Order bound dual

In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space ${\displaystyle X}$ is the set of all linear functionals on ${\displaystyle X}$ that map order intervals, which are sets of the form ${\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\},}$ to bounded sets.^[1] The order bound dual of ${\displaystyle X}$ is denoted by ${\displaystyle X^{\operatorname {b} }.}$ This space plays an important role in the theory of ordered topological vector spaces.

Canonical ordering

An element ${\displaystyle g}$ of the order bound dual of ${\displaystyle X}$ is called positive if ${\displaystyle x\geq 0}$ implies ${\displaystyle \operatorname {Re} (f(x))\geq 0.}$ The positive elements of the order bound dual form a cone that induces an ordering on ${\displaystyle X^{\operatorname {b} }}$ called the canonical ordering. If ${\displaystyle X}$ is an ordered vector space whose positive cone ${\displaystyle C}$ is generating (meaning ${\displaystyle X=C-C}$) 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 ${\displaystyle X}$ is generating and if for all positive ${\displaystyle x}$ and ${\displaystyle x}$ we have ${\displaystyle [0,x]+[0,y]=[0,x+y],}$ then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.^[1]

Suppose ${\displaystyle X}$ is a vector lattice and ${\displaystyle f}$ and ${\displaystyle g}$ are order bounded linear forms on ${\displaystyle X.}$ Then for all ${\displaystyle x\in X,}$^[1]

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

See also

• Algebraic dual space – In mathematics, vector space of linear formsPages displaying short descriptions of redirect targets
• Continuous dual space – In mathematics, vector space of linear formsPages displaying short descriptions of redirect targets
• Dual space – In mathematics, vector space of linear forms
• Order dual (functional analysis)

References

1. Schaefer & Wolff 1999, pp. 204–214.

• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.