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 ...
, 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 regularly ordered and its order is called regular if
is
Archimedean ordered and the
order dual of
distinguishes points in
.
Being a regularly ordered vector space is an important property 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 ...
s.
Examples
Every ordered locally convex space is regularly ordered.
The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.
Properties
If
is a regularly ordered
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 ...
then the
order topology on
is the finest topology on
making
into a locally convex
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 ...
.
See also
*
References
Bibliography
*
*
{{Ordered topological vector spaces
Functional analysis