Exercise 5, § 2

Suppose that the vector space E has an enumerably infinite basis $(e_n)_{n \in \mathbf{N}}$. Let C be the set of points $x = \sum_{n} \xi_n e_n$ such that for the largest index $n$ for which $\xi_n \neq 0$, we have $\xi_n > 0$. Show that C is a pointed convex cone such that $C \cap (-C) = \{0\}$ and $C \cup (-C) = E$; deduce that C is the set of elements of E that are $\geq 0$ for an order structure that is compatible with the vector space structure of E and for which E is linearly ordered. Show that on this ordered vector space the only linear positive form is 0.