Corollary 2

Suppose S is barrelled. If the sequence $(u_n)$ tends to u in $\mathscr{L}(R; S)$ and the sequence $(v_n)$ to v in $\mathscr{L}(S, T)$, then the sequence $(v_n \circ u_n)$ tends to $v \circ u$ in $\mathscr{L}(R; T)$.

In fact, the sequence $(v_n)$, being simply bounded in $\mathscr{L}(S; T)$ is equicontinuous, since S is barrelled (III, p. 25, th. 1); the corollary is then a consequence of cor. 1.