Order bound dual
In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space [math]\displaystyle{ X }[/math] is the set of all linear functionals on [math]\displaystyle{ X }[/math] that map order intervals, which are sets of the form [math]\displaystyle{ [a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \}, }[/math] to bounded sets.[1] The order bound dual of [math]\displaystyle{ X }[/math] is denoted by [math]\displaystyle{ X^{\operatorname{b}}. }[/math] This space plays an important role in the theory of ordered topological vector spaces.
Canonical ordering
An element [math]\displaystyle{ g }[/math] of the order bound dual of [math]\displaystyle{ X }[/math] is called positive if [math]\displaystyle{ x \geq 0 }[/math] implies [math]\displaystyle{ \operatorname{Re}(f(x)) \geq 0. }[/math] The positive elements of the order bound dual form a cone that induces an ordering on [math]\displaystyle{ X^{\operatorname{b}} }[/math] called the canonical ordering. If [math]\displaystyle{ X }[/math] is an ordered vector space whose positive cone [math]\displaystyle{ C }[/math] is generating (meaning [math]\displaystyle{ X = C - C }[/math]) then the order bound dual with the canonical ordering is an ordered vector space.[1]
Properties
The order bound dual of an ordered vector spaces contains its order dual.[1] If the positive cone of an ordered vector space [math]\displaystyle{ X }[/math] is generating and if for all positive [math]\displaystyle{ x }[/math] and [math]\displaystyle{ x }[/math] we have [math]\displaystyle{ [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.[1]
Suppose [math]\displaystyle{ X }[/math] is a vector lattice and [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are order bounded linear forms on [math]\displaystyle{ X. }[/math] Then for all [math]\displaystyle{ x \in X, }[/math][1]
- [math]\displaystyle{ \sup(f, g)(|x|) = \sup \{ f(y) + g(z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \} }[/math]
- [math]\displaystyle{ \inf(f, g)(|x|) = \inf \{ f(y) + g(z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \} }[/math]
- [math]\displaystyle{ |f|(|x|) = \sup \{ f(y - z) : y \geq 0, z \geq 0, \text{ and } y + z = |x| \} }[/math]
- [math]\displaystyle{ |f(x)| \leq |f|(|x|) }[/math]
- if [math]\displaystyle{ f \geq 0 }[/math] and [math]\displaystyle{ g \geq 0 }[/math] then [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are lattice disjoint if and only if for each [math]\displaystyle{ x \geq 0 }[/math] and real [math]\displaystyle{ r \gt 0, }[/math] there exists a decomposition [math]\displaystyle{ x = a + b }[/math] with [math]\displaystyle{ a \geq 0, b \geq 0, \text{ and } f(a) + g(b) \leq r. }[/math]
See also
- Dual space – In mathematics, vector space of linear forms
- Order dual (functional analysis)
References
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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