I was googling "Hilbert space" and was reading the associated Wikipedia page when I found this statement confusing :
"Let $V$ be a closed subspace of an Hilbert space $H$. Then the inclusion mapping $i_V : V\rightarrow H$ is the adjoint of the orthogonal projection $P_V : H\rightarrow V$".
I understand that means $\langle i_V(f),g\rangle=\langle f,P_V(g)\rangle $ for all $f\in V, \,g\in H$. But it is also known that an orthogonal projection is self-adjoint, so that we should have $P_V=P_V^\dagger=i_V$, which is not correct, but I can't explain why.
Could you explain where is my mistake ?
Moreover, do you know a proof for the mentioned adjoint property ?