Dual system

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a dual system, dual pair or a duality over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        
          K
        
      
    
    {\displaystyle \mathbb {K} }
  
 is a triple 
  
    
      
        (
        X
        ,
        Y
        ,
        b
        )
      
    
    {\displaystyle (X,Y,b)}
  
 consisting of two <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>, 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
, over 
  
    
      
        
          K
        
      
    
    {\displaystyle \mathbb {K} }
  
 and a non-<a href="/facts/Degenerate_bilinear_form/9TLGBvSH">degenerate</a> <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a> 
  
    
      
        b
        :
        X
        ×
        Y
        →
        
          K
        
      
    
    {\displaystyle b:X\times Y\to \mathbb {K} }
  
. 
</p><p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Duality_(mathematics)/8jGuyQIU">duality</a> is the study of dual systems and is important in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>. Duality plays crucial roles in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> because it has extensive applications to the theory of <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>.
</p>

Dual system open-in-new

Dual system