Banach lattices are Banach spaces endowed with a partial ordering that is compatible with the norm.

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 $(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots y_n) \iff x_i\leq y_i$ for every $1\leq i\leq n.$ Other examples include $L^p(\Omega) (1\leq p\leq \infty), C(K) (K:$compact), $C_0(\Omega) (\Omega:$ locally Hausdorff).