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{.}}

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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.

v t e Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Ordered vector space Partially ordered Positive cone Riesz space Upper set Young's lattice

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