Consequences of Pontryagin Duality?

Question

What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically by the theorem. But of course examples and specific consequences would be most appreciated.

Answer 1

One corollary is that every finite abelian group is a character group, namely $G$ is (isomorphic to) the character group of its dual $\widehat G$. Furthermore, for many statements involving elements of $G$ and its character group $\widehat G$ there is a dual statement by switching the roles of $G$ and $\widehat G$. For example, take the trivial statement $$\forall \chi \in \widehat G \left( \chi = 1 \Leftrightarrow \forall g \in G: \chi(g) = 1 \right).$$ Apply this statement with $\widehat G$ instead of $G$ to get $$\forall f \in \widehat{\widehat{G}} \left( f = 1 \Leftrightarrow \forall \chi \in \widehat G: f(\chi) = 1 \right),$$ which, by Pontryagin duality, is equivalent to $$\forall g \in G\left( g = 1 \Leftrightarrow \forall \chi \in \widehat G: \chi(g) = 1\right),$$ so you have shown $\bigcap_{\chi \in \widehat G} \ker \chi = 1$.

Another example: Take the statement $$\forall g \in G \left( g \in G^m \Leftrightarrow \forall \chi \in \widehat G: (\chi^m = 1 \Rightarrow \chi(g) = 1) \right)$$ which is easy to prove using the result in the first example. By switching the roles of $G$ and $\widehat G$ you obtain the dual statement $$\forall \chi \in \widehat G \left( \chi \in \widehat{G}^m \Leftrightarrow \forall g \in G: (g^m = 1 \Rightarrow \chi(g) = 1 \right).$$