Remark 5

Let $L$ be a set filtered by a filter $\mathfrak{S}$, and let $\lambda \to u_\lambda$ be a mapping of $L$ into $\mathscr{F}_\mathscr{S}(X; Y)$ which has a limit $v$ with respect to $\mathfrak{S}$; we say then that, with respect to the filter $\mathfrak{S}$, the mappings $u_\lambda$ of $X$ into $Y$ converge uniformly to $v$ [or that the family $(u_\lambda)$ is uniformly convergent to $v$] in every set of $\mathscr{S}$. If $L = \mathbf{N}$ and $\mathfrak{S}$ is the Fréchet filter, we omit mention of $\mathfrak{S}$ in this statement.

More particularly, suppose that there is a commutative and associative law of composition (written additively) defined on $Y$. If $(u_n)$ is any sequence of mappings of $X$ into $Y$, let $v_n$ be the mapping defined by

we say that the series whose general term is $u_n$ is uniformly convergent in every set of $\mathscr{G}$ if the sequence $(v_n)$ is uniformly convergent in every set of $\mathscr{G}$. Likewise we define a uniformly summable family $(u_\lambda)_{\lambda \in L}$ of mappings of $X$ into $Y$ by considering the mappings $x \to \sum_{\lambda \in J} n_\lambda(x)$ for all finite subsets $J$ of $L$ and the limit of these mappings in $\mathscr{F}_\mathscr{G}(X; Y)$ with respect to the directed set of finite subsets of $L$ (Chapter III, § 5, no. 1).