Archimedean ordered vector space explained

\leq 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 n x \leq y for all positive integers n, then necessarily x \leq 0 . An Archimedean (pre)ordered vector space is a (pre)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 ^{-1} y \leq x \leq n ^{-1} y for all positive integers n, then x = 0 .

Characterizations

(X, \leq) with an order unit u is Archimedean preordered if and only if n x \leq u for all non-negative integers n implies x \leq 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

(X, \leq) is an ordered vector space over the reals with an order unit u whose order is Archimedean and let U = [ - u, u]. Then the Minkowski functional p _{U} of U (defined by p _{U} (x) : = inf \left\{ r > 0 : x \in r [ - u, u] \right\} ) is a norm called the order unit norm. It satisfies p _{U(u)} = 1 and the closed unit ball determined by p _{U} is equal to [ - u, u] (that is, [ - u, u] = \{ x\in X : p _{U(x)} \leq 1 \}.

Examples

The space

l _{infin} (S, \R) of bounded real-valued maps on a set S with the pointwise order is Archimedean ordered with an order unit u : = 1 (that is, the function that is identically 1 on S ). The order unit norm on l _{infin} (S, \R) is identical to the usual sup norm: \|f\| : = \sup |f(S)|.

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 \R ^{n} 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.\R ^{2} over the reals with the lexicographic order is Archimedean ordered since r( 0 , 1 ) \leq ( 1 , 1 ) for every r > 0 but ( 0 , 1 ) ≠ ( 0 , 0 ).