Proposition 15

Let A be a set equipped with a filter $\mathfrak{F}$. For every $\alpha \in A$, let $(u_{\alpha,n})_{n \in \mathbf{N}}$ be a summable sequence of positive numbers, and let $\Phi_\alpha$ be the function defined on the interval ]0, 1] of \mathbf{R} by $\Phi_\alpha(x) = \sum_{n \in \mathbf{N}} u_{\alpha,n} x^n$. In order that there exist a summable sequence $(u_n)_{n \in \mathbf{N}}$ of positive numbers such that

it is necessary and sufficient that $\Phi_\alpha$ converge pointwise on ]0, 1[, with respect to $\mathfrak{F}$, to a function $\Phi$ continuous at the point 1. In this case, $\Phi(x) = \sum_{n \in \mathbf{N}} u_n x^n$ for all $x \in ]0, 1[$.

Analogous results are obtained by taking M to be the monoid $\mathbf{N}^n$, where $n$ denotes an integer > 1; the space X may then be identified with $D^n$, and one can choose $]0, 1[^n$ as the full submonoid. We leave to the reader the task of transcribing Th. 3 in this case.

Example 2). — Let us take for M the set $\mathbf{R}_+$, equipped with the law of addition and with the usual topology. Let P be the set of complex numbers z with positive real part, equipped with the topology induced by C, and with the law induced by addition in C. For every $p \in P$, denote by $f(p)$ the character $x \mapsto e^{-px}$ of $\mathbf{R}_+$; it is easily verified that $f$ is an isomorphism of the topological monoid structure of P onto that of X; we shall identify X with P by means of $f$. It is clear that $\mathbf{R}_+$ is a full submonoid of P, and Th. 3 yields the following result: