Pontryagin Duality

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.

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.

A duality between the category of locally compact abelian (LCA) topological groups and itself that assigns to each LCA group G its Pontryagin dual G^ = Hom_cont(G, S^1) (the group of continuous characters into the circle), with the property that the natural map from G to the double dual G^^ is an isomorphism for all LCA groups.