Exercise 13, § 1

Let $G_d$ be the group G endowed with the discrete topology and $\chi$ a unitary character of $G_d$. For every integer $n\geqslant 1$, all elements $x_1, . . . , x_n\in G$ and every $\varepsilon >0$, there exists $\widehat{x}\in \widehat{G}$ such that $|\langle \widehat{x}, x_i\rangle -\chi (x_i)|\leqslant \varepsilon$ for $1\leqslant i\leqslant n$. (The canonical injection of $G_d$ into G has for dual a morphism of $\widehat{G}$ into $\widehat{G}_d$ whose image is dense according to Cor. 5 of II, p. 229.)