A duality for locally compact abelian (LCA) groups that assigns to each LCA group G its character group G^ = Hom_cont(G,S^1) endowed with the compact-open (or Pontryagin) topology, and establishes a natural isomorphism G ≅ (G^)^ for all LCA groups, interchanging convolution and pointwise multiplication of characters.

Principle

Continuous characters to the circle group S^1 capture the harmonic and dual structure of an LCA group; taking the group of such characters with the appropriate topology yields a contravariant equivalence between the category of LCA groups and itself.

Demonstration

For the integers Z (discrete LCA), the Pontryagin dual is the circle group S^1; for the reals R, the dual is isomorphic to R again by identifying characters with exponential maps; finite abelian groups are self-dual via pairing with roots of unity.

Misapplication

Applying Pontryagin duality to non-abelian or non-locally-compact groups without modification, or neglecting the required topology on the character group; these omissions break the duality and the double-dual identification.

Consequence

A powerful toolbox for harmonic analysis: Fourier transforms are instances of Pontryagin duality, and many structural theorems (decompositions, Plancherel, characters) follow from the duality and the double-dual identification.

Reversal

Contrast with purely algebraic duals (Hom(G,A) for abelian groups with discrete topology): Pontryagin duality is topological and may agree with algebraic duality only in special cases; reversing to algebraic duals drops continuity and topological structure.

Boundary

Requires locally compact and abelian hypotheses and continuous homomorphisms into the circle; the theory does not extend verbatim to non-abelian groups, to groups lacking local compactness, or to non-Hausdorff settings without substantial modification.

Semantic Tension

Between 'algebraic dual' and 'topological dual': the algebraic Hom-group neglects continuity and topology, while Pontryagin duality intertwines algebraic character groups with topological structure, producing subtler identifications.

Synthesis

Pontryagin duality identifies an LCA group with the group of continuous circle-valued characters of its dual, marrying topological and algebraic features to yield a self-dual category under a contravariant equivalence that underpins Fourier analysis and harmonic decomposition.