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 $P_{\psi}^*=P_{\psi}$.

For $\xi,\eta\in H$ we have
$$\langle P_\psi(\xi),\,\eta\rangle\ =\ \big\langle \langle \psi,\xi\rangle\cdot\psi\,,\ \eta\big\rangle\ =\ \langle\psi,\xi\rangle\cdot\langle \psi,\eta\rangle$$
(assumed that conjugation happens in second variable). 
And, we get the same for $\langle \xi,\,P_\psi(\eta)\rangle\,$.
