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In mathematics, specifically in [[Order theory|order theory]] and [[Functional analysis|functional analysis]], the '''order bound dual''' of an [[Ordered vector space|ordered vector space]] <math>X</math> is the set of all linear functionals on <math>X</math> that map order intervals, which are sets of the form <math>[a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \},</math> to bounded sets.{{sfn|Schaefer|Wolff|1999|pp=204–214}} 
The order bound dual of <math>X</math> is denoted by <math>X^{\operatorname{b}}.</math>  This space plays an important role in the theory of [[Ordered topological vector space|ordered topological vector space]]s.

== Canonical ordering ==

An element <math>g</math> of the order bound dual of <math>X</math> is called '''positive''' if <math>x \geq 0</math> implies <math>\operatorname{Re}(f(x)) \geq 0.</math>
The positive elements of the order bound dual form a cone that induces an ordering on <math>X^{\operatorname{b}}</math> called the '''{{visible anchor|canonical ordering}}'''.
If <math>X</math> is an [[Ordered vector space|ordered vector space]] whose positive cone <math>C</math> is generating (meaning <math>X = C - C</math>) then the order bound dual with the canonical ordering is an ordered vector space.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

== Properties ==

The order bound dual of an ordered vector spaces contains its [[Order dual (functional analysis)|order dual]].{{sfn|Schaefer|Wolff|1999|pp=204–214}} 
If the positive cone of an [[Ordered vector space|ordered vector space]] <math>X</math> is generating and if for all positive <math>x</math> and <math>x</math> we have <math>[0, x] + [0, y] = [0, x + y],</math> then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

Suppose <math>X</math> is a vector lattice and <math>f</math> and <math>g</math> are order bounded linear forms on <math>X.</math>
Then for all <math>x \in X,</math>{{sfn|Schaefer|Wolff|1999|pp=204–214}}
# <math>\sup(f, g)(|x|) = \sup \{ f(y) + g(z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \}</math>
# <math>\inf(f, g)(|x|) = \inf \{ f(y) + g(z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \}</math>
# <math>|f|(|x|) = \sup \{ f(y - z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \}</math>
# <math>|f(x)| \leq |f|(|x|)</math>
# if <math>f \geq 0</math> and <math>g \geq 0</math> then <math>f</math> and <math>g</math> are [[Lattice disjoint|lattice disjoint]] if and only if for each <math>x \geq 0</math> and real <math>r > 0,</math> there exists a decomposition <math>x = a + b</math> with <math>a \geq 0, b \geq 0, \text{ and } f(a) + g(b) \leq r.</math>

== See also ==

* {{annotated link|Dual space}}
* {{annotated link|Order dual (functional analysis)}}

==References==

{{reflist|group=note}}
{{reflist}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->

{{Functional analysis}}

[[Category:Functional analysis]]

{{Sourceattribution|Order bound dual}}