 Weak operator topology

On $B(H)$ for a Hilbert space $H$, the weak operator topology is generated by the seminorms $T\mapsto |⟨Tx,y⟩|$, for all vectors $x,y\in H$. A net $T_i$ converges to $T$ exactly when every matrix coefficient $⟨(T_i-T)x,y⟩$ tends to zero. The strong operator topology implies the weak operator topology by the Cauchy-Schwarz inequality.

The reverse implication fails. For the unilateral shift $S$ on ${\ell }^2$, $S^n$ tends to zero in the weak operator topology. For finitely supported vectors $x,y$, the coefficient $⟨S^{n}x,y⟩$ eventually vanishes; approximation by finitely supported vectors and $\Vert S^n\Vert =1$ extend this to arbitrary $x,y$. But $\Vert S^{n}x\Vert =\Vert x\Vert$, so $S^n$ does not tend strongly to zero for nonzero $x$. Matrix coefficients can vanish while the total norm of a moving vector stays constant.