Strong dual space

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the strong dual space of a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is the <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> 
  
    
      
        
          X
          
            ′
          
        
      
    
    {\displaystyle X^{\prime }}
  
 of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 where this topology is denoted by 
  
    
      
        b
        
          (
          
            
              X
              
                ′
              
            
            ,
            X
          
          )
        
      
    
    {\displaystyle b\left(X^{\prime },X\right)}
  
 or 
  
    
      
        β
        
          (
          
            
              X
              
                ′
              
            
            ,
            X
          
          )
        
        .
      
    
    {\displaystyle \beta \left(X^{\prime },X\right).}
  
 The <a href="/facts/Comparison_of_topologies/5Q9F2fjf">coarsest</a> polar topology is called <a href="/facts/Weak_topology_(polar_topology)/XJASE9Up">weak topology</a>. 
The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. 
To emphasize that the continuous dual space, 
  
    
      
        
          X
          
            ′
          
        
        ,
      
    
    {\displaystyle X^{\prime },}
  
 has the strong dual topology, 
  
    
      
        
          X
          
            b
          
          
            ′
          
        
      
    
    {\displaystyle X_{b}^{\prime }}
  
 or 
  
    
      
        
          X
          
            β
          
          
            ′
          
        
      
    
    {\displaystyle X_{\beta }^{\prime }}
  
 may be written. 
</p>

Strong dual space open-in-new

Strong dual space