An Introduction to Functional Analysis

As a corollary of the Hahn-Banach Theorem, we show that any two convex sets can be separated using a linear functional; a key ingredient is the definition of the Minkowski functional of a convex set. This separation theorem allows us to give a characterisation of convex sets in terms of their supporting hyperplanes that will be useful later. We then define the closed convex hull of a set, introduce the notion of extreme points in a convex set, and prove the Krein-Milman Theorem: a non-empty compact convex subset of a Banach space is the closed convex hull of its extreme points.