Banach lattice

Template:Short description In the mathematical disciplines of in functional analysis and order theory, a Banach lattice Template:Math is a complete normed vector space with a lattice order, $\le$, such that for all Template:Math, the implication $$|x|\le |y|\Rightarrow \Vert x\Vert \le \Vert y\Vert$$ holds, where the absolute value Template:Math 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."Template:Sfn In particular:

• Template:Math, together with its absolute value as a norm, is a Banach lattice.
• Let Template:Mvar be a topological space, Template:Mvar a Banach lattice and Template:Math the space of continuous bounded functions from Template:Mvar to Template:Mvar with norm $$\Vert f{\Vert }_{\mathrm{\infty }}=\underset{x\in X}{\sup }\Vert f(x){\Vert }_Y\text{.}$$ Then Template:Math is a Banach lattice under the pointwise partial order: $$f\le g\iff (\mathrm{\forall }x\in X)(f(x)\le g(x))\text{.}$$

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.Template:Sfn

Every Banach lattice admits a continuous approximation to the identity.Template:Sfn

Abstract (L)-spaces

A Banach lattice satisfying the additional condition $$f,g\ge 0\Rightarrow \Vert f+g\Vert =\Vert f\Vert +\Vert g\Vert$$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of Template:Math.Template:Sfn The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.Template:Sfn

See also

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Bibliography

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• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces