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 $[a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\},$ 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,^[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 $X_{b}^{\prime }$ 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 $S\subseteq X,$ $X$ is the ordered direct sum of the band generated by $A$ and of the band $A^{\perp }$ of all elements that are disjoint from $A.$ ^[1] For any subset $A$ of $X,$ the band generated by $A$ is $A^{\perp \perp }.$ ^[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]

References

^ ^a ^b ^c ^d ^e ^f Schaefer & Wolff 1999, pp. 204–214.
^ Narici & Beckenstein 2011, pp. 139–153.
^ ^a ^b 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.