Bessel's inequality | Wikipedia audio article
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Date: 06/05/2019

This is an audio version of the Wikipedia Article:
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00:05:20 See also
00:05:35 Notes
00:05:44 External links



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SUMMARY
=======
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element 
  
    
      
        x
      
    
    {\displaystyle x}
    in a Hilbert space with respect to an orthonormal sequence.
Let 
  
    
      
        H
      
    
    {\displaystyle H}
   be a Hilbert space, and suppose that 
  
    
      
        
          e
          
            1
          
        
        ,
        
          e
          
            2
          
        
        ,
        .
        .
        .
      
    
    {\displaystyle e_{1},e_{2},...}
   is an orthonormal sequence in 
  
    
      
        H
      
    
    {\displaystyle H}
  . Then, for any 
  
    
      
        x
      
    
    {\displaystyle x}
   in 
  
    
      
        H
      
    
    {\displaystyle H}
   one has

  
    
      
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            |
            
              ⟨
              
                x
                ,
                
                  e
                  
                    k
                  
                
              
              ⟩
            
            |
          
          
            2
          
        
        ≤
        
          
            ‖
            x
            ‖
          
          
            2
          
        
        ,
      
    
    {\displaystyle \sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},}
  where 〈•,•〉 denotes the inner product in the Hilbert space 
  
    
      
        H
      
    
    {\displaystyle H}
  . If we define the infinite sum

  
    
      
        
          x
          ′
        
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          ⟨
          
            x
            ,
            
              e
              
                k
              
            
          
          ⟩
        
        
          e
          
            k
          
        
        ,
      
    
    {\displaystyle x'=\sum _{k=1}^{\infty }\left\langle x,e_{k}\right\rangle e_{k},}
  consisting of "infinite sum" of vector resolute 
  
    
      
        x
      
    
    {\displaystyle x}
   in direction 
  
    
      
        
          e
          
            k
          
        
      
    
    {\displaystyle e_{k}}
  , Bessel's inequality tells us that this series converges. One can think of it that there exists 
  
    
      
        
          x
          ′
        
        ∈
        H
      
    
    {\displaystyle x'\in H}
   that can be described in terms of potential basis 
  
    
      
        
          e
          
            1
          
        
        ,
        
          e
          
            2
          
        
        ,
        …
      
    
    {\displaystyle e_{1},e_{2},\dots }
  .
For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently 
  
    
      
        
          x
          ′
        
      
    
    {\displaystyle x'}
   with 
  
    
      
        x
      
    
    {\displaystyle x}
  ).
Bessel's inequality follows from the identity

  
    
      
        0
        ≤
        
          
            ‖
            
              x
              −
              
                ∑
          ...