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Proving that a function is an inner product in a complex space

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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
 

pasmith

Homework Helper
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369
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##.
What do you mean by [itex]\langle \cdot,\cdot \rangle[/itex] here? The question is already using that to mean the form it wants you to show is an inner product, so if (as I suspect) you mean the euclidean inner product then you need to use a different notation. So let [tex]
\langle u,v \rangle_1 = \sum_n u_n \bar{v}_n.[/tex]

Also, because this is a complex space, the correct expansion of [itex]\|u \pm v\|^2[/itex] is [tex]
\|u\|^2 \pm \langle u, v \rangle_1 \pm \langle v,u\rangle_1 + \|v\|^2
= \|u\|^2 \pm 2\mathrm{Re}\left( \langle u,v \rangle_1 \right) + \|v\|^2.
[/tex]
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
 
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