Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $I$ be a directed set, and let $(T_i)_{i\in I}$ be a net in $\mathcal{L}(H)$. Let $T\in\mathcal{L}(H)$. If $T_i\to T$ in the operator norm topology on $\mathcal{L}(H)$, then $T_i\to T$ in the strong operator topology. If $T_i\to T$ in the strong operator topology, then $T_i\to T$ in the weak operator topology.