After reading an answer to the relation between Fourier Series and Hilbert Spaces I am curious about interesting examples of Abelian Topological Groups. Specifically, the answer explains that if $f$ has a fourier expansion the domain of $f$ must be an Abelian Topological Group.
A quick example that occurred to me was the metric space $(\mathbb{R}^n,+)$ equipped with the Euclidean distance metric. A topology can be defined in terms of balls $B(x,r)$ where $r > 0$ and $x \in \mathbb{R}^n$.