Orthogonal projection into 2 different closed linear subspaces implies they are equal.

Question

Let $L,L'$ be closed linear subspace of some Hilbert space $H$, $P:H \to L$ and $P' : H \to L'$ are orthogonal projects. Show that if $Px=P'x$ for all $x \in H$, then $L = L'$.

My proof roughly goes that since $Px = P'x$, we have that $L^{\perp} = L'^{\perp}$. Then $(L^{\perp})^{\perp} = (L'^{\perp})^{\perp}$, so $L = L'$. Is this the right way to do this? If, not how would one go about showing this?

Answer 1

I would just note that $$P = P' \implies I - P = I - P' \implies \mathrm{ker}(I - P) = \mathrm{ker}(I - P') \implies L = L'.$$