I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if $A$ is a self adjoint operator and $\lambda$ is an eigenvalue for $A$, then using the Borel functional calculus $\chi_{\{\lambda\}}(A)$ is just projection onto the $\lambda$ eigenspace of $A.$
I can prove that $\chi_{\{\lambda\}}(A)\psi = \psi$ when $\psi$ is a normalized eigenvector, so all I need to show is that it takes the orthogonal complement of the eigenspace to $0.$ I have reduced this to proving that for any $\phi,$ the vector $ \chi_{\{\lambda\}}(A)\phi$ is an eigenvector with eigenvalue $\lambda,$ but I'm not sure how to prove this. (Note that $ \chi_{\{\lambda\}}(A)\phi$ may be $0,$ and in fact will be whenever $\phi$ is orthogonal to the $\lambda$ eigenspace even though I can't yet prove that.)