Introduction to topological groups

Title page......Page 1
Preface......Page 3
1. Topological spaces......Page 11
2. Metric spaces......Page 12
3. Neighborhood systems......Page 15
Appendix to Chapter......Page 16
5. Separation axioms in topological spaces......Page 17
6. Nets and filters......Page 19
7. Compact, locally compact and connected spaces......Page 20
8. Mappings......Page 22
9. Direct products......Page 24
10. Uniform spaces and Ascoli's theorem......Page 25
11. Groups and linear spaces......Page 28
12. The concept of a semitopological group......Page 37
13. Neighborhood systems of identity of a semitopological group......Page 39
14. Constructions of new semitopological groups from old......Page 40
15. Embeddings of any group in a product group......Page 42
16. S-topologies and semitopological groups......Page 43
17. Band Ctypes of semitopological groups......Page 46
18. Locally compact semitopological groups......Page 49
19. Translations in topological groups and some examp1es......Page 53
20. Neighborhood systems of identity......Page 55
21. Separation axioms in topological groups......Page 58
22. Uniform structure on a topological group......Page 62
23. Subgroups......Page 63
24. Quotient groups......Page 67
25. Products and inverse limits of groups......Page 72
26. General results on locally compact groups......Page 77
27. Classical linear groups......Page 82
28. Locally Euclidean groups......Page 89
29. Lie groups......Page 92
30. Continuous and open homomorphisms......Page 95
31. B(C) groups......Page 99
32. The open homomorphism and closed graph theorems......Page 103
VI Haar Measure......Page 111
33. Measure and integration on locally compact spaces......Page 112
34. Integration on product spaces and Fubini theorem......Page 119
35. Existence of an invariant functional......Page 120
36. Essential uniqueness of the Haar integral......Page 127
37. Computation of Haar integrals in special cases......Page 130
Cartan's proof of existence and uniqueness of Haar integral......Page 134
38. Schur's lemma......Page 137
39. Orthogonality relations......Page 140
40. Orthonormal family of functions on metrizable compact groups......Page 144
41. Integral equations on compact groups......Page 148
42. The Peter-Weyl theorem......Page 156
43. Structure of metrizable compact groups......Page 163
44. The concept and topologies of dual groups......Page 176
45. Dual groups of locally compact abelian groups......Page 178
47. Some applications of duality theory......Page 182
48. Definitions and examples of Banach algebras......Page 201
49. The Gelfand-Mazur theorem......Page 203
50. Maximal ideal space and Gelfand-Naimark theorem......Page 209
BIBLIOGRAPHY......Page 219
INDEX OF SYMBOLS......Page 221
INDEX......Page 225