6. Homomorphisms

Chapters in this book

1. Frontmatter i
2. Contents v
3. 1. Introduction 1
4. 2. Categories 3
5. 3. The Category of Groups 16
6. 4. Subgroups 24
7. 5. Normal Subgroups 29
8. 6. Homomorphisms 32
9. 7. Direct Products and Sums of Groups 35
10. 8. Relations 39
11. 9. The Category of Vector Spaces 44
12. 10. Subspaces 53
13. 11. Linear Mappings; Direct Products and Sums 58
14. 12. From Real to Complex Vector Spaces and Back 62
15. 13. Duals 65
16. 14. Multilinear Mappings; Tensor Products 71
17. 15. Example: Minkowski Vector Space 79
18. 16. Example: The Lorentz Group 87
19. 17. Functors 90
20. 18. The Category of Associative Algebras 97
21. 19. The Category of Lie Algebras 104
22. 20. Example: The Algebra of Observables 111
23. 21. Example: Fock Vector Space 114
24. 22. Representations: General Theory 120
25. 23. Representations on Vector Spaces 125
26. 24. The Algebraic Categories: Summary 132
27. 25. Subsets and Mappings 134
28. 26. Topological Spaces 136
29. 27. Continuous Mappings 147
30. 28. The Category of Topological Spaces 154
31. 29. Nets 160
32. 30. Compactness 165
33. 31. The Compact-Open Topology 172
34. 32. Connectedness 177
35. 33. Example: Dynamical Systems 183
36. 34. Homotopy 188
37. 35. Homology 199
38. 36. Homology: Relation to Homotopy 211
39. 37. The Homology Functors 214
40. 38. Uniform Spaces 217
41. 39. The Completion of a Uniform Space 225
42. 40. Topological Groups 234
43. 41. Topological Vector Spaces 240
44. 42. Categories: Summary 248
45. 43. Measure Spaces 249
46. 44. Constructing Measure Spaces 257
47. 45. Measurable Functions 259
48. 46. Integrals 262
49. 47. Distributions 270
50. 48. Hilbert Spaces 277
51. 49. Bounded Operators 285
52. 50. The Spectrum of a Bounded Operator 293
53. 51. The Spectral Theorem: Finite-dimensional Case 302
54. 52. Continuous Functions of a Hermitian Operator 306
55. 53. Other Functions of a Hermitian Operator 311
56. 54. The Spectral Theorem 319
57. 55. Operators (Not Necessarily Bounded) 324
58. 56. Self-Adjoint Operators 329
59. Index of Defined Terms 343