Locally convex vector lattice

In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space.^[1] LCVLs are important in the theory of topological vector lattices.

Lattice semi-norms

The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm $p$ such that $|y|\leq |x|$ implies $p(y)\leq p(x).$ The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.^[1]

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets.^[1]

The strong dual of a locally convex vector lattice $X$ is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of $X$ ; moreover, if $X$ is a barreled space then the continuous dual space of $X$ is a band in the order dual of $X$ and the strong dual of $X$ is a complete locally convex TVS.^[1]

If a locally convex vector lattice is barreled then its strong dual space is complete (this is not necessarily true if the space is merely a locally convex barreled space but not a locally convex vector lattice).^[1]

If a locally convex vector lattice $X$ is semi-reflexive then it is order complete and $X_{b}$ (that is, $\left(X,b\left(X,X^{\prime }\right)\right)$ ) is a complete TVS; moreover, if in addition every positive linear functional on $X$ is continuous then $X$ is of $X$ is of minimal type, the order topology $\tau _{\operatorname {O} }$ on $X$ is equal to the Mackey topology $\tau \left(X,X^{\prime }\right),$ and $\left(X,\tau _{\operatorname {O} }\right)$ is reflexive.^[1] Every reflexive locally convex vector lattice is order complete and a complete locally convex TVS whose strong dual is a barreled reflexive locally convex TVS that can be identified under the canonical evaluation map with the strong bidual (that is, the strong dual of the strong dual).^[1]

If a locally convex vector lattice $X$ is an infrabarreled TVS then it can be identified under the evaluation map with a topological vector sublattice of its strong bidual, which is an order complete locally convex vector lattice under its canonical order.^[1]

If $X$ is a separable metrizable locally convex ordered topological vector space whose positive cone $C$ is a complete and total subset of $X,$ then the set of quasi-interior points of $C$ is dense in $C.$ ^[1]

Theorem^[1] — Suppose that $X$ is an order complete locally convex vector lattice with topology $\tau$ and endow the bidual $X^{\prime \prime }$ of $X$ with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of $X^{\prime }$ ) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:

The evaluation map $X\to X^{\prime \prime }$ induces an isomorphism of $X$ with an order complete sublattice of $X^{\prime \prime }.$
For every majorized and directed subset $S$ of $X,$ the section filter of $S$ converges in $(X,\tau )$ (in which case it necessarily converges to $\sup S$ ).
Every order convergent filter in $X$ converges in $(X,\tau )$ (in which case it necessarily converges to its order limit).

Corollary^[1] — Let $X$ be an order complete vector lattice with a regular order. The following are equivalent:

$X$ is of minimal type.
For every majorized and direct subset $S$ of $X,$ the section filter of $S$ converges in $X$ when $X$ is endowed with the order topology.
Every order convergent filter in $X$ converges in $X$ when $X$ is endowed with the order topology.

Moreover, if $X$ is of minimal type then the order topology on $X$ is the finest locally convex topology on $X$ for which every order convergent filter converges.

If $(X,\tau )$ is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces $\left(X_{\alpha }\right)_{\alpha \in A}$ and a family of $A$ -indexed vector lattice embeddings $f_{\alpha }:C_{\mathbb {R} }\left(K_{\alpha }\right)\to X$ such that $\tau$ is the finest locally convex topology on $X$ making each $f_{\alpha }$ continuous.^[2]

Examples

Every Banach lattice, normed lattice, and Fréchet lattice is a locally convex vector lattice.

References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Schaefer & Wolff 1999, pp. 234–242.
^ Schaefer & Wolff 1999, pp. 242–250.

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 ⁱ ^j ^k Schaefer & Wolff 1999, pp. 234–242.

[FOOTNOTESchaeferWolff1999242–250-2] Schaefer & Wolff 1999, pp. 242–250.

[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

Lattice semi-norms

Properties

Examples

See also

References

Bibliography