Every nonempty compact convex subset of a Hausdorff locally convex space is the closed convex hull of its extreme points. In particular, it has an extreme point.
A nonempty compact face of a convex set in a Hausdorff locally convex space contains an extreme point. Chains of compact faces have nonempty compact intersection. A minimal face supplied by the Zorn lemma must be a singleton, since a separating linear functional would otherwise expose a proper maximizer face. This is the existence step in the Krein-Milman theorem.
If a compact convex set is the closed convex hull of a subset , every extreme point lies in the closure of . A neighborhood avoiding around a putative missing extreme point gives finitely many closed convex caps covering and excluding that point. The convex hull of a finite union of compact convex sets is compact: group the terms by their set and use the simplex parametrization. The point must lie in that hull, contrary to extremality. The closures are in the given Hausdorff locally convex topology.
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The Krein–Milman theorem is a fundamental result in convex analysis and functional analysis, particularly dealing with convex sets in the context of topological vector spaces. The theorem essentially provides a characterization of convex compact sets.