Proposition 4

— Let $I$ be a set and $E$ an associative, commutative, unital, linearly topologized separated complete $A$-algebra.

(i) Let $\varphi$ be a continuous homomorphism of $A[[I]]$ into $E$ and $x_i = \varphi(X_i)$. Then: (a) for all $i \in I$, $x_i^n$ tends to 0 as $n$ tends to $+ \infty$; (b) if $I$ is infinite, $x_i$ tends to 0 along the filter of complements of finite subsets of $I$.

(ii) Let $x = (x_i)_i$, $_I$ be a family of elements of $E$ satisfying a) and b) of (i). Then there exists one and only one unital continuous homomorphism $\varphi$ of $A[[I]]$ into $E$ such that $\varphi(X_i) = x_i$ for all $i \in I$.

For all $i \in I$, $X_i^n$ clearly tends to 0 in $A[[I]]$ as $n$ tends to $+ \infty$; on the other hand when $I$ is infinite, $X_i$ tends to 0 along the filter of complements of finite subsets of $I$. This proves (i).

Let $(x_i)_{i \in I}$ be a family of elements of $E$ satisfying conditions a) and b) of (i), let $\psi$ be the homomorphism $u \mapsto u((x_i)_{i \in I})$ of $A[(X_i)_{i \in I}]$ into $E$, and let $V$ be a neighbourhood of 0 of $E$ which is an ideal of $E$. By b) there exists a finite subset $J$ of $I$ such that $x_i \in V$ for all $i \in I - J$. Next there exists by a) an integer $n \geq 0$ such that $x_i^n \in V$ for all $i \in J$. Let $\beta$ be the element of $\mathbf{N}^{(I)}$ such that $\beta_i = n - 1$ for $i \in J$ and $\beta_i = 0$ for $i \in I - J$. If we define the ideal $a_\beta$ of $A[[I]]$ as at the beginning of No. 2 (IV, p. 26), then

This shows that $\psi$ is continuous if we equip $A[(X_i)_{i \in I}]$ with the topology induced by that of $A[[I]]$. Since $E$ is separated and complete, $\psi$ extends to a continuous unital homomorphism $\varphi$ of $A[[I]]$ into $E$. We have $\varphi(X_i) = \psi(X_i) = x_i$ for all $i \in I$. Finally let $\varphi'$ be a continuous unital homomorphism of $A[[I]]$ into $E$ such that $\varphi'(X_i) = x_i$. We have $\varphi'(u) = \varphi(u)$ for all $u \in A[(X_i)_{i \in I}]$, hence $\varphi' = \varphi$ because $A[(X_i)_{i \in I}]$ is dense in $A[[I]]$.

Let us keep the previous notation. If $u \in A[[I]]$, the image of $u$ by $\varphi$ is denoted by $u(x)$ or $u((x_i)_{i \in I})$ (or also $u(x_1, ..., x_n)$ if $I = (1, 2, ..., n)$) and is called the element of $E$ obtained by substitution of $x_i$ for $X_i$ in $u$, or the value of $u$ for the values $x_i$ of the $X_i$ or also the value of $u$ for $X_i = x_i$. In particular we have $u = u((X_i)_{i \in I})$.

Let $E'$ be an associative commutative and unital linearly topologized separated and complete $A$-algebra. Let $\lambda$ be a continuous unital homomorphism of $E$ into $E'$, and $(x_i)_{i \in I}$ a family of elements of $E$ satisfying conditions a) and b) of Prop. 4 (IV, p. 28). The family $(\lambda(x_i))_{i \in I}$ satisfies the same conditions a) and b). For every $u \in A[[I]]$ we have

for the mapping $u \mapsto A(u((x_i)_{i \in I}))$ is a continuous unital homomorphism of $A[[I]]$ into $E'$ which transforms $X_i$ into $\lambda(x_i)$ for all $i \in I$.

If $J$ and $K$ are two sets, we denote by $A_{J, K}$ the set of all families $(g_j)_{j \in J}$ satisfying the following conditions: (i) for all $j \in J$, $g_j$ is an element of $A[[K]]$ whose constant term is nilpotent; (ii) if $J$ is infinite, $g_j$ tends to 0 along the filter of complements of finite subsets of $J$.

We note that if $J$ is finite, every family of formal power series $(g_j)_{j \in J}$ without constant term in $A[[K]]$ belongs to $A_{J, K}$.

Let $(g_j)_{j \in J}$ be in $A_{J, K}$. By the Cor. of Prop. 3 (IV, p. 28) we have $\lim_{n \to \infty} g_j^n = 0$ for all $j \in J$. Let $f \in A[[J]]$; we can substitute $g_j$ for the variable of index $j$ in $f$ and obtain a formal power series $f((g_j)_{j \in J})$ belonging to $A[[K]]$. Moreover, the mapping $f \mapsto f((g_j)_{j \in J})$ is a continuous homomorphism of $A$-algebras $A[[J]]$ into $A[[K]]$.

In particular if $J = (1, ..., p)$ and $f \in A[[X_1, ..., X_p]]$, we can substitute for each $X_j$ a formal power series $g_j \in A[[K]]$ without constant term; the result of this substitution is written $f(g_1, ..., g_p)$.

Let $x = (x_k)_{k \in K}$ be a family of elements of $E$ satisfying conditions a) and b) of Prop. 4 (IV, p. 28). Let us apply (3), taking for $A$ the homomorphism $u \mapsto u(x)$ of $A[[K]]$ into $E$; we obtain

Let $f = (f_i)_{i \in I} \in (A[[J]])^I$ and $g = (g_j)_{j \in J} \in A_{J, K}$. We denote by $f(g)$ or $f \circ g$ the element $(f_i((g_j),_{\epsilon_J}))_{i \in I}$ of $(A[[K]])^I$. If $f \in A_{I, J}$, we have $f \circ g \in A_{,,K}$ because the mapping $f \mapsto f((g_j)_{i, J})$ of $A[[I]]$ into $A[[K]]$ is continuous.

Let $f \in (A[[J]])^I$, $g \in A_{J, K}$, $h \in A_{K, L}$. Then $g \circ h \in A_{J, L}$ and by (4), we have