Banach lattice

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, $$\leq$$, such that for all $x, y ∈ X$, the implication $${|x|\leq|y|}\Rightarrow{\|x\|\leq\|y\|}$$ holds, where the absolute value $$ is defined as $$|x| = x \vee -x := \sup\{x, -x\}\text{.}$$

Examples and constructions
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:
 * $\mathbb{R}$, together with its absolute value as a norm, is a Banach lattice.
 * Let $X$ be a topological space, $Y$ a Banach lattice and $𝒞(X,Y)$ the space of continuous bounded functions from $X$ to $Y$ with norm $$\|f\|_{\infty} = \sup_{x \in X} \|f(x)\|_Y\text{.}$$ Then $𝒞(X,Y)$ is a Banach lattice under the pointwise partial order: $${f \leq g}\Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text{.}$$

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

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

Every Banach lattice admits a continuous approximation to the identity.

Abstract (L)-spaces
A Banach lattice satisfying the additional condition $${f,g\geq0}\Rightarrow\|f+g\|=\|f\|+\|g\|$$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of $L^{1}([0,1])$. The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.