The Krein-Milman theorem in functional analysis 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 vector space 
separates points. This condition
implies that 
is a T2-space, but the converse need
not hold. 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.
