Operator topology

An operator topology specifies convergence of linear operators. Common choices include the norm, strong, and weak operator topologies.

Operators $T_n$ converge to $T$ in the strong operator topology when $T_{n}x\to Tx$ in norm for every vector $x$.

This topology on bounded operators is generated by the seminorms $(\Vert T\xi {\Vert }^2+\Vert T^{\ast }\xi {\Vert }^2{)}^{1/2}$. A net converges exactly when both it and its adjoint net converge in the strong operator topology. It is the topology used in the full-unit-ball version of the Kaplansky density theorem.

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.