Pontryagin duality
1 Pontryagin dual
Let be a locally compact abelian group (http://planetmath.org/TopologicalGroup) and the 1-torus (http://planetmath.org/NTorus), i.e. the unit circle in .
Definition - A continuous homomorphism is called a character of . The set of all characters is called the Pontryagin dual of and is denoted by .
Under pointwise multiplication is also an abelian group. Since is a group of functions we can make it a topological group under the compact-open topology (topology of convergence on compact sets).
2 Examples
-
β’
, via with .
-
β’
, via with .
-
β’
, via with .
3 Properties
The following are some important of the dual group:
Theorem - Let be a locally compact abelian group. We have that
-
β’
is also locally compact.
-
β’
is second countable if and only if is second countable.
- β’
-
β’
is discrete if and only if is compact.
-
β’
for any finite set . This isomorphism is natural.
4 Pontryagin duality
Let be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map defined by
This canonical construction preserves identity mappings and compositions, i.e. the dualization process is a functor:
Theorem - The dualization is a contravariant functor from the category of locally compact abelian groups to itself.
5 Isomorphism with the second dual
Although in general there is not a canonical identification of with its dual , there is a natural isomorphism between and its dualβs dual :
Theorem - The map defined by , where , is a natural isomorphism between and .
6 Applications
The study of dual groups allows one to visualize Fourier series, Fourier transforms and discrete Fourier transforms from a more abstract and unified view-point, providing the for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the of the of abstract abelian harmonic analysis.
Title | Pontryagin duality |
Canonical name | PontryaginDuality |
Date of creation | 2013-03-22 17:42:42 |
Last modified on | 2013-03-22 17:42:42 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 7 |
Author | asteroid (17536) |
Entry type | Theorem |
Classification | msc 43A40 |
Classification | msc 22B05 |
Classification | msc 22D35 |
Synonym | Pontrjagin duality |
Synonym | Pontriagin duality |
Related topic | DualityInMathematics |
Defines | Pontryagin dual |
Defines | Pontrjagin dual |
Defines | Pontriagin dual |
Defines | dual of an abelian group |
Defines | character |