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 C that is order bounded in A (i.e. is contained in an interval [a, b] := { z ∈ X : a ≤ z and z ≤ b } for some a and b belonging to 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,[1][2] 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.[1]

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.[1]
• If X is a locally convex topological vector lattice then the strong dual ${\displaystyle X_{b}^{\prime }}$ is an order complete locally convex topological vector lattice under its canonical order.[3]
• Every reflexive locally convex topological vector lattice is order complete and a complete TVS.[3]

Properties

• If X is an order complete vector lattice then for any subset S of X, X is the ordered direct sum of the band generated by A and of the band ${\displaystyle A^{\perp }}$ of all elements that are disjoint from A.[1] For any subset A of X, the band generated by A is ${\displaystyle A^{\perp \perp }}$.[1] 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.[1]

See also

• Vector lattice

References

1. Schaefer & Wolff 1999, pp. 204–214.
2. Narici & Beckenstein 2011, pp. 139-153.
3. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.