I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):

Suppose $E$ is a coset in $\Gamma_2$ and $\alpha$ is an affine map of $E$ into $\Gamma_1$. THen $\alpha$ can be extended to an affine map of the closure $\overline{E}$ of $E$, and $\alpha(\overline{E})$ is a closed coset in $\Gamma_1$

I think $\alpha(\overline{E})$ is not necessary a closed coset in $\Gamma_1$. The book just states that it is a corollary of the uniform continuity of $\alpha$ but consider $E = \Gamma_2 = \mathbb{Q}$ with discrete topology, $\Gamma_1 = \mathbb{R}$ and $\alpha$ is identity map. In that case, $\overline{E} = E$ and the image is not closed in $\mathbb{R}$.

Edit: Since nobody proved the statement or pointed out where my example is wrong I'm going to assume the lemma is indeed incorrect.