Pontryagin duality

Pontryagin duality states that every locally compact abelian group is naturally isomorphic to the dual of its character group, with the duality reversing arrows.

Locally compact abelian group: A Hausdorff abelian topological group whose topology is locally compact. This is the class of groups to which the duality theorem applies.
Topological group: A group equipped with a topology that makes multiplication and inversion continuous. Locally compact abelian groups are topological groups with additional properties.
Fourier transform: A transform that represents functions or measures through their frequencies or characters. Pontryagin duality provides the group-theoretic setting for Fourier transforms.
Finite abelian group: A group with finitely many elements and a commutative operation. Its character group has the same finite size and is isomorphic to it.
Lev Pontryagin: A Soviet mathematician known for work in topology, algebra, and duality theory. He established the duality theorem for locally compact abelian groups.
Character group: The group of continuous homomorphisms from a topological group to the circle group. Pontryagin duality constructs a second group from all continuous circle-valued characters.
Abelian group: A group whose operation is commutative. Commutativity is essential to the character-based duality in this form.
Fourier inversion theorem: A theorem recovering a function from its Fourier transform under suitable conditions. Duality explains why inversion exchanges a group and its character group.
Finite cyclic group: A finite group generated by a single element. Its characters form another cyclic group of the same order.
Harmonic analysis: The study of functions and measures through oscillations, characters, and related decompositions. Pontryagin duality generalizes Fourier analysis to locally compact abelian groups.

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