[math] \newcommand{\mathds}{\mathbb}[/math]

Assuming that an operator [math]T\in B(H)[/math] is normal, [math]TT^*=T^*T[/math], apply the Gelfand theorem to the [math]C^*[/math]-algebra that it generates

[[math]] \lt T \gt \subset B(H) [[/math]]

in order to deduce a diagonalization theorem for [math]T[/math].