Orthogonal Projector

Question

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

Answer 1

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

Answer 2

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 $\langle \xi,\,P_\psi(\eta)\rangle\,$.