Proposition 5

Let $f$ be a continuous homomorphism of a commutative group $G$ into a commutative group $G'$. If $(x_i)$ is a summable family in $G$, then $(f(x_i))$ is a summable family in $G'$, and we have

If $J$ is any finite subset of the index set, then $f \left( \sum_{i \in J} x_i \right) = \sum_{i \in J} f(x_i)$, and the image under $f$ of a convergent filter base is a convergent filter base (Chapter I, § 7, no. 4, Corollary 1 to Proposition 9).