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The Krein–Milman theorem

Matthias Engelmann
Foredrag for studerende
Fredag, 30 marts, 2012, at 14:30-15:30, in Aud. D4 (1531-219)
In $n$-dimensional euclidian space $\mathbb{R}^{n}$ the notion of retrieving a convex set $K$ by its extreme points, that is points which cannot be written as the middle point of a straight line segment lying completely in $K$, appears to be a natural idea. A plain triangle in $\mathbb{R}^{2}$ for example has three extreme points (its corners). Connecting these points by straight line segments gives the edges of the triangle while connecting each pair of points from the edges in a similar manner will give the interior of the triangle. 

It is unclear however, if this simple idea extends to infinite dimensional Banach spaces or more generally arbitrairy topological vector spaces $X$ carrying a locally convex topology. In this lecture I will present a first result going into this direction known as the Krein-Milman theorem. It states that for any compact and convex subset of a locally convex Hausdorff space the set of its extreme points $E(K)\neq\emptyset$ and that $K=\overline{\text{conv}(E(K))}$, where the bar denotes the closure with respect to the topology of $X$ and $\text{conv}(E(K))$ denotes the smallest convex set containing $E(K)$.

The proof mainly uses a version of the Hahn-Banach theorems for locally convex sets and Zorn's lemma. I will give a very brief and rather informal introduction to the theory of locally convex spaces and then merely state the main tools needed in the proof of the theorem. If time allows, I will also present some applications of this theorem and give some hints on how the notion of representing a convex compact set by its extreme points can be generalised by Choquet theory.
Kontaktperson: Søren Fuglede Jørgensen