Order dual (functional analysis)

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

denotes the set of all positive linear functionals on

[1] The order dual of

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

of the order dual of

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

called the canonical ordering.

is an ordered vector space whose positive cone

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

[1] The order dual is contained in the order bound dual.

[1] If the positive cone of an ordered vector space

, 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

can be finite dimension (possibly even

is an ordered vector space such that the canonical order on

into an ordered vector space.

Then the order bidual is defined to be the order dual of

If the positive cone of an ordered vector space

is an order complete vector lattice and the evaluation map

is order preserving.

is an order complete vector lattice.

[1] However, the image of this map is in general not order complete even if

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.

is order complete and of minimal type; in particular, the norm topology on this space is the finest locally convex topology for which every order convergent filter converges.

be an order complete vector lattice of minimal type.

the following are equivalent:[2] An ordered vector space

is called regularly ordered and its order is said to be regular if it is Archimedean ordered and