which topology you choose. So we could restate Krein-Milman as follows:

Let $V$ be a vector space, and $K \subset V$ a nonempty convex set. Suppose there exists a locally convex topology $\tau$ on $V$ such that $K$ is compact with respect to $\tau$. Then $K$ has extreme points (and indeed $K$ is equal to the $\tau$-closure of the convex hull of its extreme points).

Thus, taking $V = X^*$ and $K$ the closed unit ball, it suffices to come up with some locally convex topology $\tau$ in which $K$ is compact. Thanks to the Banach-Alaoglu theorem, an obvious choice is to take $\tau$ to be the weak-* topology, and then everything works.

So your proof is fine.