Definition 13.1.1 (Ordered Vector Space). Let $E$ be a vector space over $\real$ and $\le$ be a partial order on $E$, then $(E, \le)$ is a ordered vector space if

1. For any $x, y, z \in E$ with $x \le y$, $x + z \le y + z$.

2. For any $x, y \in E$ and $\lambda > 0$, $x \le y$ implies that $\lambda x \le \lambda y$.