Pontryagin duality
1 Pontryagin dual
Let $G$ be a locally compact abelian group^{} (http://planetmath.org/TopologicalGroup) and $\mathrm{\pi \x9d\x95\x8b}$ the 1torus (http://planetmath.org/NTorus), i.e. the unit circle in $\mathrm{\beta \x84\x82}$.
Definition  A continuous^{} homomorphism^{} $G\beta \x9f\u0386\mathrm{\pi \x9d\x95\x8b}$ is called a character^{} of $G$. The set of all characters is called the Pontryagin dual of $G$ and is denoted by $\widehat{G}$.
Under pointwise multiplication $\widehat{G}$ is also an abelian group. Since $\widehat{G}$ is a group of functions we can make it a topological group^{} under the compactopen topology^{} (topology^{} of convergence on compact sets).
2 Examples

β’
$\widehat{\mathrm{\beta \x84\u20ac}}\beta \x89\x85\mathrm{\pi \x9d\x95\x8b}$, via $n\beta \x86\xa6{z}^{n}$ with $z\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}$.

β’
$\widehat{\mathrm{\pi \x9d\x95\x8b}}\beta \x89\x85\mathrm{\beta \x84\u20ac}$, via $z\beta \x86\xa6{z}^{n}$ with $n\beta \x88\x88\mathrm{\beta \x84\u20ac}$.

β’
$\widehat{\mathrm{\beta \x84\x9d}}\beta \x89\x85\mathrm{\beta \x84\x9d}$, via $t\beta \x86\xa6{e}^{i\beta \x81\u2019s\beta \x81\u2019t}$ with $s\beta \x88\x88\mathrm{\beta \x84\x9d}$.
3 Properties
The following are some important of the dual group:
Theorem  Let $G$ be a locally compact abelian group. We have that

β’
$\widehat{G}$ is also locally compact.

β’
$\widehat{G}$ is second countable if and only if $G$ is second countable.
 β’

β’
$\widehat{G}$ is discrete if and only if $G$ is compact.

β’
$\widehat{({\beta \x8a\x95}_{i\beta \x88\x88J}{G}_{i})}\beta \x89\x85{\beta \x8a\x95}_{i\beta \x88\x88J}\widehat{{G}_{i}}$ for any finite set^{} $J$. This isomorphism^{} is natural.
4 Pontryagin duality
Let $f:G\beta \x9f\u0386H$ be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map $\widehat{f}:\widehat{H}\beta \x9f\u0386\widehat{G}$ defined by
$\widehat{f}\beta \x81\u2019(\mathrm{{\rm O}\x95})\beta \x81\u2019(s):=\mathrm{{\rm O}\x95}\beta \x81\u2019(f\beta \x81\u2019(s)),\mathrm{{\rm O}\x95}\beta \x88\x88\widehat{H},s\beta \x88\x88G$ 
This canonical construction preserves identity mappings and compositions, i.e. the dualization process $\widehat{}$ is a functor^{}:
Theorem  The dualization $\widehat{}:\mathrm{\pi \x9d\x90\x8b\pi \x9d\x90\x9c\pi \x9d\x90\x80}\beta \x9f\u0386\mathrm{\pi \x9d\x90\x8b\pi \x9d\x90\x9c\pi \x9d\x90\x80}$ 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 $G$ with its dual $\widehat{G}$, there is a natural isomorphism between $G$ and its dualβs dual $\widehat{\widehat{G}}$:
Theorem  The map $G\beta \x9f\u0386\widehat{\widehat{G}}$ defined by $s\beta \x86\xa6\widehat{\widehat{s}}$, where $\widehat{\widehat{s}}\beta \x81\u2019(\mathrm{{\rm O}\x95}):=\mathrm{{\rm O}\x95}\beta \x81\u2019(s)$, is a natural isomorphism between $G$ and $\widehat{\widehat{G}}$.
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 viewpoint, 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  20130322 17:42:42 
Last modified on  20130322 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 