Archimedean ordered vector space


In mathematics, specifically in order theory, a binary relation ≤ on a vector space X over the real or complex numbers is called Archimedean if for all x in X, whenever there exists some y in X such that nx ≤ y for all positive integers n, then necessarily x ≤ 0.
An Archimedean ordered vector space is a ordered vector space whose order is Archimedean.
A preordered vector space X is called almost Archimedean if for all x in X, whenever there exists a y in X such that -n−1y ≤ x ≤ n−1y for all positive integers n, then x = 0.

Characterizations

A preordered vector space with an order unit u is Archimedean preordered if and only if n x ≤ u for all non-negative integers n implies x ≤ 0.

Properties

Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS.

Order unit norm

Suppose is an ordered vector space over the reals with an order unit u whose order is Archimedean and let U = .
Then the Minkowski functional pU of U is a norm called the order unit norm.
It satisfies pU = 1 and the closed unit ball determined by pU is equal to (i.e. = \.

Examples

The space the space l∞ of bounded real-valued maps on a set S with the pointwise order is Archimedean ordered with an order unit u := 1.
The order unit norm on l∞ is identical to the usual sup norm:.

Examples

Every order complete vector lattice is Archimedean ordered.
A finite-dimensional vector lattice of dimension n is Archimedean ordered if and only if it is isomorphic to with its canonical order.
However, a totally ordered vector order of dimension > 1 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 r ≤ for every r > 0 but ≠.