Theorem 1 (“Spectral Theorem”)

Let $u$ be a normal partial operator on the Hilbert space E. There exist a locally compact topological space X, a positive measure $\mu$ on X, an isometric isomorphism $\theta$ of $L^2(X, \mu)$ onto E, and a continuous function $g$ on X, such that $u=\theta \circ m_g\circ \theta^{-1}$.

The endomorphism $b(u)$ of E is normal by definition. According to the spectral theorem for normal endomorphisms of E (Corollary 1 of IV, p. 193), there therefore exist a locally compact topological space $\widetilde{X}$, a positive measure $\mu$ on $\widetilde{X}$, an isometric isomorphism $\widetilde{\theta}$ of $L^2(\widetilde{X}, \mu)$ onto E, and a bounded continuous function $h$ on $\widetilde{X}$, such that $b(u)$ coincides with $\widetilde{\theta}\circ m_h\circ \widetilde{\theta}^{-1}$.

Let N be the closed subset of the $x\in \widetilde{X}$ such that $|h(x)|\geqslant 1$. Since the endomorphism $1-b(u)^*b(u) =\widetilde{\theta}\circ m_{1-|h|^2}\circ \widetilde{\theta}^{-1}$ is positive and injective, the set N is locally $\mu$-negligible (Lemma 7 of IV, p. 186 and cor. of Prop. 6 of IV, p. 187). Put then $X =\widetilde{X}-$ N. It is a locally compact space, and the restriction of functions to X induces an isometric isomorphism of $L^2(\widetilde{X}, \mu)$ onto $L^2(X, \mu|X)$ (Prop. 1 of IV, p. 179). The composition of $\widetilde{\theta}$ and of the restriction of functions to X therefore induces an isometric isomorphism $\theta$ of $L^2(X, \mu|X)$ onto E. As $b(u) =\theta \circ m_{h|X}\circ \theta^{-1}$, it follows that $u=\theta \circ m_{\beta^{-1}\circ(h|X)}\circ \theta^{-1}$ (Lemma 2). The theorem follows from this formula.