Banach lattice

In the mathematical disciplines of in functional analysis and order theory, a Banach lattice $(X,‖\cdot‖)$ is a complete normed vector space with a lattice order, $\leq$ , such that for all $x, y \in X$ , the implication ${|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}$ holds, where the absolute value $|\cdot|$ 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."^[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 $\|f\|_{\infty }=\sup _{x\in X}\|f(x)\|_{Y}{\text{.}}$ Then $𝒞(X, Y)$ is a Banach lattice under the pointwise partial order: ${f\leq g}\Leftrightarrow (\forall x\in X)(f(x)\leq g(x)){\text{.}}$

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

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. 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.