Order bound dual

In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space X is the set of all linear functionals on X that map order intervals (i.e. sets of the form [a, b] := { x ∈ X : a ≤ x and x ≤ b }) to bounded sets.[1] The order bound dual of X is denoted by Xb. This space plays an important role in the theory of ordered topological vector spaces.

Canonical ordering

An element f of the order bound dual of X is called positive if x ≥ 0 implies Re(f(x)) ≥ 0. The positive elements of the order bound dual form a cone that induces an ordering on Xb called the canonical ordering. If X is an ordered vector space whose positive cone C is generating (i.e. X = C - C) then the order bound dual with the canonical ordering is an ordered vector space.[1]

Properties

The order bound dual of an ordered vector spaces contains its order dual.[1] If the positive cone of an ordered vector space X is generating and if for all positive x and y we have [0, x] + [0, y] = [0, x + y], then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.[1]

Suppose X is a vector lattice and f and g are order bounded linear forms on X. Then for all x in X,[1]

  1. sup(f, g)(|x|) = sup { f(y) + g(z) : y ≥ 0, z ≥ 0, and y + z = |x| }
  2. inf(f, g)(|x|) = inf { f(y) + g(z) : y ≥ 0, z ≥ 0, and y + z = |x| }
  3. |f|(|x|) = sup { f(y - z) : y ≥ 0, z ≥ 0, and y + z = |x| }
  4. |f(x)| ≤ |f|(|x|)
  5. if f ≥ 0 and g ≥ 0 then f and g are lattice disjoint if and only if for each x ≥ 0 and real r > 0, there exists a decomposition x = a + b with a ≥ 0, b ≥ 0, and f(a) + g(b) ≤ r.

See also

References
  1. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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