Each character of has the shape for one . If one identifies with n, then is .
The group has the characters , wherein . The assignment delivers .
with addition as a link and the Euclidean topology is a locally compact Abelian group. Each character has the shape for one . If one identifies with z, one has first as sets. This applies to all and the mapping is a homeomorphism , so one also has locally compact Abelian groups.
Products by groups
If G and H are locally compact Abelian groups, so are their Cartesian products . Then define a character when you bet. In this way a group homeomorphism is obtained .
This gives you many more examples:
for every finite Abelian group G, because such is a finite product of groups of the form (see: Finite Abelian Group ).
Pontryagin's duality theorem
You have a natural image . The set of Pontrjagin states that this figure is always a topological group isomorphism is. This justifies the designation of the dual group of G, because according to the above sentence, G can be recovered from by forming a new dual group.
Relationships between group and dual group
Due to the Pontryagin duality, one expects a series of relationships between a locally compact Abelian group G and its dual group . Thereby one finds relationships between algebraic and topological properties. The following applies as an example:
A continuous homomorphism is called strict if the mapping is open , i.e. H. the picture of any open set is relatively open in the picture of . A sequence of homomorphisms is called strict if every homomorphism is strict. Finally, if one denotes the one-element group with 1 and observes , then the following theorem applies:
Let be a sequence of continuous homomorphisms between locally compact Abelian groups. Then the following statements are equivalent:
A continuous homomorphism is strict if and only if is strict.
If there is a closed subgroup, then is . Here is the dual mapping for inclusion .
Compact generated groups
The Pontryagin duality is an important tool in structural theory for locally compact Abelian groups. A locally compact group is said to be compact if there is a compact subset of G that generates G as a group . A discrete group is created compact if and only if it is finitely created.
For a locally compact Abelian group are equivalent:
G is generated compact.
, where and K is a compact group.
, where and D is a discrete group.
Addition: The numbers m and n are uniquely determined by G and K is the largest compact subgroup of G.
The Pontryagin duality, i.e. H. the above-described assignments and of locally compact Abelian groups and continuous homomorphisms is obviously a contravariant functor . Executing this functor twice in a row leads to the identical functor (more precisely: to a natural equivalence to the identical functor).
Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis , D. van Nostrand Co, 1953