Banach lattice

Short description: Banach space with a compatible structure of a 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, 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." 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.

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 [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 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. 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.