Let $N$ be an D1247: Inner product-normed vector space such that

(i) | $\Vert \cdot \Vert$ is the D504: Inner product norm in $N$ |

(ii) | $P : N \to N$ is an D36: Orthogonal projection operator in $I$ |

(iii) | $P^{\perp} : N \to N$ is the D2029: Coprojection operator of $P$ |

(iv) | $x \in N$ is a D1129: Vector in $N$ |

Then
\begin{equation}
\Vert x \Vert^2
= \Vert P x \Vert^2 + \Vert P^{\perp} x \Vert^2
\end{equation}