In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an extreme point of a convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

 S in a real vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 is a point in S which does not lie in any open line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

 joining two points of S. Intuitively, an extreme point is a "corner" of S.

• The Krein–Milman theorem states that if S is convex and compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   in a locally convex space, then S is the closed convex hull
  Convex hull
  In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
  
   of its extreme points: In particular, such a set has extreme points.

The Krein–Milman theorem is stated for locally convex topological vector spaces. The next theorems are stated for Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s with the Radon–Nikodym property:

• A theorem of Joram Lindenstrauss
  Joram Lindenstrauss
  Joram Lindenstrauss is an Israeli mathematician working in functional analysis. He is professor emeritus of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel.-Biography:...
  
   states that, in a Banach space with the Radon–Nikodym property, a closed
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
   and bounded set
  Bounded set
  In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
  
   has an extreme point. (In infinite-dimensional spaces, the property of compactness
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   is stronger than the joint properties of being closed and being bounded).
• A theorem of Gerald Edgar states that, in a Banach space with the Radon–Nikodym property, a closed and bounded set is the closed convex hull of its extreme points.

Edgar's theorem implies Lindenstrauss's theorem.