Order complete

In mathematics, specifically in order theory and functional analysis, a subset [math]\displaystyle{ A }[/math] of an ordered vector space is said to be order complete in [math]\displaystyle{ X }[/math] if for every non-empty subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ C }[/math] that is order bounded in [math]\displaystyle{ A }[/math] (meaning contained in an interval, which is a set of the form [math]\displaystyle{ [a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \}, }[/math] for some [math]\displaystyle{ a, b \in A }[/math]), the supremum [math]\displaystyle{ \sup S }[/math]' and the infimum [math]\displaystyle{ \inf S }[/math] both exist and are elements of [math]\displaystyle{ A. }[/math] 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 [math]\displaystyle{ X }[/math] is a locally convex topological vector lattice then the strong dual [math]\displaystyle{ X^{\prime}_b }[/math] 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 [math]\displaystyle{ X }[/math] is an order complete vector lattice then for any subset [math]\displaystyle{ S \subseteq X, }[/math] [math]\displaystyle{ X }[/math] is the ordered direct sum of the band generated by [math]\displaystyle{ A }[/math] and of the band [math]\displaystyle{ A^{\perp} }[/math] of all elements that are disjoint from [math]\displaystyle{ A. }[/math] For any subset [math]\displaystyle{ A }[/math] of [math]\displaystyle{ X, }[/math] the band generated by [math]\displaystyle{ A }[/math] is [math]\displaystyle{ A^{\perp \perp}. }[/math] If [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are lattice disjoint then the band generated by [math]\displaystyle{ \{x\}, }[/math] contains [math]\displaystyle{ y }[/math] and is lattice disjoint from the band generated by [math]\displaystyle{ \{y\}, }[/math] which contains [math]\displaystyle{ x. }[/math]

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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.