Lemma 1

Let $L$ be an infinite set and $(u_\lambda)_{\lambda \in L}$ a family of elements of $A[[I]]$, and put $u_\lambda = \sum \alpha_{\lambda,\nu} X^\nu$ for $\lambda \in L$. Then the following conditions are equivalent: (i) The family $(u_\lambda)_{\lambda \in L}$ is summable (Gen. Top., III, p. 262) in $A[[I]]$. (ii) We have $\lim u_\lambda = 0$, taken along the filter of complements of finite subsets of $L$. (iii) For every $\nu \in \mathbf{N}^{(I)}$ we have $\alpha_{\lambda,\nu} = 0$ except for a finite number of indices $\lambda \in L$.

When these conditions hold, the series $u = \sum_{\lambda \in L} u_\lambda$ is equal to $\sum \alpha_\nu X^\nu$ with $a_\nu = \sum_{\lambda \in L} \alpha_{\lambda,\nu}$ for each $\nu \in \mathbf{N}^{(I)}$.

The equivalence of (i) and (ii) follows from Cor. 2 of Gen. Top., III, p. 263. The equivalence of (ii) and (iii) follows from the properties of limits in a product space (Gen. Top., I, p. 55, Cor. 1). The last assertion follows from Prop. 4 of Gen. Top., III, p. 266.

Let us give some examples of summable families. a) Let $u \in A[[I]]$ and let $a_\nu$ be the coefficient of $X^\nu$ in $u$. The family $(\alpha_\nu X^\nu)_{\nu \in \mathbf{N}^{(I)}}$ is then summable, with sum $u$ (which justifies writing $u = \sum \alpha_\nu X^\nu$).

b) Let $u \in A[[I]]$; for every integer $p \geq 0$ let $u_p$ be the homogeneous component of degree $p$ of $u$. Then the family $(u_p)_{p \geq 0}$ is summable and we have $u = \sum_{p \geq 0} u_p$.

c) Let $(u_\lambda)_{\lambda \in L}$ be a family of elements of $A[[I]]$ and suppose that for every integer $n \geq 0$ the set of $\lambda \in L$ such that $\omega(u_\lambda) < n$ is finite. Then the family $(u_\lambda)_{\lambda \in L}$ is summable.