Definition 13.1.7 (Order Dual). Let $(E, \le)$ be an ordered vector space and $\Phi^{+} \in \hom(E; \real)$, then $\Phi^{+}$ is positive if for any $x \in E$ with $x \ge 0$, $\Phi^{+}(x) \ge 0$. The subspace $E^{+} \subset \hom(E; \real)$ generated by the positive linear functionals on $E$ is the order dual of $E$.