Let $H$ be a [Hilbert space](/page/Hilbert%20Space) over $\mathbb F$, where $\mathbb F\in\{\mathbb R,\mathbb C\}$, and let $M\subset H$ be a closed linear subspace. Let $P_M:H\to H$ denote the [orthogonal projection](/theorems/437) onto $M$. Then $P_M\in\mathcal L(H)$ and