Hilbert projection theorem

<p>In mathematics, the Hilbert projection theorem is a famous result of <a href="/facts/Convex_analysis/SGhUNLMK">convex analysis</a> that says that for every vector 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
  
    
      
        H
      
    
    {\displaystyle H}
  
 and every nonempty closed convex 
  
    
      
        C
        ⊆
        H
        ,
      
    
    {\displaystyle C\subseteq H,}
  
 there exists a unique vector 
  
    
      
        m
        ∈
        C
      
    
    {\displaystyle m\in C}
  
 for which 
  
    
      
        ‖
        c
        −
        x
        ‖
      
    
    {\displaystyle \|c-x\|}
  
 is minimized over the vectors 
  
    
      
        c
        ∈
        C
      
    
    {\displaystyle c\in C}
  
; that is, such that 
  
    
      
        ‖
        m
        −
        x
        ‖
        ≤
        ‖
        c
        −
        x
        ‖
      
    
    {\displaystyle \|m-x\|\leq \|c-x\|}
  
 for every 
  
    
      
        c
        ∈
        C
        .
      
    
    {\displaystyle c\in C.}

</p>

Hilbert projection theorem open-in-new

Hilbert projection theorem