Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 247: 88

see the proof below.

\begin{aligned} \|u+v\|^2+\|u-v\|^2&= \langle u+v, u+v\rangle+ \langle u-v, u-v\rangle\\ &= \langle u, u\rangle+ \langle v, v\rangle +2 \langle u, v\rangle+ \langle u, u\rangle+ \langle v, v\rangle-2\langle u, v\rangle \\ &= 2\langle u, u\rangle+ 2\langle v, v\rangle \\ &=2\|u\|^2+2\|v\|^2 . \end{aligned}