Hahn–Banach separation theorem

Remarks

For part 1, the set $C = A - B + x_0$, with $x_0 \in B - A$ fixed, is an open convex neighbourhood of $0$ that misses $x_0$. Its Minkowski functional $p$ is sublinear, and the Hahn–Banach theorem extends the functional $\lambda x_0 \mapsto \lambda$ to a linear $f \le p$; then $f < 1 = f(x_0)$ on $C$, which gives the separation. Part 2 reduces to part 1: compactness of $A$ gives a convex open neighbourhood $U$ of $0$ with $(A + U) \cap B = \emptyset$. For a complex space, apply the theorem to the underlying real space and take the real part of a complex functional. The hypotheses matter: in $\R^2$ the closed convex sets $A = \set{(x, y) : y \le 0}$ and $B = \set{(x, y) : x > 0,\ y \ge 1/x}$ are disjoint, but no line separates them strictly, so part 2 needs compactness of one set.

Going deeper

For a student

This is the geometric form of the Hahn–Banach theorem. It shows that a closed convex set in a locally convex space is the intersection of the closed half-spaces that contain it. Hence closed convex sets are closed for the weak topology, which is the key step in Mazur's lemma. It is also the main tool in the proofs of the Krein–Milman theorem, the Goldstine theorem and the bipolar theorem.