Contents xiii
1.3. Vector integrals 336
1.4. Hilbert spaces 337
§2. Operators in Hilbert spaces 339
2.1. Types of bounded operators 340
2.2. Hilbert-Schmidt and trace class operators 340
2.3. Unbounded operators 343
2.4. Spectral theory of self-adjoint operators 345
2.5. Decompositions of Hilbert spaces 350
2.6. Application to representation theory 353
§3. Mathematical model of quantum mechanics 355
Appendix V. Representation Theory 357
§1. Infinite-dimensional representations of Lie groups 357
1.1. Generalities on unitary representations 357
1.2. Unitary representations of Lie groups 363
1.3. Infinitesimal characters 368
1.4. Generalized and distributional characters 369
1.5. Non-commutative Fourier transform 370
§2. Induced representations 371
2.1. Induced representations of finite groups 371
2.2. Induced representations of Lie groups 379
2.3. ∗-representations of smooth G-manifolds 384
2.4. Mackey Inducibility Criterion 389
References 395
Index 403
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