In the mathematical disciplines of in functional analysis and order theory, a Banach lattice (X,‖·‖) is a complete normed vector space with a lattice order, ${\displaystyle \leq }$, such that for all x, y ∈ X, the implication $${\displaystyle {|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}}$$ holds, where the absolute value |·| is defined as $${\displaystyle |x|=x\vee -x:=\sup\{x,-x\}{\text{.}}}$$

Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice." In particular:

Examples of non-lattice Banach spaces are now known; James' space is one such.

The continuous dual space of a Banach lattice is equal to its order dual.

Every Banach lattice admits a continuous approximation to the identity.

A Banach lattice satisfying the additional condition $${\displaystyle {f,g\geq 0}\Rightarrow \|f+g\|=\|f\|+\|g\|}$$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of L¹([0,1]). The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.