Introduction to Hilbert Space and the Theory of Spectral Multiplicity (2nd Edition)

Originally published in 1957, this classic textbook by Prof. Halmos gives 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. With ample examples and explanations, the text is well suited for beginners.

About the author -- Excerpt from the Wikipedia:

Paul Richard Halmos (1916 - 2006) was a Hungarian-Jewish-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor.

Contents Covered:

• Preface
• Prerequisites and Notation
• 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 Spaces
  • 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
  • 68. Conclusion
• References
• Bibliography