Mapping space

<h2 id="topologies">Topologies</h2>

<p>A mapping space can be equipped with several topologies. A common one is the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>. Typically, there is then the adjoint relation
</p>

Map
        ⁡
        (
        X
        ×
        Y
        ,
        Z
        )
        ≃
        Map
        ⁡
        (
        X
        ,
        Map
        ⁡
        (
        Y
        ,
        Z
        )
        )
      
    
    {\displaystyle \operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))}

<p>and thus 
  
    
      
        Map
      
    
    {\displaystyle \operatorname {Map} }
  
 is an analog of the <a href="/facts/Hom_functor/H76BLKCD">Hom functor</a>. (For pathological spaces, this relation may fail.)
</p>
<h2 id="smooth-mappings">Smooth mappings</h2>
<p>For manifolds 
  
    
      
        M
        ,
        N
      
    
    {\displaystyle M,N}
  
, there is the subspace 
  
    
      
        
          
            
              C
            
          
          
            r
          
        
        (
        M
        ,
        N
        )
        ⊂
        Map
        ⁡
        (
        M
        ,
        N
        )
      
    
    {\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)}
  
 that consists of all the 
  
    
      
        
          
            
              C
            
          
          
            r
          
        
      
    
    {\displaystyle {\mathcal {C}}^{r}}
  
-smooth maps from 
  
    
      
        M
      
    
    {\displaystyle M}
  
 to 
  
    
      
        N
      
    
    {\displaystyle N}
  
. It can be equipped with the weak or strong topology.
</p><p>A basic approximation theorem says that 
  
    
      
        
          
            
              C
            
          
          
            W
          
          
            s
          
        
        (
        M
        ,
        N
        )
      
    
    {\displaystyle {\mathcal {C}}_{W}^{s}(M,N)}
  
 is dense in 
  
    
      
        
          
            
              C
            
          
          
            S
          
          
            r
          
        
        (
        M
        ,
        N
        )
      
    
    {\displaystyle {\mathcal {C}}_{S}^{r}(M,N)}
  
 for 
  
    
      
        1
        ≤
        s
        ≤
        ∞
        ,
        0
        ≤
        r
        <
        s
      
    
    {\displaystyle 1\leq s\leq \infty ,0\leq r<s}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

<ul><li><a href="/facts/Morris_Hirsch/grrWGgFM">Hirsch, Morris</a> (1997). <i>Differential Topology</i>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90148-5.</li>
<li>Wall, C. T. C. (4 July 2016). <a href="https://books.google.com/books?id=TT5yDAAAQBAJ"><i>Differential Topology</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781107153523.</li></ul>

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

<ol>
<li id="fn:1"><p>Hirsch 1997, Ch. 2., § 2., Theorem 2.6. - Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Mapping space open-in-new

Mapping space