Order unit

An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way the order unit generalizes the unit element in the reals.
According to H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units."

Definition

For the ordering cone in the vector space, the element is an order unit if for every there exists a such that .

The order units of an ordering cone are those elements in the algebraic interior of that is, given by

Let be the real numbers and then the unit element is an.
Let and then the unit element is an.
Each interior point of the positive cone of an ordered topological vector space is an order unit.

Each order unit of an ordered TVS is interior to the positive cone for the order topology.
If is a preordered vector space over the reals with order unit then the map is a sublinear functional.

Suppose is an ordered vector space over the reals with order unit whose order is Archimedean and let
Then the Minkowski functional of defined by is a norm called the .
It satisfies and the closed unit ball determined by is equal to that is,