Topological vector lattice

<h2 id="definition">Definition</h2>
<p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a vector lattice then by the vector lattice operations we mean the following maps:
</p>
<ol><li>the three maps 
  
    
      
        X
      
    
    {\displaystyle X}
  
 to itself defined by 
  
    
      
        x
        ↦
        
          |
        
        x
        
          |
        
      
    
    {\displaystyle x\mapsto |x|}
  
, 
  
    
      
        x
        ↦
        
          x
          
            +
          
        
      
    
    {\displaystyle x\mapsto x^{+}}
  
, 
  
    
      
        x
        ↦
        
          x
          
            −
          
        
      
    
    {\displaystyle x\mapsto x^{-}}
  
, and</li>
<li>the two maps from 
  
    
      
        X
        ×
        X
      
    
    {\displaystyle X\times X}
  
 into 
  
    
      
        X
      
    
    {\displaystyle X}
  
 defined by 
  
    
      
        (
        x
        ,
        y
        )
        ↦
        
          sup

{
        x
        ,
        y
        }
      
    
    {\displaystyle (x,y)\mapsto \sup _{}\{x,y\}}
  
 and
  
    
      
        (
        x
        ,
        y
        )
        ↦
        
          inf

{
        x
        ,
        y
        }
      
    
    {\displaystyle (x,y)\mapsto \inf _{}\{x,y\}}
  
.</li></ol>
<p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a TVS over the reals and a vector lattice, then 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is locally solid if and only if (1) its positive cone is a <a href="/facts/Normal_cone_(functional_analysis)/fYS2JVmb">normal cone</a>, and (2) the vector lattice operations are continuous.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> 
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a vector lattice and an <a href="/facts/Ordered_topological_vector_space/N1fvN8th">ordered topological vector space</a> that is a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> in which the positive cone is a <a href="/facts/Normal_cone_(functional_analysis)/fYS2JVmb">normal cone</a>, then the lattice operations are continuous.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> 
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) and an <a href="/facts/Ordered_vector_space/yN9VfKMa">ordered vector space</a> then 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is called locally solid if 
  
    
      
        X
      
    
    {\displaystyle X}
  
 possesses a neighborhood base at the origin consisting of <a href="/facts/Solid_set/0TIYIsVI">solid sets</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> 
A topological vector lattice is a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> TVS 
  
    
      
        X
      
    
    {\displaystyle X}
  
 that has a <a href="/facts/Partial_order/qb4a3FAX">partial order</a> 
  
    
      
        
        ≤
        
      
    
    {\displaystyle \,\leq \,}
  
 making it into <a href="/facts/Vector_lattice/FX7Mwfxe">vector lattice</a> that is locally solid.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> 
</p>
<h2 id="properties">Properties</h2>
<p>Every topological vector lattice has a closed positive cone and is thus an <a href="/facts/Ordered_topological_vector_space/N1fvN8th">ordered topological vector space</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 
Let 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 denote the set of all bounded subsets of a topological vector lattice with positive cone 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and for any subset 
  
    
      
        S
      
    
    {\displaystyle S}
  
, let 
  
    
      
        [
        S
        
          ]
          
            C
          
        
        :=
        (
        S
        +
        C
        )
        ∩
        (
        S
        −
        C
        )
      
    
    {\displaystyle [S]_{C}:=(S+C)\cap (S-C)}
  
 be the <a href="/facts/Cone-saturated/sIaDXRLu">
  
    
      
        C
      
    
    {\displaystyle C}
  
-saturated</a> hull of 
  
    
      
        S
      
    
    {\displaystyle S}
  
. 
Then the topological vector lattice's positive cone 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is a strict 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
-cone,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> where 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is a strict 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
-cone means that 
  
    
      
        
          {
          
            [
            B
            
              ]
              
                C
              
            
            :
            B
            ∈
            
              
                B
              
            
          
          }
        
      
    
    {\displaystyle \left\{[B]_{C}:B\in {\mathcal {B}}\right\}}
  
 is a fundamental subfamily of 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 that is, every 
  
    
      
        B
        ∈
        
          
            B
          
        
      
    
    {\displaystyle B\in {\mathcal {B}}}
  
 is contained as a subset of some element of 
  
    
      
        
          {
          
            [
            B
            
              ]
              
                C
              
            
            :
            B
            ∈
            
              
                B
              
            
          
          }
        
      
    
    {\displaystyle \left\{[B]_{C}:B\in {\mathcal {B}}\right\}}
  
).<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> 
</p><p>If a topological vector lattice 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is <a href="/facts/Order_complete/3L2EoCUu">order complete</a> then every band is closed in 
  
    
      
        X
      
    
    {\displaystyle X}
  
.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> 
</p>
<h2 id="examples">Examples</h2>
<p>The <a href="/facts/Lp_space/gbad0Rv1">Lp spaces</a> (
  
    
      
        1
        ≤
        p
        ≤
        ∞
      
    
    {\displaystyle 1\leq p\leq \infty }
  
) are <a href="/facts/Banach_lattice/pCPSsndf">Banach lattices</a> under their canonical orderings. 
These spaces are order complete for 
  
    
      
        p
        <
        ∞
      
    
    {\displaystyle p<\infty }
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Banach_lattice/pCPSsndf">Banach lattice</a> – Banach space with a compatible structure of a lattice</li>
<li><a href="/facts/Complemented_lattice/KpwmsACp">Complemented lattice</a></li>
<li><a href="/facts/Fr%25C3%25A9chet_lattice/gWAvtkd3">Fréchet lattice</a> – Topological vector lattice</li>
<li><a href="/facts/Locally_convex_vector_lattice/T6c39Lac">Locally convex vector lattice</a></li>
<li><a href="/facts/Normed_lattice/zreeqV8e">Normed lattice</a></li>
<li><a href="/facts/Ordered_vector_space/yN9VfKMa">Ordered vector space</a> – Vector space with a partial order</li>
<li><a href="/facts/Pseudocomplement/q3mSbcIY">Pseudocomplement</a></li>
<li><a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> – Partially ordered vector space, ordered as a lattice</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. 234–242. - 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, pp. 234–242. - 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>Schaefer & Wolff 1999, pp. 234–242. - 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:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Schaefer & Wolff 1999, pp. 234–242. - 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>Schaefer & Wolff 1999, pp. 234–242. - 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:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Schaefer & Wolff 1999, pp. 234–242. - 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:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Schaefer & Wolff 1999, pp. 234–242. - 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. 215–222. - 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. 234–242. - 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>
</ol>

Topological vector lattice open-in-new

Topological vector lattice