Let $H$ be a [Hilbert space](/page/Hilbert%20Space), and let $M \subseteq \mathcal{L}(H)$ be a von Neumann algebra. Let $M_*$ denote the [vector space](/page/Vector%20Space) of all normal linear functionals $\omega:M\to\mathbb C$, where normal means that $\omega$ is ultraweakly continuous, equipped with the norm inherited from the Banach dual $M^*$. Then $M_*$ is a [Banach space](/page/Banach%20Space), and the canonical evaluation map

\begin{align*} \kappa:M\to (M_*)^* \end{align*}

defined by

\begin{align*} \kappa(x)(\omega)=\omega(x) \end{align*}

for $x\in M$ and $\omega\in M_*$ is an isometric isomorphism.