Pontryagin duality

Pontryagin duality is a mathematical concept that establishes a natural relationship between certain types of groups called locally compact abelian groups and their "dual groups." The dual group consists of all continuous homomorphisms (structure-preserving maps) from the original group into the circle group (complex numbers with magnitude one). In essence, Pontryagin duality shows that each such group can be perfectly paired with its dual, and taking the dual of the dual returns you to the original. This creates a powerful symmetry, linking groups and functions in harmonic analysis and abstract algebra.