> **Krein-Milman Theorem**. In a Hausdorff, locally convex topological vector space (my one), a compact (my two) convex (my three) subset is the closed convex hull of its extreme points.

It has wonderful applications. For instance, that $L^1({\mathbb R}^n)$ is not the dual of a Banach space.

> **Baire Theorem**. In a complete metric (one) space, a denumerable (two) intersection of dense (three) subsets in dense.

It is used in the proof of

> **Banach Theorem**. Let $E$ be a Banach space (one), $F$ be a Banach space (two), $f:E\rightarrow F$ be linear, bounded (three). Then $f$ is *open* (the image of the unit ball is a neighborhood of $0_F$).