User:MikeRumex/sandbox/ordered vector space

A point x in R2 and the set of all y such that xy (in red). The order here is xy if and only if x1y1 and x2y2.

In mathematics an ordered vector space is a vector space equipped with a preorder or partial order which is compatible with the vector space operations.

Definition

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Given a vector space V over the real numbers R and a preorder ≤ on the set V, the pair (V, ≤) is called an preordered vector space if for all x,y,z in V and 0 ≤ λ in R the following two axioms are satisfied

  1. xy implies x + zy + z
  2. yx implies λ y ≤ λ x.

If, in addition, ≤ is a partial order, then (V, ≤) is called an partially ordered vector space.


Cones and preorders

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If V is a vector space over the real numbers, a subset   is called a cone if

  1.  
  2.   for all  

The set   is called a proper cone if  .

If (V, ≤) is a preordered vector space the set   is a cone, and if ≤ is a partial order,   is a proper cone. Conversely, if   is a cone and defining   to mean  , then   is a preordered vector space, and a partially ordered vector space if   is a proper cone. In this way there is a bijection between all cones (proper cones) and all preorders (partial orders) on V that are translation invariant and positively homogeneous.

Examples

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  • The real numbers with the usual order is an ordered vector space.
  • R2 is an ordered vector space with the ≤ relation defined in any of the following ways (in order of increasing strength, i.e., decreasing sets of pairs):
    • Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and bd). This is a total order. The positive cone is given by x > 0 or (x = 0 and y ≥ 0), i.e., in polar coordinates, the set of points with the angular coordinate satisfying -π/2 < θ ≤ π/2, together with the origin.
    • (a,b) ≤ (c,d) if and only if ac and bd (the product order of two copies of R with "≤"). This is a partial order. The positive cone is given by x ≥ 0 and y ≥ 0, i.e., in polar coordinates 0 ≤ θ ≤ π/2, together with the origin.
    • (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of two copies of R with "<"). This is also a partial order. The positive cone is given by (x > 0 and y > 0) or (x = y = 0), i.e., in polar coordinates, 0 < θ < π/2, together with the origin.
    • (a,b) ≤ (c,d) if and only if ac, is a preorder.
Only the second order is, as a subset of R4, closed, see partial orders in topological spaces.
For the third order the two-dimensional "intervals" p < x < q are open sets which generate the topology.
  • A Riesz space is a partially ordered vector space where the order gives rise to a lattice, i.e., for every pair of elements there exists a supremum.
  • The space of continuous functions on [0,1] where fg iff f(x) ≤ g(x) for all x in [0,1] is a Riesz space. When this space is also endowed with the uniform norm this space becomes a Banach lattice.
  • Rn becomes a partially ordered vector space when endowed with the standard order, i.e., xy if and only if xiyi for all i = 1, … , n.
  • R3 with partial order   defined from the Lorentz cone (aka ice-cream cone)  , i.e.,   means  , is a partially ordered vector space. This is one of the simplest partially ordered vector spaces that is not a vector lattice. Suprema of arbitrary pairs of elements do not necessarily exist, e.g., the pair of vectors (0,0,0) and (0,0,1) have no supremum with respect to the partial order  .

See also

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References

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  • Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
  • Schaefer, Helmut H (1999). Topological vector spaces, 2nd ed. New York: Springer. pp. 204–205. ISBN 0-387-98726-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Aliprantis, Charalambos D (2003). Locally solid Riesz spaces with applications to economics (Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0-8218-3408-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)


Category:Functional analysis Category:Ordered groups