Bounded operators on separable Hilbert spaces

Strong operator topology is generated by the semi-norms $\rho_x(T):=\lVert Tx\rVert$.

What we have to show here is that if $T$ is linear and bounded, we can find $T_n$ of finite-rank such that $\rho_x(T-T_n)\to 0$ for all $x$. Let $\{e_n\}$ an Hilbert basis of $H$, and $P_N$ the projection over $\operatorname{Span}\{e_1,\dots,e_N\}$. Let $T_N:=TP_N$. It's a finite-ranked operator. As $P_Nx\to x$ as $N\to +\infty$ and $T$ is bounded, we have convergence in the strong operator topology.

This result helps us to see that strong operator topology is really different from the topology induced by operator norm. Indeed, in the later, the closure of finite ranked operators is the set of compact operators, no matter whether the Hilbert space is separable or not.