In mathematics, specifically in order theory and functional analysis, a subset A {\displaystyle A} of an ordered vector space is said to be order complete in X {\displaystyle X} if for every non-empty subset S {\displaystyle S} of X {\displaystyle X} that is order bounded in A {\displaystyle A} (meaning contained in an interval, which is a set of the form [ a , b ] := { x ∈ X : a ≤ x and x ≤ b } , {\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\},} for some a , b ∈ A {\displaystyle a,b\in A} ), the supremum sup S {\displaystyle \sup S} and the infimum inf S {\displaystyle \inf S} both exist and are elements of A . {\displaystyle 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 {\displaystyle X} is a locally convex topological vector lattice then the strong dual X b ′ {\displaystyle X_{b}^{\prime }} 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 {\displaystyle X} is an order complete vector lattice then for any subset A ⊆ X , {\displaystyle A\subseteq X,} X {\displaystyle X} is the ordered direct sum of the band generated by A {\displaystyle A} and of the band A ⊥ {\displaystyle A^{\perp }} of all elements that are disjoint from A . {\displaystyle A.} For any subset A {\displaystyle A} of X , {\displaystyle X,} the band generated by A {\displaystyle A} is A ⊥⊥ . {\displaystyle A^{\perp \perp }.} If x {\displaystyle x} and y {\displaystyle y} are lattice disjoint then the band generated by { x } , {\displaystyle \{x\},} contains y {\displaystyle y} and is lattice disjoint from the band generated by { y } , {\displaystyle \{y\},} which contains x . {\displaystyle x.}
See also Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets
References
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.
