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.

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

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.

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

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.

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