Order complete

In mathematics, specifically in order theory and functional analysis, a subset $A$ of an ordered vector space is said to be order complete in $X$ if for every non-empty subset $S$ of $X$ that is order bounded in $A$ (meaning contained in an interval, which is a set of the form $[a, b] := {x \in X : a \leq x and x \leq b},$ for some $a, b \in A$ ), the supremum $sup S$ and the infimum $inf S$ both exist and are elements of $A .$ An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum.

Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.

Examples

The order dual of a vector lattice is an order complete vector lattice under its canonical ordering.

If $X$ is a locally convex topological vector lattice then the strong dual ${X_{b}}^{'}$ is an order complete locally convex topological vector lattice under its canonical order.

Every reflexive locally convex topological vector lattice is order complete and a complete TVS.

Properties

If $X$ is an order complete vector lattice then for any subset $A \subseteq X,$ $X$ is the ordered direct sum of the band generated by $A$ and of the band $A^{⊥}$ of all elements that are disjoint from $A .$ For any subset $A$ of $X,$ the band generated by $A$ is $A^{⊥ ⊥} .$ If $x$ and $y$ are lattice disjoint then the band generated by ${x},$ contains $y$ and is lattice disjoint from the band generated by ${y},$ which contains $x .$

References

1 2 3 4 5 6 Schaefer & Wolff 1999, pp. 204–214.
↑ Narici & Beckenstein 2011, pp. 139–153.
1 2 Schaefer & Wolff 1999, pp. 234–239.

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.

[FOOTNOTESchaeferWolff1999204–214-1] 1 2 3 4 5 6 Schaefer & Wolff 1999, pp. 204–214.

[FOOTNOTENariciBeckenstein2011139–153-2] Narici & Beckenstein 2011, pp. 139–153.

[FOOTNOTESchaeferWolff1999234–239-3] 1 2 Schaefer & Wolff 1999, pp. 234–239.

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

Order complete

Examples

Properties

See also

References

Bibliography