# Banach lattice

__: Banach space with a compatible structure of a lattice__

**Short description**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*, *y* ∈ *X*, the implication [math]\displaystyle{ {|x|\leq|y|}\Rightarrow{\|x\|\leq\|y\|} }[/math] holds, where the absolute value |·| is defined as [math]\displaystyle{ |x| = x \vee -x := \sup\{x, -x\}\text{.} }[/math]

## 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 [math]\displaystyle{ \|f\|_{\infty} = \sup_{x \in X} \|f(x)\|_Y\text{.} }[/math] Then 𝒞(*X*,*Y*) is a Banach lattice under the pointwise partial order: [math]\displaystyle{ {f \leq g}\Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text{.} }[/math]

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 [math]\displaystyle{ {f,g\geq0}\Rightarrow\|f+g\|=\|f\|+\|g\| }[/math] is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of *L*^{1}([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

## Footnotes

- ↑ Birkhoff 1948, p. 246.
- ↑ Kania, Tomasz (12 April 2017). Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022).
*Mathematics StackExchange*. StackOverflow. - ↑ Schaefer & Wolff 1999, pp. 234–242.
- ↑ Birkhoff 1948, p. 251.
- ↑ Birkhoff 1948, pp. 250,254.
- ↑ Birkhoff 1948, pp. 269-271.

## Bibliography

- Abramovich, Yuri A.; Aliprantis, C. D. (2002).
*An Invitation to Operator Theory*. Graduate Studies in Mathematics.**50**. American Mathematical Society. ISBN 0-8218-2146-6. - Birkhoff, Garrett (1948).
*Lattice Theory*. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. https://hdl.handle.net/2027/iau.31858027322886. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Original source: https://en.wikipedia.org/wiki/Banach lattice.
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