Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the closed convex hull of its extreme points. By using the Riesz-Kakutani representation theorem it is easy to see that the pure states (extreme points of $S(A)$) are exactly the point evaluations.
Is there an adhoc proof of this fact ?