In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form for some ), the supremum ' and the infimum both exist and are elements of 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, [1] [2] 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. [1]
Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
The order dual of a vector lattice is an order complete vector lattice under its canonical ordering. [1]
If is a locally convex topological vector lattice then the strong dual is an order complete locally convex topological vector lattice under its canonical order. [3]
Every reflexive locally convex topological vector lattice is order complete and a complete TVS. [3]
If is an order complete vector lattice then for any subset is the ordered direct sum of the band generated by and of the band of all elements that are disjoint from [1] For any subset of the band generated by is [1] If and are lattice disjoint then the band generated by contains and is lattice disjoint from the band generated by which contains [1]
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form for some ), the supremum ' and the infimum both exist and are elements of 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, [1] [2] 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. [1]
Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
The order dual of a vector lattice is an order complete vector lattice under its canonical ordering. [1]
If is a locally convex topological vector lattice then the strong dual is an order complete locally convex topological vector lattice under its canonical order. [3]
Every reflexive locally convex topological vector lattice is order complete and a complete TVS. [3]
If is an order complete vector lattice then for any subset is the ordered direct sum of the band generated by and of the band of all elements that are disjoint from [1] For any subset of the band generated by is [1] If and are lattice disjoint then the band generated by contains and is lattice disjoint from the band generated by which contains [1]