Let $K$ be a [Hilbert space](/page/Hilbert%20Space), let $M\subseteq\mathcal{L}(K)$ be a von Neumann algebra with unit $1$, and let $\phi:M\to\mathbb C$ be a normal state, meaning that $\phi$ is a positive complex-linear functional with $\phi(1)=1$ and, for every increasing bounded net $(x_i)_{i\in I}$ in $M_+$ with supremum $x\in M_+$, one has $\phi(x_i)\uparrow\phi(x)$. Then there exist a Hilbert space $H_\phi$, a unital complex-linear $*$-representation $\pi_\phi:M\to\mathcal{L}(H_\phi)$, which is normal in the sense that for every increasing bounded net $(x_i)_{i\in I}$ in $M_+$ with supremum $x\in M_+$, one has $\pi_\phi(x_i)\to\pi_\phi(x)$ strongly on $H_\phi$, and a vector $\Omega_\phi\in H_\phi$ cyclic for $\pi_\phi(M)$ such that $\phi(a)=(\pi_\phi(a)\Omega_\phi,\Omega_\phi)_{H_\phi}$ for every $a\in M$.