# Compton edge

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

## Definition

Given a vector space *V* over the real numbers **R** and a partial order ≤ on the set *V*, the pair (*V*, ≤) is called an **ordered 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 λ*y*≤ λ*x*.

## Notes

The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping *f*(*x*) = − *x* is an isomorphism to the dual order structure.

If ≤ is only a preorder, (*V*, ≤) is called a **preordered vector space**.

Ordered vector spaces are ordered groups under their addition operation.

## Positive cone

Given an ordered 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*. Since the partial order ≥ is antisymmetric, one can show, that *V*^{+}∩(−*V*^{+})={0}, hence *V*^{+} is a proper cone. That it is convex can be seen by combining the above two axioms with the transitivity property of the (pre)order.

If *V* is a real vector space and *C* is a proper convex cone in *V*, there exists exactly one partial order on 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 function 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.
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534