Proposition 4

Let $G = \prod_{\lambda \in L} G_\lambda$ be the product of a family of Hausdorff commutative groups. Then a family $(x_i)_{i \in I}$ of points of $G$ is summable if and only if, for each $\lambda \in L$, the family $(\mathrm{pr}_\lambda x_i)_{i \in I}$ is summable; and if $s_\lambda$ is the sum of this family, then $s = (s_\lambda)$ is the sum of the family $(x_i)$.

This follows immediately from the condition for convergence, with respect to a filter, of a function which takes its values in a product space (Chapter I, § 1, no. 6, Corollary 1 to Proposition 10); in effect, for each finite subset $J$ of $I$, we have