Krein-Milman theorem

Theorem.

Let $X$ be a locally convex topological vector space, and let $K\subset X$ be a compact convex subset (http://planetmath.org/ConvexSet). Then $K$ is the closed convex hull of its extreme points.

The closed convex hull above is defined as the intersection of all closed convex subsets of $X$ that contain $K$. This turns out to be the same as the closure of the convex hull in a topological vector space.

References

• 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988
Title Krein-Milman theorem KreinMilmanTheorem 2013-03-22 14:24:58 2013-03-22 14:24:58 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 46A03 msc 52A07 msc 52A99