Order complete

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, 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. The order dual of a vector lattice is an order complete vector lattice under its canonical ordering. 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. Every reflexive locally convex topological vector lattice is order complete and a complete TVS.