Functional Analysis/Harmonic Analysis/Topological Group

IntroductionEdit

The main algebraic structure studied in harmonic analysis is the topological group. In summary, a topological group is a group whose underlying set possesses a topology compatible with the group structure.


Preliminaries 0% developedEdit


Definition 9.1.1: A topological group is a triple  , where   is a group,   is a topological space, such that:

  1. The product map   is continuous where   is equipped with the canonical product topology.
  2. The inverse map   is continuous.

We abuse notation slightly and write   for a topological group when the product and topologies are understood from context, unless we need to be careful about a situation, for example, when talking about two different topologies on the same group.

Examples:

  1. Any group equipped with the discrete topology becomes a topological group.
  2.  , with the addition of numbers as product and the usual line topology. More generally, if   is a finite dimensional  -vector space, then   equipped with the canonical product topology and addition of vectors is a topological group.
  3. If   is a  -vector space, then the set   is linear and invertible   is a topological group equipped with map composition as product and the subspace topology inherited from the vector space  .


The following proposition gives an equivalent definition of topological group.


Proposition 9.1.2: Let   be a group and   a topological space with the same underlying set. Then   is a topological group if and only if the map  , given by   is continuous.

proof: First notice that we can write the map   as  . Suppose   is a topological group. Then, by definition 9.1.1, 1 and 2 ,   is a composition of continuous maps, and is therefore continuous.

Conversely, assume   is continuous. Since the inclusion   given by   is continuous. We can then conclude that the composition   is continuous. Finally, by a similar line of reason the product map   is continuous. QED

Definition 9.1.3: Let   and   be topological groups. A topological group homomorphism, or simply a homomorphism between   and   is a continuous group homomorphism  . To be more precise, a homomorphism of topological groups is a   such that:

  1.   for all  .
  2.   is a continuous map between the topological spaces   and  .

An isomorphism between topological groups is a bijective continuous map whose inverse is also continuous.


As with purely algebraic groups, isomorphic topological groups are seen as being the same topological group, except for very specific contexts.

Definition 9.1.5: Let   be a topological group and   a topological group such that   considered as a pure algebraic group is a subgroup of  . We call   a topological subgroup of   if the inclusion map is continuous.


Proposition 9.1.6: Let   be a homomorphism. Then   is a topological subgroup and   is a normal topological subgroup. Furthermore

Proof: If   is a homomorphism, we know from group theory that the image   is a subgroup. But we also recall from topology that the image of a continuous map is canonically equipped with the subspace topology. But the restriction of the product and inverse maps to   are continuous in the subspace topology and thus   is a topological group. Lastly, we know from topology that the subspace topology makes the inclusion map continuous and therefore   is a topological subgroup of  . The second assertion follows from the same line of reasoning.

We use the first isomorphism theorem for purely algebraic groups to conclude that   as groups, with isomorphism given by  . But since the map   is the quotient map of  , it is continuous and open. These properties together with surjectivity show that   is an isomorphism of topological groups. QED.


Lemma: The left and right translations (ref) (def of Lx, Rx, group theory) by a given element are homeomorphisms of the group with itself. More precisely, the maps   are homeomorphisms of  .

Proof: The product map is jointly continuous by assumption and therefore separately continuous. The inverses of these maps are the maps   which are continuous by the same reason. QED.

Since we shall almost exclusively deal with topological groups, we shall say homomorphism instead of homomorphism of topological groups, and if we mean pure group homomorphism we say algebraic homomorphism.


Neighborhood of the neutral element are particularly important for a topological group.

Definition: For  , denote the set of all neighborhoods of   in   by  .

Lemma: For any   we have  . In other words, the neighborhoods of a point in are the translations of the neighborhoods of the neutral element by that point.

Proof: If  , then by lemma (ref) (translations are homeos),   are neighborhoods of  . Similarly, if  , then   are neighborhoods of   such that  . QED.

This suggests that the neighborhoods of the neutral element are sufficient for the description of the topology of the group. Indeed, some topological properties of maps, groups, etc... depend only on their behaviour at the neutral element. For example we have:

Lemma: Let  , be an algebraic homomorphism. In order for   to be a homomorphism, it is necessary and sufficient for   to be continuous at  .

Proof: Necessity is clear. To show sufficiency, let   be a nonempty open set, and  . Then   is a neighborhood of the neutral element  , and by assumption   is an open neighborhood of  . For each   we have the open set   satisfying  . We claim that:

 .

Indeed if   then   since  . Consequently   is an open set and   is continuous. QED.

Proposition: For every   contained in the topological group  , we have   and  .

Proof: Let  . Then for each each  , by proposition (ref) (translations are homeos) we have  . Conversely, if  , then  . But then we can write  ,  . QED.

This lemma suggests that in order to find topologies in a group that make it into a topological group it suffices to find a "nice" base of neighborhoods for the neutral element. This is indeed true, and we have:

Theorem: Let   be a topological group and   be a class of subsets of   containing  . Then the class   is the basis for a topology making   a topological group if and only it satisfies the following properties:

  1. If   and if   then there exists   such that  


Appendices 0% developedEdit

Here, you will find a list of unsorted chapters. Some of them listed here are highly advanced topics, while others are tools to aid you on your mathematical journey. Since this is the last heading for the wikibook, the necessary book endings are also located here.