Weak and Weak* Topologies - Main Theorem

Krein-Milman Theorem provides a fundamental result about the structure of compact convex sets in locally convex spaces, showing they are determined by their extreme points.

TheoremKrein-Milman Theorem

Let $X$ be a locally convex topological vector space and $K \subset X$ a compact convex set. Then $K$ is the closed convex hull of its extreme points.

DefinitionExtreme Point

A point $x \in K$ is an extreme point of a convex set $K$ if it cannot be written as a proper convex combination: if $x = tx_1 + (1-t)x_2$ with $x_1, x_2 \in K$ and $t \in (0,1)$ , then $x = x_1 = x_2$ .

ExampleExtreme Points

Unit Ball in $\mathbb{R}^n$ : The extreme points of $B = \{x : \|x\| \leq 1\}$ form the unit sphere $\{x : \|x\| = 1\}$
Simplex: The extreme points of the standard simplex $\{(x_1, \ldots, x_n) : x_i \geq 0, \sum x_i = 1\}$ are the vertices $(1, 0, \ldots, 0), (0, 1, 0, \ldots, 0), \ldots, (0, \ldots, 0, 1)$
Unit Ball in $\ell^1$ : Extreme points are sequences with exactly one entry equal to $\pm 1$ and all others zero
State Space: In quantum mechanics, pure states are extreme points of the convex set of mixed states

TheoremApplication to Measure Theory

Let $K$ be a compact Hausdorff space. The extreme points of the unit ball in $M(K)$ (signed Borel measures with total variation $\leq 1$ ) are exactly the point masses $\pm \delta_x$ for $x \in K$ .

ExampleChoquet Theory

Beyond just existence of extreme points, Choquet theory shows that under certain conditions, every point in a compact convex set can be represented as an integral over extreme points.

For the unit ball in $\ell^1$ , every element can be written as a (possibly infinite) convex combination of extreme points.

Remark

The Krein-Milman Theorem is particularly powerful in weak* topology. For instance, the unit ball in $X^*$ with the weak* topology is compact (by Banach-Alaoglu), so its extreme points generate the entire ball. This principle is used extensively in representation theory and optimization.

This theorem has applications throughout mathematics, from functional analysis to convex geometry, optimization theory, and mathematical physics.