Кейн-Милман теоремасы және экстремалды нүктелердің жабық дөңгелек қабығышасы

On when a space equals the closed convex hull of its extreme points

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). 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

Under the Zermelo–Fraenkel set theory (ZF) axiomatic framework, the axiom of choice (AC) suffices to prove all versions of the Krein–Milman theorem given above, including statement KM and its generalization SKM. The axiom of choice also implies, but is not equivalent to, the Boolean prime ideal theorem (BPI), which is equivalent to the Banach–Alaoglu theorem. Conversely, the Krein–Milman theorem KM together with the Boolean prime ideal theorem (BPI) imply the axiom of choice. In summary, AC holds if and only if both KM and BPI hold. Earlier, proved that if is 3 dimensional then equals the convex hull of the set of its extreme points. This assertion was expanded to the case of any finite dimension by
The Krein–Milman theorem generalizes this to arbitrary locally convex ; however, to generalize from finite to infinite dimensional spaces, it is necessary to use the closure.