Lemma 7

Let $\lambda = (\lambda_i)_{i\in I}$ be a family of complex numbers. For each $t\in \mathbf{C}^*$, let $n_t$ be the cardinal of the set of $i\in I$ such that $\lambda_i=t$. The family $(\lambda_i)_{i\in I}$ is summable if and only if $n_t$ is finite for each $t\in \mathbf{C}^*$ and the family $(n_tt)_{t\in\mathbf{C}^*}$ is summable. In this case, the sums of these two families are equal.

Suppose that the family $(\lambda_i)_{i\in I}$ is summable. For each $t\in \mathbf{C}$, let $I_t$ be the set of $i\in I$ such that $\lambda_i=t$. By TG, III, p. 39, Theorem 2, applied to the partition of I by the sets $I_t$, the set $I_t$ is finite for each $t\in \mathbf{C}^*$, and moreover the family $(n_tt)_{t\in\mathbf{C}^*}$ is summable, with sum equal to that of the family $(\lambda_i)_{i\in I}$.

Conversely, suppose that $n_t$ is finite for each $t\in \mathbf{C}^*$ and that the family $(n_tt)_{t\in\mathbf{C}^*}$ is summable. Let J be a finite subset of I and Λ the set of the $\lambda_i$ for $i\in J$. We have

hence the family $(\lambda_i)_{i\in I}$ is summable (TG, VII, p. 17, corollary).