Archimedean ordered vector space

<h2 id="characterizations">Characterizations</h2>
<p>A pre<a href="/facts/Ordered_vector_space/yN9VfKMa">ordered vector space</a> 
  
    
      
        (
        X
        ,
        ≤
        )
      
    
    {\displaystyle (X,\leq )}
  
 with an <a href="/facts/Order_unit/YPkVmwvR">order unit</a> 
  
    
      
        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.}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be an <a href="/facts/Ordered_vector_space/yN9VfKMa">ordered vector space</a> 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.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="order-unit-norm">Order unit norm</h2>
<p>Suppose 
  
    
      
        (
        X
        ,
        ≤
        )
      
    
    {\displaystyle (X,\leq )}
  
 is an ordered vector space over the reals with an <a href="/facts/Order_unit/YPkVmwvR">order unit</a> 
  
    
      
        u
      
    
    {\displaystyle u}
  
 whose order is Archimedean and let 
  
    
      
        U
        =
        [
        −
        u
        ,
        u
        ]
        .
      
    
    {\displaystyle U=[-u,u].}
  
 
Then the <a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> 
  
    
      
        
          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\}.}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Examples</h3>
<p>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)|.}
  
<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="examples">Examples</h2>
<p>Every <a href="/facts/Order_complete/3L2EoCUu">order complete</a> <a href="/facts/Vector_lattice/FX7Mwfxe">vector lattice</a> is Archimedean ordered.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> 
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.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> 
However, a totally ordered vector order of dimension 
  
    
      
        
        >
        1
      
    
    {\displaystyle \,>1}
  
 can not be Archimedean ordered.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> 
There exist ordered vector spaces that are almost Archimedean but not Archimedean.
</p><p>The <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
  
    
      
        
          
            R
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} ^{2}}
  
 over the reals with the <a href="/facts/Lexicographic_order/xQANy8Br">lexicographic order</a> 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).}
  
<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Archimedean_property/61pC7UP6">Archimedean property</a> – Mathematical property of algebraic structures</li>
<li><a href="/facts/Ordered_vector_space/yN9VfKMa">Ordered vector space</a> – Vector space with a partial order</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Archimedean ordered vector space open-in-new

Archimedean ordered vector space