Proposition 2

Let $n\geqslant 0$ be an integer and let $G_1, . . . ,G_n$ be locally compact commutative groups. Let G be the product group of the groups $G_j$ for $1\leqslant j\leqslant n$. For $1\leqslant j\leqslant n$, let $\lambda_j$ be the injection of $G_j$ into G which associates to $x\in G_j$ the element $(x_k)$ such that $x_k=e$ if $k\not=j$ and $x_j=x$. The mapping

is a group isomorphism.

Let $m$ be the product mapping of $\widehat{G}^n$ into $\widehat{G}$, and for every $j$ such that $1\leqslant j\leqslant n$, let $\pi_j$ be the projection of G onto $G_j$. Let $\mu$ be the morphism of topological groups $m\circ (\widehat{\pi}_j)_j$ from $\prod\widehat{G}_j$ into $\widehat{G}$, such that

The mapping $\mu$ is continuous, and one verifies that $\mu$ and $(\widehat{\lambda}_j)$ are reciprocal bijections of one another. The proposition follows.