Strong-star operator topology ($T_i\xi \to T\xi ,T_i^{\ast }\xi \to T^{\ast }\xi (\xi \in H)$)

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.