Order complete specifically in order theory and functional analysis , a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A , the supremum sup S and the infimum inf S both exist and are elements of A . ordered vector space is called order complete , Dedekind completecomplete 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 . countably order complete if each countable subset that is bounded above has a supremum.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 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 .Properties If X is an order complete vector lattice then for any subset S of X , X is the ordered direct sum of the band generated by A and of the band of all elements that are disjoint from A . For any subset A of X , the band generated by A is. If x and y are lattice disjoint then the band generated by contains y and is lattice disjoint from the band generated by, which contains x .
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