Archimedean ordered vector space

In mathematics, specifically in order theory, a binary relation ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} on a vector space ${\displaystyle X}${\displaystyle X} over the real or complex numbers is called Archimedean if for all ${\displaystyle x\in X,}${\displaystyle x\in X,} whenever there exists some ${\displaystyle y\in X}${\displaystyle y\in X} such that ${\displaystyle nx\leq y}${\displaystyle nx\leq y} for all positive integers ${\displaystyle n,}${\displaystyle n,} then necessarily ${\displaystyle x\leq 0.}${\displaystyle x\leq 0.} An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean.^[1] A preordered vector space ${\displaystyle X}${\displaystyle X} is called almost Archimedean if for all ${\displaystyle x\in X,}${\displaystyle x\in X,} whenever there exists a ${\displaystyle y\in X}${\displaystyle y\in X} such that ${\displaystyle -n^{-1}y\leq x\leq n^{-1}y}${\displaystyle -n^{-1}y\leq x\leq n^{-1}y} for all positive integers ${\displaystyle n,}${\displaystyle n,} then ${\displaystyle x=0.}${\displaystyle x=0.}^[2]

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

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

Suppose ${\displaystyle (X,\leq )}${\displaystyle (X,\leq )} is an ordered vector space over the reals with an order unit ${\displaystyle u}${\displaystyle u} whose order is Archimedean and let ${\displaystyle U=[-u,u].}${\displaystyle U=[-u,u].} Then the Minkowski functional ${\displaystyle p_{U}}${\displaystyle p_{U}} of ${\displaystyle U}${\displaystyle U} (defined by ${\displaystyle p_{U}(x):=\inf \left\{r>0:x\in r[-u,u]\right\}}${\displaystyle p_{U}(x):=\inf \left\{r>0:x\in r[-u,u]\right\}}) is a norm called the order unit norm. It satisfies ${\displaystyle p_{U}(u)=1}${\displaystyle p_{U}(u)=1} and the closed unit ball determined by ${\displaystyle p_{U}}${\displaystyle p_{U}} is equal to ${\displaystyle [-u,u]}${\displaystyle [-u,u]} (that is, ${\displaystyle [-u,u]=\{x\in X:p_{U}(x)\leq 1\}.}${\displaystyle [-u,u]=\{x\in X:p_{U}(x)\leq 1\}.}^[3]

The space ${\displaystyle l_{\infty }(S,\mathbb {R} )}${\displaystyle l_{\infty }(S,\mathbb {R} )} of bounded real-valued maps on a set ${\displaystyle S}${\displaystyle S} with the pointwise order is Archimedean ordered with an order unit ${\displaystyle u:=1}${\displaystyle u:=1} (that is, the function that is identically ${\displaystyle 1}${\displaystyle 1} on ${\displaystyle S}${\displaystyle S}). The order unit norm on ${\displaystyle l_{\infty }(S,\mathbb {R} )}${\displaystyle l_{\infty }(S,\mathbb {R} )} is identical to the usual sup norm: ${\displaystyle \|f\|:=\sup _{}|f(S)|.}${\displaystyle \|f\|:=\sup _{}|f(S)|.}^[3]

Every order complete vector lattice is Archimedean ordered.^[5] A finite-dimensional vector lattice of dimension ${\displaystyle n}${\displaystyle n} is Archimedean ordered if and only if it is isomorphic to ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} with its canonical order.^[5] However, a totally ordered vector order of dimension ${\displaystyle ,円>1}${\displaystyle ,円>1} can not be Archimedean ordered.^[5] There exist ordered vector spaces that are almost Archimedean but not Archimedean.

The Euclidean space ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}} over the reals with the lexicographic order is not Archimedean ordered since ${\displaystyle r(0,1)\leq (1,1)}${\displaystyle r(0,1)\leq (1,1)} for every ${\displaystyle r>0}${\displaystyle r>0} but ${\displaystyle (0,1)\neq (0,0).}${\displaystyle (0,1)\neq (0,0).}^[3]

• Archimedean property – Mathematical property of algebraic structures
• Ordered vector space – Vector space with a partial order

• 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.

• Functional analysis
• Order theory

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