Remark 4

Suppose that the uniform space $Y$ carries a commutative and associative law of composition, written additively, such that the mapping $(y, y') \to y + y'$ is continuous on $Y \times Y$. Then, if $(u_n)$ is a sequence of continuous mappings of $X$ into $Y$ such that the series whose general term is $u_n$ is uniformly convergent in $X$, the sum of the series is continuous on $X$.

We leave it to the reader to state the corresponding result for uniformly summable families (no. 1, Remark 5) of continuous mappings.