Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $I$ be a directed set, and let $(T_i)_{i\in I}$ be a net in $\mathcal{L}(H)$ such that each $T_i$ is self-adjoint, meaning $T_i=T_i^*$ where $T_i^*$ denotes the Hilbert-space adjoint. Suppose that $(T_i)_{i\in I}$ is increasing in the operator order, meaning that $T_j-T_i$ is positive whenever $i\le j$, and suppose that there exists $C>0$ such that

\begin{align*} \|T_i\|_{\mathcal{L}(H)}\le C \end{align*}

for every $i\in I$. Then there exists a self-adjoint operator $T\in\mathcal{L}(H)$ such that $T_i\to T$ in the strong operator topology, meaning that $\|T_i\xi-T\xi\|_H\to 0$ for every $\xi\in H$. If, moreover, $M\subseteq\mathcal{L}(H)$ is a von Neumann algebra and $T_i\in M$ for every $i\in I$, then $T\in M$.