In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space is the set of all linear functionals on that map order intervals, which are sets of the form to bounded sets.[1] The order bound dual of is denoted by 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 bound dual of   is called positive if   implies   The positive elements of the order bound dual form a cone that induces an ordering on   called the canonical ordering. If   is an ordered vector space whose positive cone   is generating (meaning  ) then the order bound dual with the canonical ordering is an ordered vector space.[1]

Properties

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The order bound dual of an ordered vector spaces contains its order dual.[1] If the positive cone of an ordered vector space   is generating and if for all positive   and   we have   then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.[1]

Suppose   is a vector lattice and   and   are order bounded linear forms on   Then for all  [1]

  1.  
  2.  
  3.  
  4.  
  5. if   and   then   and   are lattice disjoint if and only if for each   and real   there exists a decomposition   with  

See also

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References

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  1. ^ a b c d e Schaefer & Wolff 1999, pp. 204–214.
  • 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.