Remark

— Suppose that $I$ is finite. For every integer $n \geq 0$ let $b_n$ be the set of formal power series $u \in A[[I]]$ such that $\omega(u) \geq n$. The sequence $(b_n)_{n \geq 0}$ is a fundamental system of neighbourhoods of 0 in $\mathbf{A}[[\mathbf{I}]]$. Therefore a family of elements $u_\lambda$ of $\mathbf{A}[[\mathbf{I}]]$ ($\lambda \in L$) is summable if and only if for every $n \in \mathbf{N}$ the set of $\mathbf{A} \in L$ such that $\omega(u_\lambda) < n$ is finite.