Pontryagin duality

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this theorem. The subject is named after Lev Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940. Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups: Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series; Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and Complex-valued functions on a finite abelian group have discrete Fourier transforms, which are functions on the dual group, which is a (non-canonically) isomorphic group.