Krein–Milman theorem

Krein–Milman theorem Given a convex shape {displaystyle K} (light blue) and its set of extreme points {displaystyle B} (red), the convex hull of {displaystyle B} is {displaystyle K.} In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs).

Krein–Milman theorem — A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points.

This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane {displaystyle mathbb {R} ^{2}.} Contents 1 Statement and definitions 1.1 Preliminaries and definitions 1.2 Statement 2 More general settings 3 Related results 4 Relation to the axiom of choice 5 History 6 See also 7 Citations 8 Bibliography Statement and definitions Preliminaries and definitions A convex set in light blue, and its extreme points in red.

Throughout, {displaystyle X} will be a real or complex vector space.

For any elements {displaystyle x} and {displaystyle y} in a vector space, the set {displaystyle [x,y]:={tx+(1-t)y:0leq tleq 1}} is called the closed line segment or closed interval between {displaystyle x} and {displaystyle y.} The open line segment or open interval between {displaystyle x} and {displaystyle y} is {displaystyle (x,x):=varnothing } when {displaystyle x=y} while it is {displaystyle (x,y):={tx+(1-t)y:0

Si quieres conocer otros artículos parecidos a Krein–Milman theorem puedes visitar la categoría Convex hulls.