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 int ...
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, norm, topology, etc.) and the linear functions defined o ...
, the order bound dual 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 the set of all
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s on
that map order intervals, which are sets of the form
to bounded sets.
The order bound dual of
is denoted by
This space plays an important role in the theory of
ordered topological vector spaces.
Canonical ordering
An element
of the order bound dual of
is called positive if
implies
The positive elements of the order bound dual form a cone that induces an ordering on
called the .
If
is 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 ...
whose positive cone
is generating (meaning
) then the order bound dual with the canonical ordering is an ordered vector space.
Properties
The order bound dual of an ordered vector spaces contains its
order dual.
If the positive cone 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 generating and if for all positive
and
we have
then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.
Suppose
is 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 ''Su ...
and
and
are order bounded linear forms on
Then for all
#
#
#
#
# if
and
then
and
are
lattice disjoint if and only if for each
and real
there exists a decomposition
with
See also
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References
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{{Functional analysis
Functional analysis