Milman's theorem in functional analysis states that if 
is a locally convex topological
vector space, 
is a compact subset of
, and the closed convex hull 
of 
is also compact, then 
contains all the extreme
points of 
.
The importance of Milman's theorem is subtle but enormous. One well-known fact from functional analysis is that 
where 
denotes the set of extreme
points of 
.
Ostensibly, however, one may have that 
has extreme points
which are not in 
.
This behavior is considered a pathology, and Milman's theorem states that this pathology
cannot exist whenever 
is compact (e.g., when 
is a subset of a Fréchet
space 
).
Milman's theorem should not be confused with the Krein-Milman theorem which says that every nonempty compact convex set 
in 
necessarily satisfies the identity 
.
