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Proving the parallelogram law and polarization identity

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If x,y are elements  of a Hilbert space H, then prove that:            
             [norm of (x + y)]^2 + [norm of (x - y)]^2 = 2(norm of x)^2 + (norm of y)^2                                               
Alternatively, prove that in any Hilbert space H, the parallelogram law holds.                                        
Also, prove the polarization identity, that is, if x,y are elements of a Hilbert space H, then:                      
              4(x , y) = [norm of (x + y)]^2 - [norm of (x - y)]^2 + i[norm of (x + iy)]^2 - i[norm of (x - iy)]^2                              

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Solution Summary

This solution explains the proof of the parallelogram law and polarization identity in Hilbert space.  The solution is given in detail. This is mainly for solving the problem on functional analysis.

Solution provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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