Let H be a Hilbert space and M be a closed linear subspace of H.
Than for each $ x\in H$ can be uniquely expressed as $x=x_1+x_2$ where $x_1 \in M$ and $x_2\in M^\perp$.
The operator $p:H\rightarrow M$ defined by $p(x)=x_1\, \forall x\in H$ is called the projection operator of H on M.
How to prove that
1.$p$ is onto,
2.$\ker p=M^\perp$ and