Order complete

Template:Short description 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 $[a,b]:=\{x\in X:a\le x\text{ and }x\le b\},$ 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,Template:SfnTemplate:Sfn 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.Template:Sfn

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.Template:Sfn

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.Template:Sfn

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

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 $A.$Template:Sfn For any subset $A$ of $X,$ the band generated by $A$ is $A^{\perp \perp }.$Template:Sfn 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 $x.$Template:Sfn

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces