Polarization identity

<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the polarization identity is any one of a family of formulas that express the <a href="/facts/Inner_product/HoyjElEy">inner product</a> of two <a href="/facts/Vector_(geometry)/bCLrdksy">vectors</a> in terms of the <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> of a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a>. 
If a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. 
</p><p>The norm associated with any <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> satisfies the <a href="/facts/Parallelogram_law/JI1LwPxd">parallelogram law</a>: 
  
    
      
        ‖
        x
        +
        y
        
          ‖
          
            2
          
        
        +
        ‖
        x
        −
        y
        
          ‖
          
            2
          
        
        =
        2
        ‖
        x
        
          ‖
          
            2
          
        
        +
        2
        ‖
        y
        
          ‖
          
            2
          
        
        .
      
    
    {\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.}
  
 
In fact, as observed by <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a>, the parallelogram law characterizes those norms that arise from inner products. 
Given a <a href="/facts/Normed_vector_space/rmDljfyE">normed space</a> 
  
    
      
        (
        H
        ,
        ‖
        ⋅
        ‖
        )
      
    
    {\displaystyle (H,\|\cdot \|)}
  
, the parallelogram law holds for 
  
    
      
        ‖
        ⋅
        ‖
      
    
    {\displaystyle \|\cdot \|}
  
 if and only if there exists an inner product 
  
    
      
        ⟨
        ⋅
        ,
        ⋅
        ⟩
      
    
    {\displaystyle \langle \cdot ,\cdot \rangle }
  
 on 
  
    
      
        H
      
    
    {\displaystyle H}
  
 such that 
  
    
      
        ‖
        x
        
          ‖
          
            2
          
        
        =
        ⟨
        x
        ,
         
        x
        ⟩
      
    
    {\displaystyle \|x\|^{2}=\langle x,\ x\rangle }
  
 for all 
  
    
      
        x
        ∈
        H
        ,
      
    
    {\displaystyle x\in H,}
  
 in which case this inner product is uniquely determined by the norm via the polarization identity.
</p>

Polarization identity open-in-new

Polarization identity