Extreme points of the dual unit ball of C(K) ($\{\alpha {\delta }_x:|\alpha |=1,x\in K\}$)

For complex $C(K)$ on a compact Hausdorff space, the extreme points of its dual unit ball are precisely unimodular multiples of Dirac measures. A measure whose variation measure has mass on two disjoint sets splits as a nontrivial convex combination of normalized restrictions and is not extreme. This description is the key to the Banach–Stone theorem.