Exercise 8, § 5

In a Hausdorff locally convex space, let A and B be two closed convex sets, without common points, such that $C_A \cap C_B = \{ 0 \}$ (II, p. 67, exerc. 14), and such that B is locally compact. Show that there exists a closed hyperplane separating A from B ($cf.$ II, p. 67, exerc. 16). Similarly, if A and B are two closed convex cones of vertex 0, such that $A \cap B = \{ 0 \}$ and B is locally compact, then there exists a closed hyperplane passing through 0 and separating A from B (use lemma 1 of II, p. 39).