In mathematics, specifically in order theory, a binary relation
on a vector space
over the real or complex numbers is called Archimedean if for all
An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean.
[1] A preordered vector space
is called almost Archimedean if for all
for all positive integers
[2] A preordered vector space
is Archimedean preordered if and only if
for all non-negative integers
be an ordered vector space over the reals that is finite-dimensional.
is Archimedean if and only if the positive cone of
is closed for the unique topology under which
is a Hausdorff TVS.
is an ordered vector space over the reals with an order unit
Then the Minkowski functional
) is a norm called the order unit norm.
and the closed unit ball determined by
of bounded real-valued maps on a set
(that is, the function that is identically
The order unit norm on
is identical to the usual sup norm:
[3] Every order complete vector lattice is Archimedean ordered.
[5] A finite-dimensional vector lattice of dimension
is Archimedean ordered if and only if it is isomorphic to
with its canonical order.
[5] However, a totally ordered vector order of dimension
[5] There exist ordered vector spaces that are almost Archimedean but not Archimedean.
The Euclidean space
over the reals with the lexicographic order is not Archimedean ordered since