Proposition 3

If the normed space Y is complete, then every series $(u_n)$ of bounded mappings of X into Y which is absolutely convergent in the normed space $\mathscr{B}(X; Y)$ (i.e. which is such that $\sum_{n=0}^\infty ||u_n|| < +\infty$; cf. Chapter IX, § 3, no. 6) is uniformly convergent in X.

For since $\mathscr{B}(X; Y)$ is complete (no. 1, Proposition 2, Corollary 1), the result follows from Chapter IX, § 3, no. 6, Proposition 11 and the definition of a uniformly convergent series.