Chapter 9 begins with a brief review of the basic notions of convex space theory. It defines various nonstandard notions of convexity and proves a series of lemmas about them that are relevant for establishing the Separating Hyperplane Theorem. After introducing the standard proof of this theorem in a detailed and complete manner, it proceeds to demonstrate a nonstandard version of it. The chapter then explains in detail the meaning and application of the Loeb–Lyapunov Theorem in economic theory. It concludes with a proof of the nonstandard version of the Fan–Glicksberg Fixed Point Theorem and derives Kakutani’s theorem as a corollary.