A Banach lattice is a complete normed vector lattice such that the ordering and norm are compatible.
A Banach lattice is a complete normed vector lattice such that $|x| \leq |y| \implies \|x\| \leq \|y\|$.
For example, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm and order given by $$(x_1,x_2,\ldots,x_n) \leq (y_1,y_2,\ldots y_n) \iff x_i\leq y_i \text{ for every } 1\leq i\leq n,$$ are Banach lattices. Other examples include $L^p(\Omega) (1\leq p\leq \infty)$, $C(K)$ ($K$: compact), $C_0(\Omega)$ ($\Omega$: locally Hausdorff).
