Corollary 2

The topological group $\widehat{G}$ is locally compact.

In fact, $\mathsf{X}(L^1(G))$ is locally compact (corollary of Theorem 1 of I, p. 29).

We shall identify $\widehat{G}$ with $\mathsf{X}(L^1(G))$ and $\mathsf{X}$(Stell(G)). For $x\in G$ and $\chi \in \widehat{G}$, we shall denote by $\langle \chi , x\rangle$ the complex number $\chi (x)$, which belongs to $\mathbf{U}$.

We say that $x$ and $\chi$ are orthogonal if $\langle \chi , x\rangle = 1$. Let A be a subset of G (resp. of $\widehat{G}$); the set of elements of $\widehat{G}$ (resp. of G) orthogonal to A is a closed subgroup of $\widehat{G}$ (resp. of G), called the orthogonal of A and denoted by $A^{\bot}$. The orthogonal of G is reduced to $e$.

For $x\in G$, let us denote by $\eta (x)$ the mapping of $\widehat{G}$ into $\mathbf{U}$ defined by $\chi \mapsto$ $\langle \chi , x\rangle$. By definition of the multiplication in $\widehat{G}$, the mapping $\eta (x)$ is a group homomorphism. It is continuous since the mapping $(x, \chi )\mapsto \langle \chi , x\rangle$ of $G\times \widehat{G}$ into $\mathbf{U}$ is continuous (TG, X, p. 28, th. 3). Thus we have defined a mapping $\eta$, called canonical, of G into the bidual group $\widehat{\widehat{G}}$; it is a group homomorphism. Moreover, the mapping $\eta$ is continuous (TG, X, p. 28, th. 3). We shall prove later (II, p. 220, th. 2) that $\eta$ is a group isomorphism of G onto $\widehat{\widehat{G}}$.

Let G and H be locally compact commutative groups, and let $\varphi : G\rightarrow H$ be a morphism of topological groups. For every $\chi \in \widehat{H}$, the mapping $\chi \circ \varphi$ is a character of G, denoted by $\widehat{\varphi}(\chi )$. This definition is expressed by the formula

for every $\chi \in \widehat{H}$ and every $x\in G$. It follows that $\widehat{\varphi}$ is a morphism of the topological group $\widehat{H}$ into the topological group $\widehat{G}$; one says that $\widehat{\varphi}$ is the dual of the morphism $\varphi$.

Let K be a locally compact commutative group and let $\psi : H\rightarrow K$ be a morphism of topological groups. The definition shows that $\widehat{\psi}\circ \varphi =\widehat{\varphi}\circ \widehat{\psi}$. If $\varphi$ is the identity mapping of G, then $\widehat{\varphi}$ is the identity mapping of $\widehat{G}$. In particular, if $\varphi$ is a group isomorphism, the same is true of $\widehat{\varphi}$, and $\widehat{\varphi}^{-1}$ is the dual of $\varphi^{-1}$.