A duality principle that associates to each locally compact abelian (LCA) topological group G its character group G^ (the group of continuous homomorphisms from G to the circle group), and that identifies G naturally with the double dual G^^ under the evaluation pairing.

Principle

Continuous characters into the circle group separate points of LCA groups and the evaluation pairing yields a canonical isomorphism between an LCA group and the double dual of its character group; dualization reverses arrows and converts products to coproducts in the LCA category.

Demonstration

The integers Z (discrete) have dual the circle group T; R^n is self-dual via the Fourier transform; a finite abelian group A is identified with Hom(A, S^1) and the double dual recovers A—these illustrate how characters and evaluation produce the canonical identification.

Misapplication

Treating Pontryagin duality as valid for arbitrary (nonabelian or non–locally-compact) topological groups, or using purely algebraic character groups without the topology, leads to incorrect identifications and loss of continuity information.

Consequence

When applied correctly, Pontryagin duality underpins harmonic analysis on LCA groups, yields equivalences between categories of LCA groups and their duals, and explains the inversion formulas for Fourier transforms and Plancherel theory in that setting.

Reversal

The inversion would be to assert every group of homomorphisms into S^1 corresponds to an underlying LCA group without checking topology or abelian property; this fails in general and highlights the directional role of continuity and local compactness.

Boundary

Applies only to locally compact abelian topological groups and continuous characters into the circle group; it excludes nonabelian groups, groups lacking adequate topology, and duals formed with noncompact target groups.

Semantic Tension

Competes with purely algebraic dual notions (vector-space duals, Pontryagin dual in finite-group algebra) — those ignore topology and may coincide only in specific discrete/compact cases; the tension is continuity versus algebraic homomorphisms.

Synthesis

Pontryagin duality is the topological identification of an LCA group with the double dual of its continuous character group, organizing harmonic analysis by turning groups into their spaces of continuous phase-valued characters and recovering structure through evaluation.