Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $T\in\mathcal L(H)$ be normal, and let $E:\mathcal B(\sigma(T))\to\mathcal L(H)$ be the spectral measure of $T$. For every pair of bounded Borel functions $f,g:\sigma(T)\to\mathbb C$ and every pair of scalars $\alpha,\beta\in\mathbb C$, the [Borel functional calculus](/theorems/2696) satisfies

\begin{align*} (\alpha f+\beta g)(T)=\alpha f(T)+\beta g(T). \end{align*}

Also,

\begin{align*} (fg)(T)=f(T)g(T). \end{align*}

Finally, if $\overline f:\sigma(T)\to\mathbb C$ denotes the bounded Borel function $z\mapsto \overline{f(z)}$, then

\begin{align*} (\overline f)(T)=f(T)^*. \end{align*}

If $(f_n)_{n=1}^{\infty}$ is a sequence of bounded Borel functions $f_n:\sigma(T)\to\mathbb C$ such that there is a constant $M>0$ with $|f_n(z)|\le M$ for every $n\in\mathbb N$ and every $z\in\sigma(T)$, and if $q:\sigma(T)\to\mathbb C$ is a bounded Borel function such that $f_n(z)\to q(z)$ for every $z\in\sigma(T)$, then

\begin{align*} f_n(T)h\to q(T)h \end{align*}

in $H$ for every $h\in H$.