Explore BrainMass

Explore BrainMass

    Proving the parallelogram law and polarization identity

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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                              

    © BrainMass Inc. brainmass.com October 10, 2019, 7:20 am ad1c9bdddf

    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.