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