It has been asked ~here~ how to understand the mapping of a period function onto its Fourier coefficients, which has led me to look into Pontryagin duality. It's fascinating stuff, except I don't buy it.
My problem with this is, suppose you have a function $\phi(x)$ that is continuous on a finite interval $[0,1]$ with Dirichlet boundary conditions for simplicity. It has the Fourier expansion $\phi(x) = \sum c_n \sin(n \pi x) $. These $c_n$'s constitute the same information as $\phi(x)$: knowledge of either one implies knowledge of both. Yet, when you think of $\phi(x)$ and the sequence $\{c_n\}$ as vectors in infinite-dimensional spaces, one of them appears to be uncountably infinite while the other is countably infinite.
This reminds me of the motivation for having different cardinalities, namely, the mapping of integers onto rationals on a finite interval. How is it possible that a countable set ( $\{c_n\}$ ) could be used as a stand-in for $\phi(x)$?
The Wikipedia article on Pontryagin duality contains a clue: "[A] group $G$ and its dual group $\widehat{G}$ are not in general isomorphic, but their group algebras are." I am coming from physics/engineering background, though, so I am not even sure if this a relevant statement.