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, ${\displaystyle \leq }$, 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 $${\displaystyle |x|=x\vee -x:=\sup\{x,-x\}{\text{.}}}$$

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:

• 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 $${\displaystyle \|f\|_{\infty }=\sup _{x\in X}\|f(x)\|_{Y}{\text{.}}}$$ 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 $${\displaystyle {f,g\geq 0}\Rightarrow \|f+g\|=\|f\|+\|g\|}$$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of L¹([0,1]).^[5] The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.^[6]

See also

• Banach space – Normed vector space that is complete
• Normed vector lattice
• Riesz space – Partially ordered vector space, ordered as a lattice
• Lattice (order) – Set whose pairs have minima and maxima