Order dual (functional analysis)

In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space

X

{\displaystyle X}

is the set

Pos

⁡

(

X

∗

)

−

Pos

⁡

(

X

∗

)

{\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)}

where

Pos

⁡

(

X

∗

)

{\displaystyle \operatorname {Pos} \left(X^{*}\right)}

denotes the set of all positive linear functionals on

X

{\displaystyle X}

, where a linear function

f

{\displaystyle f}

on

X

{\displaystyle X}

is called positive if for all

x

∈

X

,

{\displaystyle x\in X,}

x

≥

0

{\displaystyle x\geq 0}

implies

f

(

x

)

≥

0.

{\displaystyle f(x)\geq 0.}

The order dual of

X

{\displaystyle X}

is denoted by

X

+

{\displaystyle X^{+}}

.

Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.

Canonical ordering

An element

f

{\displaystyle f}

of the order dual of

X

{\displaystyle X}

is called positive if

x

≥

0

{\displaystyle x\geq 0}

implies

Re

⁡

f

(

x

)

≥

0.

{\displaystyle \operatorname {Re} f(x)\geq 0.}

The positive elements of the order dual form a cone that induces an ordering on

X

+

{\displaystyle X^{+}}

called the canonical ordering.

If

X

{\displaystyle X}

is an ordered vector space whose positive cone

C

{\displaystyle C}

is generating (that is,

X

=

C

−

C

{\displaystyle X=C-C}

) then the order dual with the canonical ordering is an ordered vector space.

The order dual is the span of the set of positive linear functionals on

X

{\displaystyle X}

.

Properties

The order dual is contained in the order bound dual.

If the positive cone of an ordered vector space

X

{\displaystyle X}

is generating and if

[

0

,

x

]

+

[

0

,

y

]

=

[

0

,

x

+

y

]

{\displaystyle [0,x]+[0,y]=[0,x+y]}

holds for all positive

x

{\displaystyle x}

and

y

{\displaystyle y}

, then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.

The order dual of a vector lattice is an order complete vector lattice.