Orthogonal projection

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

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.