Proposition 19

Let K be a locally compact non-discrete field, not necessarily commutative, and let G be the additive group of K, whose group law is denoted additively. Let $\chi$ be a unitary character of G distinct from 1. For $x, y\in G$, put $\theta (x, y) =\chi (xy)$. Then G is in duality with itself relative to $\theta$.

For $y\in G$, let $\chi_y$ be the mapping of G into $\mathbf{U}$ such that $\chi_y(x) =$ $\chi (xy)$. We have $\chi_y\in \widehat{G}$, and it is necessary to prove that $\beta :y\mapsto \chi_y$ is an isomorphism of topological groups of G onto $\widehat{G}$.

The map $\beta$ is an injective homomorphism of G into $\widehat{G}$; it is continuous (TG, X, p. 28, Theorem 3 applied to the continuous map $\theta$ of $G\times G$ into $\mathbf{C}$). Let us prove that $\theta$ is a homeomorphism onto its image. It is enough (Lemma 2 of II, p. 200) to prove that for every neighbourhood U of 0 in K, there exists a neighbourhood V of $e$ in $\widehat{G}$ such that $\overset{-1}{\beta}(V)\subset U$. Let $x\mapsto |x|$ be an absolute value on K defining the topology of K (AC, VI, §9, No$^o1$, Proposition 1), and let $x_0\in K$ be such that $\chi (x_0)\not= 1$; let us set $\eta =|\chi (x_0)-1|>0$. Let U be a neighbourhood of 0 in K. There exists $\delta >0$ such that U contains the set of $y\in K$ such that $|y|< \delta$. Let V be the set of characters $\xi \in \widehat{G}$ such that $|\langle \xi , x\rangle -1|< \eta$ for every element $x\in K$ satisfying $|x|\leqslant |x_0|/\delta$. This is a neighbourhood of $e$ in $\widehat{G}$. If $y\not= 0$ is such that $\beta (y)$ belongs to V, we therefore have $|\chi (xy)-1|<|\chi (x_0)-1|$ for every $x$ such that $|x|\leqslant |x_0|/\delta$. Consequently, we have $|x_0y^{-1}|>|x_0|/\delta$, and therefore $|y|< \delta$, so that $y\in U$.

Since $\beta$ is a homeomorphism onto its image, the latter is closed in $\widehat{G}$ (TG, III, p. 22, Corollary 2). But moreover the orthogonal of the image of $\beta$ is the set of elements $x$ of G such that $\chi (xy) = 1$ for all $y\in G$, and is therefore reduced to $\{0\}$. The image of $\beta$ is therefore dense in $\widehat{G}$ (Corollary 3 of II, p. 228). We conclude that $\beta$ is surjective.