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Concept
Order dual (functional analysis)
Summary
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all implies
The order dual of is denoted by .
Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.
An element of the order dual of is called positive if implies
The positive elements of the order dual form a cone that induces an ordering on called the canonical ordering.
If is an ordered vector space whose positive cone is generating (that is, ) 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 .
The order dual is contained in the order bound dual.
If the positive cone of an ordered vector space is generating and if holds for all positive and , 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.
The order dual of a vector lattice can be finite dimension (possibly even ) even if is infinite-dimensional.
Suppose that is an ordered vector space such that the canonical order on makes into an ordered vector space.
Then the order bidual is defined to be the order dual of and is denoted by .
If the positive cone of an ordered vector space is generating and if holds for all positive and , then is an order complete vector lattice and the evaluation map is order preserving.
In particular, if is a vector lattice then is an order complete vector lattice.
If is a vector lattice and if is a solid subspace of that separates points in , then the evaluation map defined by sending to the map given by , is a lattice isomorphism of onto a vector sublattice of .
However, the image of this map is in general not order complete even if is order complete.
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