 Orthogonal projection

The orthogonal projection $P_F$ onto a closed subspace $F$ sends $x$ to the component in the decomposition $x=P_{F}x+(I-P_F)x$ with $(I-P_F)x\in F^{\perp }$. It is a contraction: $\Vert P_{F}x\Vert \le \Vert x\Vert$.