Archimedean ordered vector space

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.[1] A preordered vector space is called almost Archimedean if for all whenever there exists a such that for all positive integers then[2]

CharacterizationsEdit

A preordered vector space   with an order unit   is Archimedean preordered if and only if   for all non-negative integers   implies  [3]

PropertiesEdit

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.[4]

Order unit normEdit

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,  [3]

ExamplesEdit

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:  [3]

ExamplesEdit

Every order complete vector lattice is Archimedean ordered.[5] A finite-dimensional vector lattice of dimension   is Archimedean ordered if and only if it is isomorphic to   with its canonical order.[5] However, a totally ordered vector order of dimension   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   for every   but  [3]

See alsoEdit

ReferencesEdit

  1. ^ Schaefer & Wolff 1999, pp. 204–214.
  2. ^ Schaefer & Wolff 1999, p. 254.
  3. ^ a b c d Narici & Beckenstein 2011, pp. 139-153.
  4. ^ Schaefer & Wolff 1999, pp. 222–225.
  5. ^ a b c Schaefer & Wolff 1999, pp. 250–257.

BibliographyEdit

  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.