Krein–Milman theorem

The Krein–Milman theorem states that every compact convex subset of a locally convex Hausdorff topological vector space is the closed convex hull of its extreme points.

Locally convex topological vector space: A topological vector space whose topology is generated by a family of seminorms. This is the ambient setting in which the theorem guarantees enough extreme points.
Continuous linear functional: A linear map from a topological vector space to the scalar field that is continuous. Functionals separate points and define the maximizing faces used in proof strategies.
Choquet theory: The study of representing points of compact convex sets by probability measures on their extreme points. Choquet theory strengthens convex-hull generation into measure-valued representations.
Carathéodory's theorem: A theorem that points in a finite-dimensional convex hull need at most one more point than the dimension. It gives finite convex representations under finite-dimensional assumptions, unlike the closed-hull conclusion here.
Compact space: A topological space in which every open cover has a finite subcover. Compactness supplies the key finiteness and existence properties in the theorem.
Supporting hyperplane: A hyperplane that meets a convex set while leaving the set in one closed half-space. Supporting hyperplanes expose boundary structure that helps identify extreme points.
Banach–Alaoglu theorem: The theorem that the closed unit ball in a dual space is compact in the weak-star topology. It supplies compact convex sets where Krein–Milman produces extreme points.
Minkowski theorem (convex geometry): A finite-dimensional result expressing compact convex sets as convex hulls of their extreme points. This finite-dimensional counterpart avoids the general theorem's topological-vector-space framework.
Convex set: A subset of a real vector space containing the segment between any two of its points. The theorem reconstructs such a set using convex combinations of its extreme points.
Face (convex geometry): A convex subset of a convex set that contains both endpoints whenever it contains an interior point of their segment. Maximizers of a linear functional form faces, which can be narrowed toward extreme points.

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Convex geometryRelated: It shows how extreme points determine an entire compact convex set.