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 open (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 $0_F$).
