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 {{math|(X,‖·‖)}} is a complete normed vector space with a lattice order, $\leq$ , such that for all {{math|x, y ∈ X}}, the implication

x|\leq|y

\Rightarrow{\|x\|\leq\|y\|} holds, where the absolute value {{math|{{pipe}}·{{pipe}}}} 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."{{sfn|Birkhoff|1948|p=246}} In particular:

{{math|{{mathbb|R}}}}, together with its absolute value as a norm, is a Banach lattice.
Let {{mvar|X}} be a topological space, {{mvar|Y}} a Banach lattice and {{math|𝒞(X,Y)}} the space of continuous bounded functions from {{mvar|X}} to {{mvar|Y}} with norm $\|f\|_{\infty} = \sup_{x \in X} \|f(x)\|_Y\text{.}$ Then {{math|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.Kania, Tomasz (12 April 2017). [https://math.stackexchange.com/a/2230649 Answer] to "Banach space that is not a Banach lattice" (accessed 13 August 2022). Mathematics StackExchange. StackOverflow.

Properties

The continuous dual space of a Banach lattice is equal to its order dual.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

Every Banach lattice admits a continuous approximation to the identity.{{sfn|Birkhoff|1948|p=251}}

Abstract (L)-spaces

A Banach lattice satisfying the additional condition ${f,g\geq0}\Rightarrow\|f+g\|=\|f\|+\|g\|$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of {{math|L¹({{closed-closed|0,1}})}}.{{sfn|Birkhoff|1948|pp=250,254}} The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.{{sfn|Birkhoff|1948|pp=269-271}}

Footnotes

Bibliography

{{cite book|last=Abramovich|first=Yuri A.|author2=Aliprantis, C. D.|year=2002|title=An Invitation to Operator Theory|series= Graduate Studies in Mathematics|volume= 50|publisher=American Mathematical Society|location=|isbn=0-8218-2146-6}}
{{cite book|series=AMS Colloquium Publications 25|title=Lattice Theory|last=Birkhoff|first=Garrett|author-link=Garrett Birkhoff|edition=Revised|publisher=AMS|location=New York City|year=1948|hdl=2027/iau.31858027322886 |via=HathiTrust|url=https://hdl.handle.net/2027/iau.31858027322886}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}

Category:Functional analysis

Category:Order theory