Orthogonal decomposition by a closed subspace
= Orthogonal decomposition by a closed subspace
= Orthogonal decomposition
{synonym}
For a closed subspace $F$ of a Hilbert space,
$$
H=F\oplus F^\perp.
$$
The closest point $P_Fx$ gives the first component, and differentiating $\|x-P_Fx-tu\|^2$ at zero shows that the remainder is orthogonal to every $u\in F$.