Book:Paul R. Halmos/Finite-Dimensional Vector Spaces
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Paul Halmos: Finite-Dimensional Vector Spaces
Published $\text {1942}$, Springer
- ISBN 0-387-90093-4
This book is part of Springer's Undergraduate Texts in Mathematics series.
Subject Matter
Contents
- Preface
- I. SPACES
- 1. Fields
- 2. Vector spaces
- 3. Examples
- 4. Comments
- 5. Linear dependence
- 6. Linear combinations
- 7. Bases
- 8. Dimension
- 9. Isomorphism
- 10. Subspaces
- 11. Calculus of subspaces
- 12. Dimension of a subspace
- 13. Dual spaces
- 14. Brackets
- 15. Dual bases
- 16. Reflexivity
- 17. Annihilators
- 18. Direct sums
- 19. Dimension of a direct sum
- 20. Dual of a direct sum
- 21. Quotient spaces
- 22. Dimension of a quotient space
- 23. Bilinear forms
- 24. Tensor products
- 25. Product bases
- 26. Permutations
- 27. Cycles
- 28. Parity
- 29. Multilinear forms
- 30. Alternating forms
- 31. Alternating forms of maximal degree
- II. TRANSFORMATIONS
- 32. Linear transformations
- 33. Transformations as vectors
- 34. Products
- 35. Polynomials
- 36. Inverses
- 37. Matrices
- 38. Matrices of transformations
- 39. Invariance
- 40. Reducibilty
- 41. Projections
- 42. Combinations of projections
- 43. Projections and invariance
- 44. Adjoints
- 45. Adjoints of projections
- 46. Change of basis
- 47. Similarity
- 48. Quotient transformations
- 49. Range and nullspace
- 50. Rank and nullity
- 51. Transformations of rank one
- 52. Tensor products of transformations
- 53. Determinants
- 54. Proper values
- 55. Multiplicity
- 56. Triangular form
- 57. Nilpotence
- 58. Jordan form
- III. ORTHOGONALITY
- 59. Inner products
- 60. Comples inner products
- 61. Inner product spaces
- 62. Orthogonality
- 63. Completeness
- 64. Schwarz's inequality
- 65. Complete orthonormal sets
- 66. Projection theorem
- 67. Linear functionals
- 68. Parentheses versus brackets
- 69. Natural isomorphisms
- 70. Self-adjoint transformations
- 71. Polarization
- 72. Positive transformations
- 73. Isometries
- 74. Change of orthonormal basis
- 75. Perpendicular projections
- 76. Combinations of perpendicular projections
- 77. Complexification
- 78. Characterization of spectra
- 79. Spectral theorem
- 80. Normal transformations
- 81. Orthogonal transformations
- 82. Functions of transformations
- 83. Polar decomposition
- 84. Commutativity
- 85. Self-adjoint transformations of rank one
- IV. ANALYSIS
- 86. Convergence of vectors
- 87. Norm
- 88. Expressions for the norm
- 89. Bounds of a self-adjoint transformation
- 90. Minimax principle
- 91. Convergence of linear transformations
- 92. Ergodic theorem
- 93. Power series
- Appendix. HILBERT SPACE
- Recommended Reading
- Index of Terms
- Index of Symbols