Vector space with inner product/Theorem of Pythagoras/Fact

Theorem of Pythagoras (inner product)

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let ${}u,v\in V$ be vectors that are orthogonal to each other.

Then

${}\Vert {u+v}\Vert ^{2}=\Vert {u}\Vert ^{2}+\Vert {v}\Vert ^{2}\,$

holds.

Proof, Write another proof