Order complete

In mathematics, specifically in order theory and functional analysis, a subset ${\displaystyle A}$ of an ordered vector space is said to be order complete in ${\displaystyle X}$ if for every non-empty subset ${\displaystyle S}$ of ${\displaystyle C}$ that is order bounded in ${\displaystyle A}$ (meaning contained in an interval, which is a set of the form ${\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\},}$ for some ${\displaystyle a,b\in A}$), the supremum ${\displaystyle \sup S}$' and the infimum ${\displaystyle \inf S}$ both exist and are elements of ${\displaystyle A.}$ 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 ${\displaystyle X}$ 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 ${\displaystyle X}$ is an order complete vector lattice then for any subset ${\displaystyle S\subseteq X,}$${\displaystyle X}$ is the ordered direct sum of the band generated by ${\displaystyle A}$ and of the band ${\displaystyle A^{\perp }}$ of all elements that are disjoint from ${\displaystyle A.}$ ^[1] For any subset ${\displaystyle A}$ of ${\displaystyle X,}$ the band generated by ${\displaystyle A}$ is ${\displaystyle A^{\perp \perp }.}$ ^[1] If ${\displaystyle x}$ and ${\displaystyle y}$ are lattice disjoint then the band generated by ${\displaystyle \{x\},}$ contains ${\displaystyle y}$ and is lattice disjoint from the band generated by ${\displaystyle \{y\},}$ which contains ${\displaystyle x.}$ ^[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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN   978-1-4612-7155-0. OCLC   840278135.