Order dual (functional analysis)

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 [1] 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.

Canonical ordering

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

Properties

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The order dual is contained in the order bound dual.[1] 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.[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  ) even if   is infinite-dimensional.[1]

Order bidual

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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.[1] In particular, if   is a vector lattice then   is an order complete vector lattice.[1]

Minimal vector lattice

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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  .[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.[1]

Examples

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For any  , the Banach lattice   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.[2]

Properties

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Let   be an order complete vector lattice of minimal type. For any   such that   the following are equivalent:[2]

  1.   is a weak order unit.
  2. For every non-0 positive linear functional   on  ,  
  3. For each topology   on   such that   is a locally convex vector lattice,   is a quasi-interior point of its positive cone.
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An ordered vector space   is called regularly ordered and its order is said to be regular if it is Archimedean ordered and   distinguishes points in  .[1]

See also

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  • Algebraic dual space – In mathematics, vector space of linear forms
  • Continuous dual space – In mathematics, vector space of linear forms
  • Dual space – In mathematics, vector space of linear forms
  • Order bound dual – Mathematical concept

References

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  1. ^ a b c d e f g h i j k l Schaefer & Wolff 1999, pp. 204–214.
  2. ^ a b Schaefer & Wolff 1999, pp. 234–242.

Bibliography

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  • 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.