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

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

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#### pasmith

Homework Helper
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 $\langle \cdot,\cdot \rangle$ 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 $$\langle u,v \rangle_1 = \sum_n u_n \bar{v}_n.$$

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

Last edited:

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

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