Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $I_H\in\mathcal{L}(H)$ denote the identity operator, let $\mathcal{L}(H)^*$ denote the Banach dual of $\mathcal{L}(H)$ with operator norm $\|\cdot\|_{\mathcal{L}(H)^*}$, and let $\mathcal{T}(H)$ be the [Banach space](/page/Banach%20Space) of trace-class operators on $H$ with trace norm $\|\cdot\|_{\mathcal{T}(H)}$. Let $\mathcal{L}(H)_*$ denote the Banach space of normal linear functionals on the von Neumann algebra $\mathcal{L}(H)$, where normality is equivalently ultraweak continuity and, for positive functionals, preservation of bounded increasing strong limits of positive operators. For each $T\in\mathcal{T}(H)$ define

\begin{align*} \omega_T:\mathcal{L}(H)&\to\mathbb C \end{align*}

\begin{align*} a&\mapsto \operatorname{Tr}(Ta). \end{align*}

Then the map $T\mapsto \omega_T$ is an isometric isomorphism from $\mathcal{T}(H)$ onto $\mathcal{L}(H)_*$. Equivalently, every normal linear functional on $\mathcal{L}(H)$ is uniquely of the form $a\mapsto \operatorname{Tr}(Ta)$ for a trace-class operator $T\in\mathcal{T}(H)$. Moreover, the normal states on $\mathcal{L}(H)$ are exactly the functionals $\omega_T$ with $T\in\mathcal{T}(H)$, $T\ge 0$, and $\operatorname{Tr}(T)=1$.