# Ordered vector space

In mathematics, an **ordered vector space** or **partially ordered vector space** is a vector space equipped with a partial order that is compatible with the vector space operations.

## Definition

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

*x*≤*y*implies*x*+*z*≤*y*+*z**y*≤*x*implies*λy*≤*λx*.

If ≤ is a partial order, (*V*, ≤) is called an **ordered vector space**. The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping *x* ↦ −*x* is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.

## Positive cone

Given a preordered vector space *V*, the subset *V*^{+} of all elements *x* in *V* satisfying *x* ≥ 0 is a convex cone, called the **positive cone** of *V*. If *V* is an ordered vector space, then *V*^{+} ∩ (−*V*^{+}) = {0}, and hence *V*^{+} is a proper cone.

If *V* is a real vector space and *C* is a proper convex cone in *V*, there exists a unique partial order on *V* that makes *V* into an ordered vector space such *V*^{+} = *C*. This partial order is given by

*x*≤*y*if and only if*y*−*x*is in*C*.

Therefore, there exists a one-to-one correspondence between the partial orders on a vector space *V* that are compatible with the vector space structure and the proper convex cones of *V*.

## Examples

- The real numbers with the usual order is an ordered vector space.
**R**^{2}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*b*≤*d*). 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*a*≤*c*and*b*≤*d*(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.

- Lexicographical order: (

- Only the second order is, as a subset of
**R**^{4}, 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.

**R**^{n}is an ordered vector space with the ≤ relation defined similarly. For example, for the second order mentioned above:*x*≤*y*if and only if*x*_{i}≤*y*_{i}for*i*= 1, ...,*n*.

- A Riesz space is an ordered vector space where the order gives rise to a lattice.
- The space of continuous functions on [0,1] where
*f*≤*g*iff*f*(*x*) ≤*g*(*x*) for all*x*in [0,1].

## Remarks

- An interval in a partially ordered vector space is a convex set. If [
*a*,*b*] = {*x*:*a*≤*x*≤*b*}, from axioms 1 and 2 above it follows that*x*,*y*in [*a*,*b*] and λ in (0,1) implies λ*x*+(1-λ)*y*in [*a*,*b*].

## See also

## References

- Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
- Schaefer, Helmut H; Wolff, M.P. (1999).
*Topological vector spaces, 2nd ed*. New York: Springer. pp. 204–205. ISBN 0-387-98726-6. - Aliprantis, Charalambos D; Burkinshaw, Owen (2003).
*Locally solid Riesz spaces with applications to economics*(Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0-8218-3408-8.