Let 
be a locally convex topological
vector space and let 
be a compact subset
of 
.
In functional analysis, Milman's theorem is
a result which says that if 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 
.
