Orthogonal Projector Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$.
$P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. 
I have to prove that $P_{\psi}$ is an orthogonal projector on $H$. 
I can prove $P_{\psi}P_{\psi}=P_{\psi}$, but I can't prove $P_{\psi}^*=P_{\psi}$.

$$
\langle P_\psi^*\phi,\eta\rangle = \langle \phi,P_\psi\eta\rangle=\langle\,\phi,\langle\psi,\eta\rangle\,\psi\rangle\,
=\langle\psi,\eta\rangle\,\langle\phi,\psi\rangle
=\langle\,\langle\psi,\phi\rangle\,\psi,\eta\rangle=\langle P_\psi\phi,\eta\rangle.
$$
Note that we are using that the inner product is conjugate linear in the first component, and that $\langle\psi,\phi\rangle=\overline{\langle\phi,\psi\rangle}$. 
