Solution
= Solution
Expand both squared norms using the <inner product>:
$$
\|x\pm y\|^2=\|x\|^2+\|y\|^2\pm2\operatorname{Re}\langle x,y\rangle.
$$
Adding cancels the cross terms, giving the <parallelogram law>
$$
\boxed{\|x-y\|^2+\|x+y\|^2=2(\|x\|^2+\|y\|^2).}
$$
The real part may be omitted in a real inner-product space.