Proposition 4

Let $A, B$ be two linearly topologized commutative rings, $B$ being Hausdorff and complete, and $u : A \to B$ a continuous homomorphism. For every family $b = (b_i)_{1 \leq i \leq p}$ of elements of $B$, there exists a unique continuous homomorphism

such that $\tilde{u}(a) = u(a)$ for all $a \in A$ and $\tilde{u}(X_i) = b_i$ for $1 \leq i \leq p$.

There exists a unique homomorphism $v : A[X_1, \ldots, X_p] \to B$ such that $v(a) = u(a)$ for $a \in A$ and $v(X_i) = b_i$ for $1 \leq i \leq p$. Moreover, if $\mathfrak{J}$ is a neighbourhood of 0 in $B$ which is an ideal, $\overline{u}^{-1}(\mathfrak{J}) = 3$ is an ideal of $A$ which is a neighbourhood of 0 and, for every polynomial $P \in \mathbf{N}_\mathfrak{J}$, clearly $v(P) \in \mathfrak{J}$ and hence $v$ is continuous. As $A[X_1, \ldots, X_p]$ is dense in $A\{X_1, \ldots, X_p\}$, the existence and uniqueness of $\tilde{u}$ follow from General Topology, Chapter 111,§ 3, no. 3, Proposition 5 and the principle of extension of identities.

In the special case where $A = B$ and $u$ is the identity mapping we shall write $f(b_1, \ldots, b_p)$ or $f(b)$ for the value of $\tilde{u}(f)$ for every restricted formal power series $f \in A\{X_1, \ldots, X_p\}$.

Remarks