Weak operator topology

Operators $T_n$ converge to $T$ in the weak operator topology when $⟨T_{n}x,y⟩\to ⟨Tx,y⟩$ for every pair of vectors $x,y$.

For a Hilbert space $H$, the linear span of the matrix-coefficient functionals $T\mapsto ⟨Tx,y⟩$ is a predual of $\mathcal{B}(H)$. The unit ball is compact in the resulting weak operator topology by the Tychonoff theorem, and the compact norming dual-pair criterion identifies $\mathcal{B}(H)$ with the dual of that span.