Order dual (functional analysis)

An element ${\displaystyle f}$ of the order 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 dual form a cone that induces an ordering on ${\displaystyle X^{+}}$ called the canonical ordering. If ${\displaystyle X}$ is an ordered vector space whose positive cone ${\displaystyle C}$ is generating (that is, ${\displaystyle X=C-C}$) then the order dual with the canonical ordering is an ordered vector space.[1] The order dual is the span of the set of positive linear functionals on ${\displaystyle X}$.[1]

The order dual is contained in the order bound dual.[1] If the positive cone of an ordered vector space ${\displaystyle X}$ is generating and if ${\displaystyle [0,x]+[0,y]=[0,x+y]}$ holds for all positive ${\displaystyle x}$ and ${\displaystyle y}$, then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.[1]

The order dual of a vector lattice is an order complete vector lattice.[1] The order dual of a vector lattice ${\displaystyle X}$ can be finite dimension (possibly even ${\displaystyle \{0\}}$) even if ${\displaystyle X}$ is infinite-dimensional.[1]

Suppose that ${\displaystyle X}$ is an ordered vector space such that the canonical order on ${\displaystyle X^{+}}$ makes ${\displaystyle X^{+}}$ into an ordered vector space. Then the order bidual is defined to be the order dual of ${\displaystyle X^{+}}$ and is denoted by ${\displaystyle X^{++}}$. If the positive cone of an ordered vector space ${\displaystyle X}$ is generating and if ${\displaystyle [0,x]+[0,y]=[0,x+y]}$ holds for all positive ${\displaystyle x}$ and ${\displaystyle y}$, then ${\displaystyle X^{++}}$ is an order complete vector lattice and the evaluation map ${\displaystyle X\to X^{++}}$ is order preserving.[1] In particular, if ${\displaystyle X}$ is a vector lattice then ${\displaystyle X^{++}}$ is an order complete vector lattice.[1]

If ${\displaystyle X}$ is a vector lattice and if ${\displaystyle G}$ is a solid subspace of ${\displaystyle X^{+}}$ that separates points in ${\displaystyle X}$, then the evaluation map ${\displaystyle X\to G^{+}}$ defined by sending ${\displaystyle x\in X}$ to the map ${\displaystyle E_{x}:G^{+}\to \mathbb {C} }$ given by ${\displaystyle E_{x}(f):=f(x)}$, is a lattice isomorphism of ${\displaystyle X}$ onto a vector sublattice of ${\displaystyle G^{+}}$.[1] However, the image of this map is in general not order complete even if ${\displaystyle X}$ is order complete. Indeed, a regularly ordered, order complete vector lattice need not be mapped by the evaluation map onto a band in the order bidual. An order complete, regularly ordered vector lattice whose canonical image in its order bidual is order complete is called minimal and is said to be of minimal type.[1]

An ordered vector space ${\displaystyle X}$ is called regularly ordered and its order is said to be regular if it is Archimedean ordered and ${\displaystyle X^{+}}$ distinguishes points in ${\displaystyle X}$.[1]

• 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 bound dual

• 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.