Pontryagin dual

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Definition

General definition for a locally compact abelian group

Suppose G is a locally compact abelian group. The Pontryagin dual of G, denoted \hat{G}, is defined as follows:

  • As an abstract group, it is the group of all continuous homomorphisms from G to the circle group, with pointwise multiplication (these homomorphisms are called characters, though that term has other related meanings too).
  • The topology on the set is that of uniform convergence on compact sets.

Definition for a finite group

Suppose G is a finite abelian group. The Pontryagin dual of G, denoted \hat{G} is the finite group of all homomorphisms from G to the circle group.

Note that treating G as a discrete topological group, this definition agrees with the previous definition, and \hat{G} also comes with the discrete topology.

Facts

  • The Pontryagin dual of a locally compact abelian group is also a locally compact abelian group. Thus, the operation of taking Pontryagin duals can be iterated.
  • Pontryagin duality theorem: The canonical homomorphism from a locally compact abelian group to its double Pontryagin dual (i.e., the dual of its dual) is an isomorphism. Thus, being Pontryagin dual is a symmetric relationship.
  • Finite abelian group is isomorphic to its Pontryagin dual: Note, however, that this isomorphism is not canonical.
  • The Pontryagin dual of a compact abelian group is discrete, and the Pontryagin dual of a discrete abelian group is compact. In particular, in the infinite case, it is not necessarily true that a group be isomorphic to its Pontryagin dual.