Definition 13.2.1 (Banach Lattice). Let $(E, \normn{\cdot}_{E})$ be a Banach space over $\real$ and $\le$ be a partial order on $E$, then $(E, \normn{\cdot}_{E}, \le)$ is a Banach lattice if for any $x, y \in E$, $|x| \le |y|$ implies that $\normn{x}_{E} \le \normn{x}_{E}$.