Order bound dual

Template:Short description In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space $X$ is the set of all linear functionals on $X$ that map order intervals, which are sets of the form $[a,b]:=\{x\in X:a\le x\text{ and }x\le b\},$ to bounded sets.Template:Sfn The order bound dual of $X$ is denoted by $X^{\mathrm{b}}.$ This space plays an important role in the theory of ordered topological vector spaces.

Canonical ordering

An element $g$ of the order bound dual of $X$ is called positive if $x\ge 0$ implies $\mathrm{Re}(f(x))\ge 0.$ The positive elements of the order bound dual form a cone that induces an ordering on $X^{\mathrm{b}}$ called the Template:Visible anchor. If $X$ is an ordered vector space whose positive cone $C$ is generating (meaning $X=C-C$) then the order bound dual with the canonical ordering is an ordered vector space.Template:Sfn

Properties

The order bound dual of an ordered vector spaces contains its order dual.Template:Sfn If the positive cone of an ordered vector space $X$ is generating and if for all positive $x$ and $x$ we have $[0,x]+[0,y]=[0,x+y],$ then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.Template:Sfn

Suppose $X$ is a vector lattice and $f$ and $g$ are order bounded linear forms on $X.$ Then for all $x\in X,$Template:Sfn

1. $\sup (f,g)(|x|)=\sup \{f(y)+g(z):y\ge 0,z\ge 0,\text{ and }y+z=|x|\}$
2. $\inf (f,g)(|x|)=\inf \{f(y)+g(z):y\ge 0,z\ge 0,\text{ and }y+z=|x|\}$
3. $|f|(|x|)=\sup \{f(y-z):y\ge 0,z\ge 0,\text{ and }y+z=|x|\}$
4. $|f(x)|\le |f|(|x|)$
5. if $f\ge 0$ and $g\ge 0$ then $f$ and $g$ are lattice disjoint if and only if for each $x\ge 0$ and real $r>0,$ there exists a decomposition $x=a+b$ with $a\ge 0,b\ge 0,\text{ and }f(a)+g(b)\le r.$

See also

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References

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces