= Weak operator topology
{wiki}
On $B(H)$ for a <Hilbert space> $H$, the <weak operator topology> is generated by the <seminorms> $T\mapsto|\langle Tx,y\rangle|$, for all <vectors> $x,y\in H$. A <net> $T_i$ converges to $T$ exactly when every matrix coefficient $\langle(T_i-T)x,y\rangle$ 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 $\langle S^nx,y\rangle$ eventually vanishes; approximation by finitely supported <vectors> and $\|S^n\|=1$ extend this to arbitrary $x,y$. But $\|S^nx\|=\|x\|$, 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.
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