Borel graph theorem

<h2 id="statement">Statement</h2>
<p>A <a href="/facts/Topological_space/v2ut7CbK">topological space</a> is called a <a href="/facts/Polish_space/CkUFcGap">Polish space</a> if it is a <a href="/facts/Separable_space/2bmU73bk">separable</a> complete metrizable space and that a <a href="/facts/Souslin_space/CkUFcGap">Souslin space</a> is the continuous image of a Polish space. The weak <a href="/facts/Continuous_dual_space/4Uw4Knz1">dual</a> of a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Fr%C3%A9chet_space/9gXkQu2u">Fréchet space</a> and the <a href="/facts/Strong_dual_space/4RI1SYl6">strong dual</a> of a separable Fréchet–Montel space are Souslin spaces. Also, the space of distributions and all <a href="/facts/Lp-space/gbad0Rv1">Lp-spaces</a> over open subsets of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 be <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> spaces and let 
  
    
      
        u
        :
        X
        →
        Y
      
    
    {\displaystyle u:X\to Y}
  
 be linear. If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is the <a href="/facts/Inductive_limit/V3JuHnYK">inductive limit</a> of an arbitrary family of <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>, if 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a Souslin space, and if the graph of 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is a Borel set in 
  
    
      
        X
        ×
        Y
        ,
      
    
    {\displaystyle X\times Y,}
  
 then 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is continuous.
<h2 id="generalization">Generalization</h2>
<p>An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is called a 
  
    
      
        
          K
          
            σ
            δ
          
        
      
    
    {\displaystyle K_{\sigma \delta }}
  
 if it is the countable intersection of countable unions of <a href="/facts/Compact_space/d0cgXJH7">compact sets</a>. A Hausdorff topological space 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is called K-analytic if it is the continuous image of a 
  
    
      
        
          K
          
            σ
            δ
          
        
      
    
    {\displaystyle K_{\sigma \delta }}
  
 space (that is, if there is a 
  
    
      
        
          K
          
            σ
            δ
          
        
      
    
    {\displaystyle K_{\sigma \delta }}
  
 space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 onto 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
). Every compact set is K-analytic so that there are non-separable K-analytic spaces.
Also, every Polish, Souslin, and <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a> <a href="/facts/Fr%C3%A9chet_space/9gXkQu2u">Fréchet space</a> is K-analytic as is the weak dual of a Fréchet space. The generalized theorem states:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 be locally convex Hausdorff spaces and let 
  
    
      
        u
        :
        X
        →
        Y
      
    
    {\displaystyle u:X\to Y}
  
 be linear. If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is the inductive limit of an arbitrary family of Banach spaces, if 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a K-analytic space, and if the graph of 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is closed in 
  
    
      
        X
        ×
        Y
        ,
      
    
    {\displaystyle X\times Y,}
  
 then 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is continuous.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Closed_graph_property/canolygI">Closed graph property</a> – Graph of a map closed in the product space</li>
<li><a href="/facts/Closed_graph_theorem/CK2ra00W">Closed graph theorem</a> – Theorem relating continuity to graphs</li>
<li><a href="/facts/Closed_graph_theorem_(functional_analysis)/l01mnNoJ">Closed graph theorem (functional analysis)</a> – Theorems connecting continuity to closure of graphs</li>
<li><a href="/facts/Graph_of_a_function/htYX0BGX">Graph of a function</a> – Representation of a mathematical function</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Fran%C3%A7ois_Tr%C3%A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul>
<h2 id="external-links">External links</h2>

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

<ol>
<li id="fn:1"><p>Trèves 2006, p. 549. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Trèves 2006, p. 549. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Trèves 2006, p. 549. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Trèves 2006, pp. 557–558. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Borel graph theorem open-in-new

Borel graph theorem