Pontryagin dual - Revision history

Vipul: Created page with "==Definition== ===General definition for a locally compact abelian group=== Suppose $G$ is a locally compact abelian group. The '''Pontryagin dual''' of
2012-08-16T02:08:04Z

Created page with "==Definition== ===General definition for a locally compact abelian group=== Suppose <math>G</math> is a locally compact abelian group. The '''Pontryagin dual''' of <math..."

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==Definition==

===General definition for a locally compact abelian group===

Suppose <math>G</math> is a [[locally compact abelian group]]. The '''Pontryagin dual''' of <math>G</math>, denoted <math>\hat{G}</math>, is defined as follows:

* As an abstract group, it is the group of all continuous homomorphisms from <math>G</math> to the [[circle group]], with pointwise multiplication (these homomorphisms are called [[character of a locally compact abelian group|character]]s, 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 <math>G</math> is a [[finite abelian group]]. The '''Pontryagin dual''' of <math>G</math>, denoted <math>\hat{G}</math> is the finite group of all homomorphisms from <math>G</math> to the [[circle group]].

Note that treating <math>G</math> as a [[discrete topological group]], this definition agrees with the previous definition, and <math>\hat{G}</math> 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.