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 whenever there exists some such that for all positive integers then necessarily
An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean.
A preordered vector space is called almost Archimedean if for all whenever there exists a such that for all positive integers then
Suppose is an ordered vector space over the reals with an order unit whose order is Archimedean and let
Then the Minkowski functional of (defined by ) is a norm called the order unit norm.
It satisfies and the closed unit ball determined by is equal to (that is,
Every order complete vector lattice is Archimedean ordered.
A finite-dimensional vector lattice of dimension is Archimedean ordered if and only if it is isomorphic to with its canonical order.
However, a totally ordered vector order of dimension can not be Archimedean ordered.
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 for every but