= Strong operator topology
{wiki}
On the <bounded linear operators> $B(X,Y)$ between <normed vector spaces>, the <strong operator topology> is generated by the <seminorms> $p_x(T)=\|Tx\|_Y$, one for every $x\in X$. Thus a <net> $T_i$ converges to $T$ precisely when $\|(T_i-T)x\|_Y\to0$ 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 $\|(T_i-T)x\|\leq\|T_i-T\|\|x\|$. 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 $P_nx\to x$ for every $x$, but $\|P_n-I\|=1$. An increasing sequence of finite-dimensional approximations can therefore converge strongly without uniformly approximating every unit <vector>.
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