# A possible mistake in Walter Rudin, "Fourier analysis on groups"

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.

• What locally compact group $G_2$ has its dual group isomorphic to $\Gamma_2 = \mathbb{Q}$ with the discrete topology? Feb 28, 2016 at 19:24
• @NateEldredge $G_2$ is the dual of $\Gamma_2$, by the Pontrjagin duality. Feb 28, 2016 at 19:33
• @NateEldredge Is this question rhetorical? Every discrete abelian group has a compact Pontrjagin dual. Without thinking much, I imagine that the dual of ${\mathbb Q}_d$ has some kind of solenoidal flavour (it certainly isn't anything like compact Lie, of course) Feb 28, 2016 at 19:44
• @YemonChoi: No, it wasn't rhetorical, just naive :-) Feb 28, 2016 at 19:52
• The dual of $\mathbb Q$ is indeed a solenoid. You might call it "the" solenoid because it is the biggest, but that name is usually taken by the dual of $\mathbb Z[\frac12]$. Another name for the dual of $\mathbb Q$ is the adele quotient: $\mathbb A/\mathbb Q$. Similarly, the binary solenoid is $(\mathbb R\times \mathbb Q_2)/\mathbb Z[\frac12]$. Feb 28, 2016 at 20:47