Krein–Milman theorem

The Krein - Milman (after Mark Grigoryevich Krein and David Milman ) is a statement from the mathematical branch of functional analysis.

Statement

If a locally convex space and a compact and convex subset of it, so is equal to the closed convex hull of its extreme points.

This set has a partial reversal, which is often called set of Milman: Is a compact, convex set and is such that the closed convex hull of is, the conclusion must of all extreme points of contain.

The set of Choquet exacerbated the Krein - Milman. In finite-dimensional spaces is considered with the Minkowski theorem and the theorem of Carathéodory even much sharper, dimension-dependent statement.

Application

The Banach space of real or complex zero sequences with the supremum is not a dual space.

If he were a dual space, the unit ball would be compact by the theorem of Banach - Alaoglu in the weak -* topology, would thus according to the above set of Krein - Milman extreme points. But is an arbitrary point of the unit sphere, then there's an index, because the sequence converges to 0 is now defined by and for, it is and and that is the arbitrarily given point is not an extreme point. So, the unit ball of no extreme points and therefore can not be a dual space.