Proposition 7

— Let $(y_i)_{i \in I}$, be a family of elements of E. When I is infinite, we assume that $y_i$ tends to 0 along the filter of complements of finite subsets of I. There exists then a unique continuous A-derivation D of $A[[I]]$ into the $A[[I]]$*-module* E such that $D(X_i) = y_i$ for all $i \in I$. We have

Since 0 admits in E a fundamental system of neighbourhoods consisting of ideals, the family $((D_i u)(x) \cdot y_i)_{i \in I}$ is summable in E for all $u \in A[[I]]$ (Gen. Top., III, p. 263, Cor. 2). Formula (16) thus defines an A-linear mapping $D : A[[I]] \to E$. We leave the reader to verify that D is a continuous derivation.

Let $D_1$ be a continuous A-derivation of $A[[I]]$ into E, such that $D_1(X_i) = y_i$ for all $i \in I$. The kernel of the continuous derivation $D - D_1$ is a closed subalgebra B of $\mathbf{A}[[\mathbf{I}]]$ containing 1 and the indeterminates $X_i$. Since the polynomial algebra $\mathbf{A}[(X_i)_{i \in I}]$ is dense in $\mathbf{A}[[\mathbf{I}]]$, we have $B = \mathbf{A}[[\mathbf{I}]]$ and so $D_1 = D$.