Let \( V \) be an inner product space , and let \(u,v\in V\) be orthogonal . Then
\[ \lVert u+v\rVert^2 = \lVert u\rVert^2 + \lVert v\rVert^2, \]where \(\lVert u\rVert=\langle u, u\rangle^{1/2}\).
Proof
Let \(u,v\in V\). Then
\[ \lVert x+y\rVert^2 = \langle x+y, x+y\rangle = \langle x, x\rangle + \langle y, y\rangle = \lVert x\rVert^2 + \lVert y\rVert^2, \]where we used the orthogonality condition in the second equality.