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 $A$ (meaning contained in an interval, which is a set of the form $$, 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, 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.

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.

If $X$ is a locally convex topological vector lattice then the strong dual $X^_b$ is an order complete locally convex topological vector lattice under its canonical order.

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

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 $A^$ of all elements that are disjoint from $A.$ For any subset $A$ of $X,$ the band generated by $A$ is $A^.$ If $x$ and $y$ are lattice disjoint then the band generated by $\,$ contains $y$ and is lattice disjoint from the band generated by $\,$ which contains $x.$

See also

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References

Bibliography

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{{Ordered topological vector spaces

Functional analysis