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Proving a function is an inner product in a complex space

  1. Feb 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Prove the following form for an inner product in a complex space V:
    ##\langle u,v \rangle## = ##\frac 1 4####\left\|u+v\right\|##2 - ##\frac 1 4####\left\|u-v\right\|##2 + ##\frac 1 4####\left\|u+iv\right\|##2 - ##\frac 1 4####\left\|u-iv\right\|##2

    2. Relevant equations
    ##\langle u,v \rangle## = uTA##\overline v## (where A is Hermitian)

    3. The attempt at a solution
    by opening the expressions and canceling equals I've managed to bring the expression
    ##\left\|u+v\right\|##2 - ##\left\|u-v\right\|##2 +##\left\|u+iv\right\|##2 - ##\left\|u-iv\right\|##2
    into the form 4##\langle u,v \rangle## +4##\langle u, iv \rangle##. Dividing by 4 means the original expression may be written as ##\langle u,v \rangle## +##\langle u, iv \rangle##. This is where I got stuck, I have managed to reach this expression yet I do not know how to show it follows the three axioms or alternatively express it using a Hermitian matrix (uTA##\overline v##). This probably stems from a sort of misunderstanding I have regarding the fundamental nature of complex inner products.

    Any help would be greatly appriciated
     
  2. jcsd
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