On page $263$ in Conway's book "A Course in Functional Analysis", the spectral theorem for normal operators is stated. It says that if $N$ is a normal operator, there is a unique spectral measure $E$ on the Borel sets of $\sigma(N)$ such that $N = \int z \, dE(z)$, where $z$ is the inclusion map on the spectrum of $N$. The proof in Conway's book relies upon the theory of $C^*$ algebras, which I want to avoid. Is there a proof that does not use this approach? Can anyone point me to a good source if one exists?
Thanks.
You could try Brian Hall's book "Quantum Theory for Mathematicians" (2013): from what I recall his proof of the spectral theorem for bounded self-adjoint operators (Ch.8) as well as the modifications to the proof for the case of normal operators (Theorem 10.20 ff.) avoid technical arguments from $C^*$-algebra theory.