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.
Definition 9.1.1: A topological group is a triple , where is a group, is a topological space, such that:
- The product map is continuous where is equipped with the canonical product topology.
- 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.
- 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 .
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.
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:
- If and if then there exists such that
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.