In mathematics, specifically in

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, a subset

A

of an

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. Definition Given a vector space ''X'' over the real numbers R and a pr ...

is said to be order complete in

X

if for every non-empty subset

S

C

that is order bounded in

A

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

, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

:= \, for some

a, b \in A

), the

supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...

\sup S

' and the

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...

\inf S

both exist and are elements of

A.

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 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 ...

. 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 lattice In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order \,\leq\, making it into vector lattice that is possesses a neighborhood ba ...

Examples

The order dual of a

is an order complete vector lattice under its canonical ordering. If

X

is a

locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...

then the strong dual

X^_b

is an order complete locally convex topological vector lattice under its canonical order. Every reflexive

is order complete and a complete TVS.

Properties

X

is an order complete

then for any subset

S \subseteq X,

X

is the ordered direct sum of the band generated by

A

and of the band

A^

of all elements that are disjoint from

A.

For any subset

A

X,

the band generated by

A

A^.

x

and

y

are

lattice disjoint In mathematics, specifically in order theory and functional analysis, two elements ''x'' and ''y'' of a vector lattice ''X'' are lattice disjoint or simply disjoint if \inf \left\ = 0, in which case we write x \perp y, where the absolute value of '' ...

then the band generated by

\,

contains

y

and is lattice disjoint from the band generated by

\,

which contains

x.

References

Bibliography

* * {{Ordered topological vector spaces Functional analysis

Examples

Properties

See also

References

Bibliography