Book:Paul R. Halmos/Finite-Dimensional Vector Spaces

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Paul Halmos: Finite-Dimensional Vector Spaces

Published $1942$, 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