Definition 17.2.2 (Order Dual).label Let $(E, \le)$ be an ordered vector space over $K \in \RC$ and $\Phi^{+} \in \hom(E; K)$, then $\Phi^{+}$ is positive if for any $x \in E$ with $x \ge 0$, $\text{Re}\dpn{x, \Phi^+}{E}\ge 0$. The subspace $E^{+} \subset \hom(E; K)$ generated by the positive linear functionals on $E$ is the order dual of $E$.