Functional Analysis/Harmonic Analysis/Topological Group

Introduction.
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.
formula_1
Definition 9.1.1: "A topological group is a triple formula_2, where formula_3 is a group, formula_4 is a topological space, such that:" 
We abuse notation slightly and write formula_8 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:
The following proposition gives an equivalent definition of topological group.
Proposition 9.1.2: "Let formula_18 be a group and formula_19 a topological space with the same underlying set. Then formula_20 is a topological group if and only if the map formula_21, given by formula_22 is continuous."
"proof:" First notice that we can write the map formula_23 as formula_24. Suppose formula_25 is a topological group. Then, by definition 9.1.1, 1 and 2 , formula_26 is a composition of continuous maps, and is therefore continuous.
Conversely, assume formula_27 is continuous. Since the inclusion formula_28 given by formula_29 is continuous. We can then conclude that the composition formula_30 is continuous. Finally, by a similar line of reason the product map formula_31 is continuous. QED
 Definition 9.1.3: "Let formula_32 and formula_33 be topological groups. A topological group homomorphism, or simply a homomorphism between formula_34 and formula_35 is a continuous group homomorphism formula_36. To be more precise, a homomorphism of topological groups is a formula_36 such that: "
" 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 formula_8 be a topological group and formula_44 a topological group such that formula_44 considered as a pure algebraic group is a subgroup of formula_46. We call formula_44 a topological subgroup of formula_48 if the inclusion map is continuous.
Proposition 9.1.6: "Let formula_49 be a homomorphism. Then formula_50 is a topological subgroup and formula_51 is a normal topological subgroup. Furthermore"
Proof: If formula_52 is a homomorphism, we know from group theory that the image formula_50 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 formula_54 are continuous in the subspace topology and thus formula_55 is a topological group. Lastly, we know from topology that the subspace topology makes the inclusion map continuous and therefore formula_54 is a topological subgroup of formula_57. The second assertion follows from the same line of reasoning. 
We use the first isomorphism theorem for purely algebraic groups to conclude that formula_58 as groups, with isomorphism given by formula_59. But since the map formula_60 is the quotient map of formula_61, it is continuous and open. These properties together with surjectivity show that formula_62 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 formula_63 are homeomorphisms of formula_46."
Proof: The product map is jointly continuous by assumption and therefore separately continuous. The inverses of these maps are the maps formula_65 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 formula_66, denote the set of all neighborhoods of formula_67 in formula_48 by formula_69. 
Lemma: "For any formula_70 we have formula_71. In other words, the neighborhoods of a point in are the translations of the neighborhoods of the neutral element by that point.
Proof: If formula_72, then by lemma (ref) (translations are homeos), formula_73 are neighborhoods of formula_74. Similarly, if formula_75, then formula_76 are neighborhoods of formula_77 such that formula_78. 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 formula_36, be an algebraic homomorphism. In order for formula_52 to be a homomorphism, it is necessary and sufficient for formula_52 to be continuous at formula_77."
Proof: Necessity is clear. To show sufficiency, let formula_83 be a nonempty open set, and formula_84. Then formula_85 is a neighborhood of the neutral element formula_86, and by assumption formula_87 is an open neighborhood of formula_88. For each formula_89 we have the open set formula_90 satisfying formula_91. We claim that:
 formula_92. 
Indeed if formula_93 then formula_94 since formula_95. Consequently formula_96 is an open set and formula_97 is continuous. QED. 
Proposition: "For every formula_98 contained in the topological group formula_46, we have formula_100 and formula_101. 
Proof: Let formula_102. Then for each each formula_72, by proposition (ref) (translations are homeos) we have formula_104. Conversely, if formula_105, then formula_106. But then we can write formula_107, formula_108. 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 formula_48 be a topological group and formula_110 be a class of subsets of formula_46 containing formula_77. Then the class formula_113 is the basis for a topology making formula_48 a topological group if and only it satisfies the following properties:
Appendices.
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.