order complete

{{Short description|Property of subsets of ordered vector spaces}}

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,{{sfn|Schaefer|Wolff|1999|pp=204–214}}{{sfn|Narici|Beckenstein|2011|pp=139-153}} 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.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

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.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

If $X$ is a locally convex topological vector lattice then the strong dual $X^{\prime}_b$ is an order complete locally convex topological vector lattice under its canonical order.{{sfn|Schaefer|Wolff|1999|pp=234–239}}

Every reflexive locally convex topological vector lattice is order complete and a complete TVS.{{sfn|Schaefer|Wolff|1999|pp=234–239}}

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 $A.$ {{sfn|Schaefer|Wolff|1999|pp=204–214}} For any subset $A$ of $X,$ the band generated by $A$ is $A^{\perp \perp}.$ {{sfn|Schaefer|Wolff|1999|pp=204–214}} 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 $x.$ {{sfn|Schaefer|Wolff|1999|pp=204–214}}

References

Bibliography

{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}

Category:Functional analysis

order complete

Examples

Properties

See also

References

Bibliography