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:

• , 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]

## Footnotes

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

## Bibliography

• Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. ISBN 0-8218-2146-6.
• Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust.
• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.