Pontryagin duality

Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:

As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.

One important application of Pontryagin duality is the following characterization of compact abelian topological groups:

Theories built to date are divided into two main groups: the theories where the dual object has the same nature as the source one (like in the Pontryagin duality itself), and the theories where the source object and its dual differ from each other so radically that it is impossible to count them as objects of one class.