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, $\leq$, such that for all x, y ∈ X, the implication

$${|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}$$

holds, where the absolute value |·| 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." 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.

Properties

The continuous dual space of a Banach lattice is equal to its order dual.

Every Banach lattice admits a continuous approximation to the identity.

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 L¹([0,1]). The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.

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

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.