Definition 2

Given a commutative topological ring $A$, a formal power series

in the ring $A[[X_1, \ldots, X_p]]$ is called restricted if, for every neighbourhood $V$ of 0 in $A$, there is only a finite number of coefficients $c_{n_1 n_2 \ldots n_p}$ not belonging to $V$ (in other words, the family $(c_{n_1 n_2 \ldots n_p})$ tends to 0 in $A$ with respect to the filter of complements of finite subsets of $\mathbf{N}^p$).

If $A$ is linearly topologized, the restricted formal power series in $A[[X_1, \ldots, X_p]]$ form a subring of $A[[X_1, \ldots, X_p]]$, denoted by $A\{X_1, \ldots, X_p\}$: for if

are two restricted formal power series and $\mathfrak{J}$ a neighbourhood of 0 in $A$ which is an ideal of $A$, there exists an integer $m$ such that $c_{n_1 \ldots n_p} \in \mathfrak{z}$ and $c'_{n_1 \ldots n_p} \in \mathfrak{g}$ for every system $(n_1, \ldots, n_p)$ such that $n_k \geq m$ for at least one index $k$; now, if

then $c''_{n_1 \ldots n_p} = \sum c_{r_1 \ldots r_p} c'_{s_1 \ldots s_p}$ for all systems $(r_k), (s_k)$ such that $r_k + s_k = n_k$ for $1 \leq k \leq p$; we conclude that if $n_k \geq 2m$, then $r_k \geq m$ or $s_k \geq m$ and hence, since $\mathfrak{z}$ is an ideal, $c''_{n_1 \ldots n_p} \in \mathfrak{g}$ so long as $n_k \geq 2m$ for at least one $k$, which establishes our assertion. Moreover, every derivative $\partial T / \partial X_i$ ($1 \leq i \leq p$) of a restricted formal power series is restricted, as follows immediately from the definition and the fact that the neighbourhoods $\mathfrak{z} \in \mathscr{B}$ are additive subgroups of $\mathbf{A}$.

If $\mathbf{A}$ is discrete, the ring of restricted formal power series is just the polynomial ring $\mathbf{A}[X_1, \ldots, X_n]$.

Let us always assume that $\mathbf{A}$ is linearly topologized and let $\mathscr{B}$ be a fundamental system of neighbourhoods of 0 in $\mathbf{A}$ consisting of ideals of $\mathbf{A}$; for all $\mathfrak{g} \in \mathscr{B}$, let $p_\mathfrak{g}: \mathbf{A} \to \mathbf{A}/\mathfrak{g}$ be the canonical homomorphism. By definition, for every restricted formal power series $T \in \mathbf{A}\{X_1, \ldots, X_n\}$,

Clearly

is an inverse system of rings (relative to the directed indexing set $\mathscr{B}$) and $(\bar{p}_\mathfrak{g})$ is an inverse system of homomorphisms $\mathbf{A}\{X_1, \ldots, X_n\} \to (\mathbf{A}/\mathfrak{g})[X_1, \ldots, X_p]$; as every polynomial is a restricted formal power series, $\bar{p}_\mathfrak{g}$ is surjective; its kernel $N_\mathfrak{g}$ is the ideal of $\mathbf{A}\{X_1, \ldots, X_n\}$ consisting of the restricted formal power series all of whose coefficients belong to $\mathfrak{g}$; we shall give $\mathbf{A}\{X_1, \ldots, X_n\}$ the (linear) topology for which the $N_\mathfrak{g}$ (for $\mathfrak{g} \in \mathscr{B}$) form a fundamental system of neighbourhoods of 0 (a topology which obviously depends only on that on $\mathbf{A}$). Then it follows from General Topology, Chapter III, § 7, no. 3, Proposition 2 that

is a strict morphism whose kernel is the closure of $\{0\}$ in $\mathbf{A}\{X_1, \ldots, X_n\}$ and whose image is dense in