A clear, readable introductory treatment of Hilbert Space. The multiplicity theory of continuous spectra is treated, for the first time in English, in full generality.
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978-0-8218-1378-2 (9780821813782)
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Schweitzer Klassifikation
The Geometry of Hilbert Space: 1. Linear functionals 2. Bilinear functionals 3. Quadratic forms 4. Inner product and norm 5. The inequalities of Bessel and Schwarz 6. Hilbert space 7. Infinite sums 8. Conditions for summability 9. Examples of Hilbert space 10. Subspaces 11. Vectors in and out of subspaces 12. Orthogonal complements 13. Vector sums 14. Bases 15. A non-closed vector sum 16. Dimension 17. Boundedness 18. Bounded bilinear functionals The Algebra of Operators: 19. Operators 20. Examples of operators 21. Inverses 22. Adjoints 23. Invariance 24. Hermitian operators 25. Normal and unitary operators 26. Projections 27. Projections and subspaces 28. Sums of projections 29. Products and differences of projections 30. Infima and suprema of projections 31. The spectrum of an operator 32. Compactness of spectra 33. Transforms of spectra 34. The spectrum of a Hermitian operator 35. Spectral Heuristics 36. Spectral measures 37. Spectral integrals 38. Regular spectral measures 39. Real and complex spectral measures 40. Complex spectral integrals 41. Description of the spectral subspaces 42. Characterization of the spectral subspaces 43. The spectral theorem for Hermitian operators 44. The spectral theorem for normal operators The Analysis of Spectral Measures: 45. The Problem of unitary equivalence 46. Multiplicity functions in finite-dimensional spaces 47. Measures 48. Boolean operations on measures 49. Multiplicity functions 50. The canonical example of a spectral measure 51. Finite-dimensional spectral measures 52. simple finite-dimensional spectral measures 53. The commutator of a set of projections 54. Pairs of commutators 55. Columns 56. Rows 57. Cycles 58. Separable projections 59. Characterizations of rows 60. Cycles and rows 61. The existence of rows 62. Orthogonal systems 63. The power of a maximal orthogonal system 64. Multiplicities 65. Measures from vectors 66. Subspaces from measures 67. The multiplicity function of a spectral measure 69. Conclusion References Bibliography.
