Exercise 7, § 2

Let E be a Riesz space of finite dimension $n$.

a) If E is archimedean ($§ 1$, Exer. 5), show that the cone P of elements $\geq 0$ of E is closed and has an interior point. From this, deduce that the intersection of the support hyperplanes of P reduces to 0 (make use of the fact that $P \cap (-P) = \{0\}$).

b) Using a) and Exer. 6, show that every archimedean Riesz space of dimension $n$ is isomorphic to the product space $\mathbf{R}^n$ (cf. $§ 1$, Exer. 13 e)).

c) Give an example of a totally ordered Riesz space of dimension 2.