Remark 2

Let $(u_n)$ be a sequence of continuous linear mappings from a Banach space E into a Banach space F; it may happen that $(u_n(x))$ has a limit at every point of an everywhere dense vector subspace T of E, without the sequence $(u_n)$ being bounded in the normed space $\mathscr{L}(E; F)$. For example, take E to be the space of all continuous numerical functions on $\mathbf{R}$, tending to zero at infinity, with the norm $\|f\| = \sup_{x \in \mathbf{R}} |f(x)|$ and let T be the subspace of continuous numerical functions with compact support. The sequence of continuous linear mappings $f \mapsto nf(n)$ from E into $\mathbf{R}$ converges to 0 for all $f \in T$, but is not bounded in $\mathscr{L}_b(E; \mathbf{R})$. The same example shows that in the space $\mathscr{L}(T; \mathbf{R})$, a sequence $(v_n)$ may be simply convergent and non-bounded for the topology of bounded convergence.

On the other hand, the sequence of continuous linear mappings $f \mapsto \sum_{k=1}^n f(k)$ is a Cauchy sequence in $\mathscr{L}(T; \mathbf{R})$ for the topology of simple convergence, but does not tend to a limit in $\mathscr{L}(T; \mathbf{R})$ for this topology.