Let $M \subset H$ be a closed subset of a Hilbert space H, show that $M^\perp=Ker(P_M)$.

Question

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space and $M \subset H$ a closed subspace of $H$. Let $u \in H$. So, there are unique $v \in M$ and $w \in M^\perp$ such that $ u=v +w$. Consider a ortogonal projection $P_M: H \longrightarrow M$ given by $$P_M(u)=v,\;\; \forall \; u \in H.$$

I want to show

$$M^\perp=Ker(P_M).$$

I could show only the inclusion $Ker(P_M) \subset M^\perp$. How to show reverse inclusion?

Answer 1

If $x\in M^{\perp}, x=0+x, 0\in M, x\in M^{\perp}$ implies that $P_M(x)=0$