| Preface |
|
vii | |
|
|
|
1 | (10) |
|
|
|
2 | (2) |
|
|
|
4 | (1) |
|
|
|
5 | (1) |
|
The Axiomatic Method. The Process of Abstraction |
|
|
6 | (1) |
|
|
|
7 | (4) |
|
|
|
11 | (32) |
|
|
|
12 | (2) |
|
|
|
14 | (3) |
|
|
|
17 | (1) |
|
|
|
18 | (1) |
|
Equivalence Relations and Partitions |
|
|
19 | (3) |
|
|
|
22 | (7) |
|
|
|
29 | (9) |
|
|
|
38 | (5) |
|
|
|
43 | (116) |
|
|
|
44 | (1) |
|
Introduction to Metric Spaces |
|
|
45 | (1) |
|
Metric Spaces: Definition |
|
|
45 | (2) |
|
Examples of Metric Spaces |
|
|
47 | (9) |
|
Subspaces and Product Spaces |
|
|
56 | (5) |
|
|
|
61 | (8) |
|
|
|
69 | (5) |
|
A Connection Between Continuity and Convergence |
|
|
74 | (3) |
|
Some Deeper Metric Space Concepts |
|
|
77 | (1) |
|
|
|
77 | (5) |
|
|
|
82 | (10) |
|
|
|
92 | (5) |
|
Examples of Homeomorphic Metric Spaces |
|
|
97 | (4) |
|
Closed Sets and the Closure Operations |
|
|
101 | (11) |
|
|
|
112 | (8) |
|
Completion of Metric Spaces |
|
|
120 | (5) |
|
|
|
125 | (9) |
|
Total Boundedness and Approximations |
|
|
134 | (7) |
|
|
|
141 | (18) |
|
|
|
159 | (54) |
|
|
|
160 | (1) |
|
Introduction to Linear Spaces |
|
|
161 | (1) |
|
Linear Spaces and Linear Subspaces |
|
|
161 | (4) |
|
|
|
165 | (6) |
|
|
|
171 | (2) |
|
|
|
173 | (3) |
|
Linear Independence and Dependence |
|
|
176 | (7) |
|
Hamel Bases and Dimension |
|
|
183 | (5) |
|
The Use of Matrices to Represent Linear Transformations |
|
|
188 | (4) |
|
Equivalent Linear Transformations |
|
|
192 | (4) |
|
|
|
196 | (1) |
|
|
|
196 | (5) |
|
|
|
201 | (3) |
|
Linear Functionals and the Algebraic Conjugate of a Linear Space |
|
|
204 | (4) |
|
Transpose of a Linear Transformation |
|
|
208 | (5) |
|
Combined Topological and Algebraic Structure |
|
|
213 | (182) |
|
|
|
214 | (1) |
|
|
|
215 | (1) |
|
|
|
215 | (3) |
|
Examples of Normal Linear Spaces |
|
|
218 | (6) |
|
|
|
224 | (5) |
|
|
|
229 | (5) |
|
Continuous Linear Transformations |
|
|
234 | (9) |
|
Inverses and Continuous Inverses |
|
|
243 | (4) |
|
|
|
247 | (10) |
|
Equivalence of Normed Linear Spaces |
|
|
257 | (7) |
|
Finite-Dimensional Spaces |
|
|
264 | (6) |
|
Normed Conjugate Space and Conjugate Operator |
|
|
270 | (2) |
|
|
|
272 | (1) |
|
Inner Product and Hilbert Spaces |
|
|
272 | (6) |
|
|
|
278 | (4) |
|
|
|
282 | (10) |
|
Orthogonal Complements and the Projection Theorem |
|
|
292 | (8) |
|
|
|
300 | (5) |
|
Orthogonal Sets and Bases: Generalized Fourier Series |
|
|
305 | (17) |
|
Examples of Orthonormal Bases |
|
|
322 | (9) |
|
Unitary Operators and Equivalent Inner Product Spaces |
|
|
331 | (9) |
|
Sums and Direct Sums of Hilbert Spaces |
|
|
340 | (4) |
|
Continuous Linear Functionals |
|
|
344 | (8) |
|
|
|
352 | (1) |
|
|
|
352 | (15) |
|
Normal and Self-Adjoint Operators |
|
|
367 | (12) |
|
|
|
379 | (9) |
|
Foundations of Quantum Mechanics |
|
|
388 | (7) |
|
Analysis of Linear Operators (Compact Case) |
|
|
395 | (90) |
|
|
|
396 | (1) |
|
|
|
397 | (1) |
|
Geometric Analysis of Operators |
|
|
397 | (2) |
|
Geometric Analysis. The Eigenvalue-Eigenvector Problem |
|
|
399 | (2) |
|
A Finite-Dimensional Problem |
|
|
401 | (10) |
|
|
|
411 | (1) |
|
The Spectrum of Linear Transformations |
|
|
411 | (3) |
|
|
|
414 | (17) |
|
Properties of the Spectrum |
|
|
431 | (8) |
|
|
|
439 | (1) |
|
Resolutions of the Identity |
|
|
439 | (10) |
|
Weighted Sums of Projections |
|
|
449 | (1) |
|
Spectral Properties of Compact, Normal, and Self-Adjoint Operators |
|
|
449 | (10) |
|
|
|
459 | (9) |
|
Functions of Operators (Operational Calculus) |
|
|
468 | (2) |
|
Applications of the Spectral Theorem |
|
|
470 | (6) |
|
|
|
476 | (9) |
|
Analysis of Unbounded Operators |
|
|
485 | (63) |
|
|
|
486 | (2) |
|
|
|
488 | (5) |
|
|
|
493 | (2) |
|
Examples of Symmetric Operators |
|
|
495 | (3) |
|
Sturm-Liouville Operators |
|
|
498 | (7) |
|
|
|
505 | (5) |
|
Elliptic Partial Differential Operators |
|
|
510 | (6) |
|
|
|
516 | (7) |
|
The Heat Equations and Wave Equation |
|
|
523 | (4) |
|
|
|
527 | (6) |
|
|
|
533 | (6) |
|
Quantum Mechanics, Revisited |
|
|
539 | (2) |
|
Heisenberg Uncertainty Principle |
|
|
541 | (2) |
|
|
|
543 | (5) |
| Appendix A The Holder, Schwartz, and Minkowski Inequalities |
|
548 | (4) |
| Appendix B Cardinality |
|
552 | (4) |
| Appendix C Zorn's Lemma |
|
556 | (2) |
| Appendix D Integration and Measure Theory |
|
558 | (41) |
|
|
|
558 | (1) |
|
|
|
559 | (5) |
|
3. A Problem with the Riemann Integral |
|
|
564 | (1) |
|
|
|
564 | (2) |
|
|
|
566 | (3) |
|
6. Convergence Almost Everywhere |
|
|
569 | (3) |
|
|
|
572 | (4) |
|
|
|
576 | (5) |
|
|
|
581 | (5) |
|
10. Other Definitions of the Integral |
|
|
586 | (3) |
|
11. The Lebesgue Space, Lp |
|
|
589 | (2) |
|
12. Dense Subspaces of Lp, 1 ≤ p < ∞ |
|
|
591 | (2) |
|
|
|
593 | (3) |
|
14. The Radon-Nikodym Theorem |
|
|
596 | (2) |
|
|
|
598 | (1) |
| Appendix E Probability Spaces and Stochastic Processes |
|
599 | (16) |
|
|
|
599 | (1) |
|
2. Random Variables and Probability Distributions |
|
|
600 | (2) |
|
|
|
602 | (1) |
|
4. Stochastic Independence |
|
|
603 | (1) |
|
5. Conditional Expectation Operator |
|
|
604 | (3) |
|
|
|
607 | (8) |
| Index of Symbols |
|
615 | (2) |
| Index |
|
617 | |
