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Concept
Archimedean ordered vector space
Summary
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
A preordered vector space with an order unit is Archimedean preordered if and only if for all non-negative integers implies
Let be an ordered vector space over the reals that is finite-dimensional. Then the order of is Archimedean if and only if the positive cone of is closed for the unique topology under which is a Hausdorff TVS.
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,
The space of bounded real-valued maps on a set with the pointwise order is Archimedean ordered with an order unit (that is, the function that is identically on ).
The order unit norm on is identical to the usual sup norm:
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.
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