 Strong operator topology

On the bounded linear operators $B(X,Y)$ between normed vector spaces, the strong operator topology is generated by the seminorms $p_x(T)=\Vert Tx{\Vert }_Y$, one for every $x\in X$. Thus a net $T_i$ converges to $T$ precisely when $\Vert (T_i-T)x{\Vert }_Y\to 0$ for each fixed vector $x$. This defines a topology, including cases where sequences alone do not characterize it.

Convergence in operator norm implies convergence in the strong operator topology, since $\Vert (T_i-T)x\Vert \le \Vert T_i-T\Vert \Vert x\Vert$. The converse fails even for orthogonal projections. On the Hilbert space ${\ell }^2$, let $P_n$ retain the first $n$ coordinates. The tail of a square-summable sequence tends to zero, so $P_{n}x\to x$ for every $x$, but $\Vert P_n-I\Vert =1$. An increasing sequence of finite-dimensional approximations can therefore converge strongly without uniformly approximating every unit vector.