Vector space with inner product/Theorem of Pythagoras/Fact/Proof

Proof

We have

${}\Vert {u+v}\Vert ^{2}=\left\langle u+v,u+v\right\rangle =\left\langle u,u\right\rangle +\left\langle u,v\right\rangle +\left\langle v,u\right\rangle +\left\langle v,v\right\rangle =\Vert {u}\Vert ^{2}+\Vert {v}\Vert ^{2}\,.$

To fact