# Banach lattice

Short description: Banach space with a compatible structure of a 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, such that for all x, yX, the implication $\displaystyle{ {|x|\leq|y|}\Rightarrow{\|x\|\leq\|y\|} }$ holds, where the absolute value |·| is defined as $\displaystyle{ |x| = x \vee -x := \sup\{x, -x\}\text{.} }$

## 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."[1] In particular:

• Template:Mathbb, 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 $\displaystyle{ \|f\|_{\infty} = \sup_{x \in X} \|f(x)\|_Y\text{.} }$ Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order: $\displaystyle{ {f \leq g}\Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text{.} }$

Examples of non-lattice Banach spaces are now known; James' space is one such.[2]

## Properties

The continuous dual space of a Banach lattice is equal to its order dual.[3]

Every Banach lattice admits a continuous approximation to the identity.[4]

## Abstract (L)-spaces

A Banach lattice satisfying the additional condition $\displaystyle{ {f,g\geq0}\Rightarrow\|f+g\|=\|f\|+\|g\| }$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of L1([0,1]).[5] The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.[6]