= Cyclic multiplication model for a normal operator
{title2=$T|_{H_v}\simeq M_z\text{ on }L^2(K,\mu_v)$}
The <continuous functional calculus> of a bounded <normal operator> $T$ is a representation of $C(K)$, with $K=\sigma(T)$. The <Riesz-Markov-Kakutani representation theorem> represents $f\mapsto\langle f(T)v,v\rangle$ by a positive regular measure $\mu_v$. The assignment $f\mapsto f(T)v$ is isometric for the $L^2(\mu_v)$ <norm>, and extends to a unitary map onto the <closed> cyclic subspace it generates. Multiplication by $z$ and $\overline z$ represent $T$ and $T^*$. Orthogonal decomposition into cyclic <reducing subspaces>, allowing an uncountable family, constructs the <Borel functional calculus for a normal operator> without requiring separability.
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