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IntroductionEdit


Definition:

Proposition:

Proof: QED.


Theorem:

Proof: QED.


Lemma:

Proof: QED.


Notation  Edit

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.

Notation And Previous Definitions 0% developedEdit

Definition: A subset   of a group   is called symmetric if  .

Definition: Let   be a map between two sets. For any subset  , we define  . In particular, if   are subsets of a group   we have:

  1.  
  2.  
  3. If   is a singleton, we denote   and  .


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 (jointly) continuous where   is equipped with the canonical product topology.
  2. The inversion 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  .
  4. If   is a topological group, the opposite or reversed topological group is the group  , where  .
  5. Let   be a nonempty set and consider   the set of all bijections from   to  . If equipped with the product  , i.e., the composition of maps, then   becomes a group.


From now on, when we use a topological property or charateristic to describe a group, one should understand that we are talking about a topological group. Therefore instead of saying connected topological group or locally compact topological group we say connected group or locally compact group.

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. In other words, it is a group homomorphism which is also a homeomorphism of topological spaces


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



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.


Corollary: Any topological group   is a homogeneous topological space.

Proof: If  , then by lemma (ref) (translations are homeos) the map   is an homeomorphism of   sending   to  .

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. Until the end of this chapter fix a topological group  .

SubgroupsEdit

Proposition: If   is an algebraic subgroup of   equipped with the subspace topology, then   is a topological group.

Proof: This follows from the fact that the product and inversion, which are continuous maps, of   restricted to   remain continuous QED.


Definition: Let   be a topological group and   an algebraic subgroup, which is also a topological group by (ref) . In this case   is called a subgroup of  . We denote   if   is a subgroup of  .

Recall from topology (ref) that the inclusion of subspaces are continuous, and from abstract algebra that the inclusion of groups (ref) is a group homomorphism. Combining this information we conclude that the inclusion of a subgroup is a homomorphism of topological groups.



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.


Proposition: Suppose   is a   topological space. If  , then  . Furthermore:

  1.   is abelian if and only if   is abelian.
  2. If   is normal, then   is normal.


Proof: Indeed, let  , and let   be neighborhoods. Then  . Using proposition (ref) we find a symmetric   such that  . Then   is a neighborhood of   and therefore there exists   since  . Similarly we find  . But then  . Thus  .

If M_2 is a symmetric neighborhood of   such that  , then  . Similarly, there exists   which means that  . Thus  .

To prove (1.) let   be abelian, and let  . Take   nets in   converging to   respectively. Then the net   converges to   and to  . Since   is  ,  . The reverse implication is clear.

To prove (2.) let  ,  , and let   be a net in   converging to  . For every    . But  . QED.

Lemma: Let   be any collection of subgroups of a given group  . Then the intersection of all the subgroups in   is a subgroup of  . Symbolically,  .

Proof: Since all   are subgroups, the neutral element is contained in all of them, and therefore in their intersection. If   for each   then   and   for each  , and therefore   QED.


Proposition: Given any subset  , there exists a unique subgroup   containing   which is minimal with this property. This unique subgroup is called the subgroup generated by  .

Proof: To show existence we use the previous lemma on the collection   of all subgroups containing  , which is nonempty since  . To prove uniqueness, denote  . If   is another subgroup containing   with the minimality property then, by the minimality property   and  . QED


Topological Quotient SpacesEdit

Throughout this subsection, fix   a closed subgroup, and denote by   the topology of  . We shall study the quotient space   of left equivalence classes modulo  . The canonical topology on this set is the quotient topology, i.e., the topology  .



Proposition: A subset   is open if and only if   is open. The quotient map   is open and closed.

HA-TOPGP-TQS-001

Proof: QED.

Proposition: If   is a closed subgroup of  , then the space   is Hausdorff (satisfies the   axiom of separability).


HA-TOPGP-TQS-002

Proof: Let  . Since   is closed,   is an open neighborhood of  . QED.


From this proposition we may conclude that if   possesses a topological property   preserved on continuous images, then   also has this property. For example, if   is compact or connected, then so is  . Unfortunately, local compactness is not preserved by continuous images. However, the quotient map is also open, so we have:



Proposition: If   is locally compact, then so is  .

HA-TOPGP-TQS-

Proof: If  , let   be a compact neighborhood of  . Since   is open and continuous,   is a compact neighborhood of  . QED.




Proposition: '

HA-TOPGP-TQS-

Proof: QED.

Proposition: '

HA-TOPGP-TQS-

Proof: QED.

Proposition: '

HA-TOPGP-TQS-

Proof: QED.


Neighborhoods of the Neutral ElementEdit

Neighborhoods 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.

Proposition: For any  , we have

 

Proof: 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. For every   there exists   such that  
  2. For every   there exists   such that  .
  3. For every   there exists   such that  .
  4. For every   and every   there exists   such that  .
  5. For every   and every   there exists   such that  .

Furthermore, this topology is   (ref) (def of t1 space) if and only if  .

Proof: QED.

Proposition: Every topological group possesses a base of neighborhoods of the neutral element composed of symmetric open sets.

Proof: QED.

Proposition Every topological group is a Tychonoff space, i.e., completely regular and Hausdorff

Proof QED.


Proposition: If   is a neighborhood of a subset   of  , then there exists   such that  .

Proof: QED.



Functions on Topological GroupsEdit

Proposition: Let   be a locally connected group. Then the connected component   containing the neutral element is a subgroup of  .

Proof: By assumption,   contains the neutral element.

Proposition: An open subgroup is also closed.


Operations With SubsetsEdit

Proposition: Let   be subsets of  .

  1. If   is open, then   is open
  2. If   are connected, then   and   are connected
  3. If   is closed and   is compact, then both   and   are closed.
  4. If   are compact, then   is compact.


Proof: Exercise (ref). QED.

Proposition: If   is a closed subgroup of the LCG  , then the topological space   is locally compact.

Proof: QED.


Topological Vector Spaces Associated to Topological Groups 0% developedEdit

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.


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Actions of Topological Groups 0% developedEdit

Definition: An action of a topological group   is a pair   where   is a topological space (ref) and   is a continuous group action (ref). In other words it satisfies, for all   and    :

  1.   .
  2.  .
  3. The map   is continuous.


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Locally Compact Groups 0% developedEdit

In this section we define the most well-known class of topological groups, namely locally compact groups. This class includes compact groups which in turn includes all finite groups, finite-dimensional Lie groups, etc.

Notation 0% developedEdit

Preliminaries 0% developedEdit

Definition 9.2.1: A locally compact group is a topological group whose underlying topological space is locally compact.

Examples:

  1. All compact, and therefore all finite groups are locally compact.
  2. A discrete group is always locally compact.
  3. Any finite-dimensional vector space is a locally compact group (equipped with addition).


The Hilbert space   is not locally compact in the norm topology.

Proposition: An open subgroup of a locally compact group is always closed. A closed subgroup of a locally compact group is locally compact.

Proof: Indeed, let   be an open subgroup of  . Choose a set  , one   for each class in  , but choosing   for the class of  . We then have the disjoint union  . Since left multiplication by a given element   is a homeomorphism between   and  , we have that each such set is open in  . Therefore the complement of   is open in   and therefore   is also closed.

If now   be an closed subgroup of  , let  . There exists a compact neighborhood   of   in  . But then the intersection   is a compact neighborhood of   in  . QED.

Combining the statements in the last proposition we conclude that an open subgroup of a locally compact group is also locally compact.

Proposition: Let   be a topological group. In order for   to be locally compact it is necessary and sufficient that the neutral element   possesses a compact neighborhood.

Proof: Indeed, if   is a compact neighborhood of  , then   is a compact neighborhood of   for any  , since   is the image of a continuous map by lemma (ref) (left and right multiplication maps). QED.

Abelian Groups 0% developedEdit

Examples 0% developedEdit

1) Sn and symmetries of polygons

2)   and its subgroups

3)  adic numbers

4) Isometries of a metric space.

5) Complex functions on a topological group, finite, N, Z, R, C, Q, Qua, GL,

6)

1) Finite Groups.

The canonical topology on finite groups is the discrete topology. Consider, then, the symmetric groups  , where   is any set with   elements.


2) The General Linear Group.

Let   be a field and  . Consider the topological vector space   of linear maps from   into itself with the operator norm topology, i.e., for each  ,  .

Definition: The general linear group of   and   is the group of invertible linear operators  .

Proposition: Equipped with the subspace topology inherited from the topological vector space  ,   is a topological group.

Proof: QED.


3)


Exercises 0% developedEdit

1) If A is open and B is any set, then AB is open

2) find an example in which A and B are closed but AB is not closed.



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.