order bound dual

An element $g$ of the order bound dual of $X$ is called positive if $x\; \backslash geq\; 0$ implies $\backslash operatorname\{Re\}(f(x))\; \backslash geq\; 0.$

The positive elements of the order bound dual form a cone that induces an ordering on $X^\{\backslash operatorname\{b\}\}$ called the {{visible anchor|canonical ordering}}.

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.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

The order bound dual of an ordered vector spaces contains its order dual.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

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.{{sfn|Schaefer|Wolff|1999|pp=204–214}}

Suppose $X$ is a vector lattice and $f$ and $g$ are order bounded linear forms on $X.$

Then for all $x\; \backslash in\; X,${{sfn|Schaefer|Wolff|1999|pp=204–214}}

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

• {{annotated link|Algebraic dual space}}
• {{annotated link|Continuous dual space}}
• {{annotated link|Dual space}}
• {{annotated link|Order dual (functional analysis)}}

{{reflist|group=note}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Ordered topological vector spaces}}

{{Functional analysis}}

Category:Functional analysis