# User:Htttbooks/sandbox

## 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 Edit

**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: *

*If is a singleton, we denote and .*

## Preliminaries Edit

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

*The product map is (jointly) continuous where is equipped with the canonical product topology*.*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:**

- Any group equipped with the discrete topology becomes a topological group.
- , 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.
- 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 .
- If is a topological group, the opposite or reversed topological group is the group , where .
- 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: *

*for all .**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:*

*is abelian if and only if is abelian.**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:*

- For every there exists such that
- For every there exists such that .
- For every there exists such that .
- For every and every there exists such that .
- 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 .*

- If is open, then is open
- If are connected, then and are connected
- If is closed and is compact, then both and are closed.
- 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 Edit

## Appendices Edit

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 Edit

**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 :*

*.**The map is continuous*.

_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

## Locally Compact Groups Edit

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 Edit

## Preliminaries Edit

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

**Examples:**

- All compact, and therefore all finite groups are locally compact.
- A discrete group is always locally compact.
- 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 Edit

## Examples Edit

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 Edit

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 Edit

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.