Need help to prove an identity for inner product space

Question

Show that the following identity holds for vectors in any inner product space: $$ \langle \mathbf{u}, \mathbf{v} \rangle = \frac{1}{4}\left(\| \mathbf{u} + \mathbf{v}\|\right)^2 - \frac{1}{4}\left(\| \mathbf{u} - \mathbf{v}\|\right)^2. $$

The "in any inner product space" is quite confusing for me.

Answer 1

HINT

By definition we have

$$\left|u+v \right|^2=\left< u+v,u+v\right>=\left< u,u+v\right>+\left< v,u+v\right>=$$

$$=\left< u,u\right>+\left< u,v\right>+\left< v,u\right>+\left< v,v\right>$$