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.

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Related concepts (2)

Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied implies implies If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure.

Riesz space

In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem.