Order unit

An order unit is an element of an ordered vector space which can be used to bound all elements from above.

[1] In this way (as seen in the first example below) 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.

"[2] For the ordering cone

in the vector space

, the element

is an order unit (more precisely a

there exists a

λ

[3] The order units of an ordering cone

are those elements in the algebraic interior of

be the real numbers and

then the unit element

is an order unit.

then the unit element

is an order unit.

Each interior point of the positive cone of an ordered topological vector space is an order unit.

[2] Each order unit of an ordered TVS is interior to the positive cone for the order topology.

is a preordered vector space over the reals with order unit

then the map

is a sublinear functional.

is an ordered vector space over the reals with order unit

whose order is Archimedean and let

Then the Minkowski functional

defined by

is a norm called the order unit norm.

It satisfies

and the closed unit ball determined by