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 (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean.[1] 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.[2]
Characterizations
A preordered vector space (X, ≤) with an order unit u is Archimedean preordered if and only if n x ≤ u for all non-negative integers n implies x ≤ 0.[3]
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.[4]
Order unit norm
Suppose (X, ≤) 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 pU of U (defined by ) is a norm called the order unit norm. It satisfies pU(u) = 1 and the closed unit ball determined by pU is equal to [-u, u] (i.e. [-u, u] = \{ x \in X : pU(x) ≤ 1 \}.[3]
Examples
The space of bounded real-valued maps on a set S with the pointwise order is Archimedean ordered with an order unit u := 1 (i.e. the function that is identically 1 on S). The order unit norm on is identical to the usual sup norm: .[3]
Examples
Every order complete vector lattice is Archimedean ordered.[5] A finite-dimensional vector lattice of dimension n is Archimedean ordered if and only if it is isomorphic to with its canonical order.[5] However, a totally ordered vector order of dimension > 1 can not be Archimedean ordered.[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 r(0, 1) ≤ (1, 1) for every r > 0 but (0, 1) ≠ (0, 0).[3]
See also
References
- Schaefer & Wolff 1999, pp. 204–214.
- Schaefer & Wolff 1999, p. 254.
- Narici 2011, pp. 139-153.
- Schaefer & Wolff 1999, pp. 222–225.
- Schaefer & Wolff 1999, pp. 250–257.
Sources
- Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 978-1-58488-866-6. OCLC 144216834.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)