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

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

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.

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.

is an order complete vector lattice then for any subset

is the ordered direct sum of the band generated by

of all elements that are disjoint from

the band generated by

are lattice disjoint then the band generated by

and is lattice disjoint from the band generated by