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Archimedean ordered vector space
A binary relation on a vector space

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

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Characterizations

A preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} with an order unit u {\displaystyle u} is Archimedean preordered if and only if n x ≤ u {\displaystyle nx\leq u} for all non-negative integers n {\displaystyle n} implies x ≤ 0. {\displaystyle x\leq 0.} 3

Properties

Let X {\displaystyle X} be an ordered vector space over the reals that is finite-dimensional. Then the order of X {\displaystyle X} is Archimedean if and only if the positive cone of X {\displaystyle X} is closed for the unique topology under which X {\displaystyle X} is a Hausdorff TVS.4

Order unit norm

Suppose ( X , ≤ ) {\displaystyle (X,\leq )} is an ordered vector space over the reals with an order unit u {\displaystyle u} whose order is Archimedean and let U = [ − u , u ] . {\displaystyle U=[-u,u].} Then the Minkowski functional p U {\displaystyle p_{U}} of U {\displaystyle U} (defined by p U ( x ) := inf { r > 0 : x ∈ r [ − u , u ] } {\displaystyle 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 {\displaystyle p_{U}(u)=1} and the closed unit ball determined by p U {\displaystyle p_{U}} is equal to [ − u , u ] {\displaystyle [-u,u]} (that is, [ − u , u ] = { x ∈ X : p U ( x ) ≤ 1 } . {\displaystyle [-u,u]=\{x\in X:p_{U}(x)\leq 1\}.} 5

Examples

The space l ∞ ( S , R ) {\displaystyle l_{\infty }(S,\mathbb {R} )} of bounded real-valued maps on a set S {\displaystyle S} with the pointwise order is Archimedean ordered with an order unit u := 1 {\displaystyle u:=1} (that is, the function that is identically 1 {\displaystyle 1} on S {\displaystyle S} ). The order unit norm on l ∞ ( S , R ) {\displaystyle l_{\infty }(S,\mathbb {R} )} is identical to the usual sup norm: ‖ f ‖ := sup | f ( S ) | . {\displaystyle \|f\|:=\sup _{}|f(S)|.} 6

Examples

Every order complete vector lattice is Archimedean ordered.7 A finite-dimensional vector lattice of dimension n {\displaystyle n} is Archimedean ordered if and only if it is isomorphic to R n {\displaystyle \mathbb {R} ^{n}} with its canonical order.8 However, a totally ordered vector order of dimension > 1 {\displaystyle \,>1} can not be Archimedean ordered.9 There exist ordered vector spaces that are almost Archimedean but not Archimedean.

The Euclidean space R 2 {\displaystyle \mathbb {R} ^{2}} over the reals with the lexicographic order is not Archimedean ordered since r ( 0 , 1 ) ≤ ( 1 , 1 ) {\displaystyle r(0,1)\leq (1,1)} for every r > 0 {\displaystyle r>0} but ( 0 , 1 ) ≠ ( 0 , 0 ) . {\displaystyle (0,1)\neq (0,0).} 10

See also

Bibliography

  • 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

References

  1. Schaefer & Wolff 1999, pp. 204–214. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  2. Schaefer & Wolff 1999, p. 254. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  3. Narici & Beckenstein 2011, pp. 139–153. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

  4. Schaefer & Wolff 1999, pp. 222–225. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  5. Narici & Beckenstein 2011, pp. 139–153. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

  6. Narici & Beckenstein 2011, pp. 139–153. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

  7. Schaefer & Wolff 1999, pp. 250–257. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  8. Schaefer & Wolff 1999, pp. 250–257. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  9. Schaefer & Wolff 1999, pp. 250–257. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

  10. Narici & Beckenstein 2011, pp. 139–153. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩