Orthogonal projection

Let $M$ be a closed linear subspace of a Hilbert space $H$ . The decomposition $H=M\oplus M^\perp$ , using the orthogonal complement, defines the orthogonal projection $P_M:H\to H$ by $P_M(m+n)=m$ .

Operator characterization

It is the unique bounded operator satisfying $P_M^2=P_M$ , $P_M^*=P_M$ , and $\operatorname{range}(P_M)=M$ . Conversely, the range of every self-adjoint idempotent bounded operator is a closed subspace.