Finite-Dimensional Vector Spaces

P.R. Halmos (Autor)

Buch | Softcover

202 Seiten

2011
Springer-Verlag New York Inc.
978-1-4612-6389-0 (ISBN)

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“The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. All in all this is an excellent work, of equally high value for both student and teacher.” Zentralblatt für Mathematik

“The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. The presentation is never awkward or dry, as it sometimes is in other “modern” textbooks; it is as unconventional as one has come to expect from the author. The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher.” Zentralblatt für Mathematik

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. Reducibility.- 41. Projections.- 42. Combinations of pro¬jections.- 43. Projections and invariance.- 44. Adjoints.- 45. Adjoints of projections.- 46. Change of basis.- 47. Similarity.- 48. Quotient transformations.- 49. Range and null-space.- 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. Complex 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.

Reihe/Serie	Undergraduate Texts in Mathematics
Zusatzinfo	VIII, 202 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Endlichdimensionaler Vektorraum • Transformation • Vector
ISBN-10	1-4612-6389-1 / 1461263891
ISBN-13	978-1-4612-6389-0 / 9781461263890
Zustand	Neuware