Topological vector lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) $X$ that has a partial order $\,\leq \,$ making it into vector lattice that possesses a neighborhood base at the origin consisting of solid sets.^[1] Ordered vector lattices have important applications in spectral theory.

Definition

If $X$ is a vector lattice then by the vector lattice operations we mean the following maps:

the three maps $X$ to itself defined by $x\mapsto |x|$ , $x\mapsto x^{+}$ , $x\mapsto x^{-}$ , and
the two maps from $X\times X$ into $X$ defined by $(x,y)\mapsto \sup _{}\{x,y\}$ and $(x,y)\mapsto \inf _{}\{x,y\}$ .

If $X$ is a TVS over the reals and a vector lattice, then $X$ is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.^[1]

If $X$ is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.^[1]

If $X$ is a topological vector space (TVS) and an ordered vector space then $X$ is called locally solid if $X$ possesses a neighborhood base at the origin consisting of solid sets.^[1] A topological vector lattice is a Hausdorff TVS $X$ that has a partial order $\,\leq \,$ making it into vector lattice that is locally solid.^[1]

Properties

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.^[1] Let ${\mathcal {B}}$ denote the set of all bounded subsets of a topological vector lattice with positive cone $C$ and for any subset $S$ , let $[S]_{C}:=(S+C)\cap (S-C)$ be the $C$ -saturated hull of $S$ . Then the topological vector lattice's positive cone $C$ is a strict ${\mathcal {B}}$ -cone,^[1] where $C$ is a strict ${\mathcal {B}}$ -cone means that $\left\{[B]_{C}:B\in {\mathcal {B}}\right\}$ is a fundamental subfamily of ${\mathcal {B}}$ that is, every $B\in {\mathcal {B}}$ is contained as a subset of some element of $\left\{[B]_{C}:B\in {\mathcal {B}}\right\}$ ).^[2]

If a topological vector lattice $X$ is order complete then every band is closed in $X$ .^[1]

Examples

The L^p spaces ( $1\leq p\leq \infty$ ) are Banach lattices under their canonical orderings. These spaces are order complete for $p<\infty$ .

References

^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 234–242.
^ Schaefer & Wolff 1999, pp. 215–222.

Bibliography

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.

[FOOTNOTESchaeferWolff1999234–242-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 234–242.

[FOOTNOTESchaeferWolff1999215–222-2] Schaefer & Wolff 1999, pp. 215–222.

[1]

[2]

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 Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Definition

Properties

Examples

See also

References

Bibliography