Theorem 1

Let E, F, G be three complete Hausdorff commutative groups; let A be a dense subgroup of E and let B be a dense subgroup of F. If f is a $\mathbf{Z}$ continuous $\mathbf{Z}$-bilinear (*) mapping of $A \times B$ into G, then f can be extended by continuity to a continuous $\mathbf{Z}$-bilinear mapping of $E \times F$ into G.

Let $(x_0, y_0)$ be an arbitrary point of $E \times F$, and let $\mathfrak{U}, \mathfrak{B}$ be the traces on A, B respectively of the neighbourhood filters of $x_0, y_0$ ($\mathfrak{U}, \mathfrak{B}$ are filters, by hypothesis). To show that f can be extended by continuity, it is enough to show that $f(\mathfrak{U} \times \mathfrak{B})$ is a Cauchy filter-base on G (Chapter II, § 3, no. 6, Proposition 11). Consider the identity

We shall show that by taking $(x, y)$ and $(x', y')$ in a sufficiently small set of $\mathfrak{U} \times \mathfrak{B}$, and by choosing $x_1$ and $y_1$ suitably, we can make each of the terms on the right-hand side very small. Let W be any neighbourhood of 0 in G; since f is continuous at $(0, 0) \in A \times B$, there is a set $U \in \mathfrak{B}$ and a set $V \in \mathfrak{B}$ such that $f(x' - x, y' - y) \in W$ whenever $x \in U, x' \in U, y \in V, y' \in V$. Take a point $x_1 \in U$ and a point $y_1 \in V$; then for all $x, x'$ in U and all $y, y'$ in V we shall have

On the other hand, the partial mapping $x \to f(x, y_1)$ is continuous on A; hence there is a set $U' \subset U$, belonging to $\mathfrak{U}$, and such that, whenever $x$ and $x'$ belong to $U'$, we have $f(x' - x, y_1) \in W$. Likewise there exists $V' \subset V$ belonging to $\mathfrak{B}$ such that, whenever $y$ and $y'$ belong to $V'$, we have $f(x_1, y' - y) \in W$. Consequently, if $(x, y)$ and $(x', y')$ are any two points of $U' \times V'$, then

this proves the existence of the extension $\overline{f}$ of f. The fact that $\overline{f}$ is $\mathbf{Z}$-bilinear is an immediate consequence of the principle of extension of identities (Chapter I, § 8, no. 1, Corollary 1 to Proposition 2).

Q.E.D.

In the application of this theorem to a Hausdorff topological ring A, we take E, F and G to be $\hat{A}$, A and B to be the ring A, and f to be

(*) $f$ is said to be $\mathbf{Z}$-bilinear if, for all elements $x, x'$ of A and all elements $y, y' \in B$ we have

and

the $\mathbf{Z}$-bilinear mapping $(x, y) \to xy$, which by hypothesis is continuous. We denote again by $xy$ the value of the extended function on $\hat{A} \times \hat{A}$; this function is a law of composition on $\hat{A}$, and to say that it is $\mathbf{Z}$-bilinear means that it is distributive on both sides with respect to addition; and it is also associative, by the principle of extension of identities. Consequently: