In the field of functional analysis, the Krein-Milman theorem is a result which characterizes all (nonempty) compact convex subsets 
of "sufficiently nice" topological
vector spaces 
in terms of the so-called extreme points of 
.
To be more precise, suppose that 
is a topological vector space on which the continuous
dual space 
separates points (i.e., is T2-space). The Krein-Milman
theorem says that every nonempty compact convex set 
in 
is necessarily the closed convex hull of the set 
of its extreme points, i.e., that
 |
Intuitively speaking, the Krein-Milman theorem says that, despite the name "extreme point" being suggestive of a subset which is perhaps relatively small, the actuality
may be that the collection 
is quite large relative to 
.
It should be noted that the Krein-Milman theorem is different from Milman's theorem, a separate result in functional analysis which says that a compact set
in a locally convex topological vector space 
contains every extreme point of 
provided that 
is compact.
