مرجع فى التحليل الدالى

محتويات الكتاب

CONTENTS
Chapter 1. Metric Spaces . . . .
1.1 Metric Space 2
1.2 Further Examples of Metric Spaces 9
1.3 Open Set, Closed Set, Neighborhood 17
1.4 Convergence, Cauchy Sequence, Completeness 25
1.5 Examples. Completeness Proofs 32
1.6 Completion of Metric Spaces 41
1
Chapter 2. Normed Spaces. Banach Spaces. . . . . 49
2.1 Vector Space 50
2.2 Normed Space. Banach Space 58
2.3 Further Properties of Normed Spaces 67
2.4 Finite Dimensional Normed Spaces and Subspaces 72
2.5 Compactness and Finite Dimension 77
2.6 Linear Operators 82
2.7 Bounded and Continuous Linear Operators 91
2.8 Linear Functionals 103
2.9 Linear Operators and Functionals on Finite Dimen-sional Spaces 111
2.10 Normed Spaces of Operators. Dual Space 117
Chapter 3. Inner Product Spaces. Hilbert Spaces. . .127
3.1 Inner Product Space. Hilbert Space 128
3.2 Further Properties of Inner Product Spaces 136
3.3 Orthogonal Complements and Direct Sums 142
3.4 Orthonormal Sets and Sequences 151
3.5 Series Related to Orthonormal Sequences and Sets 160
3.6 Total Orthonormal Sets and Sequences 167
3.7 Legendre, Hermite and Laguerre Polynomials 175
3.8 Representation of Functionals on Hilbert Spaces 188
3.10 Self-Adjoint, Unitary and Normal Operators 201
x ( 'on/olts
Chapter 4. Fundamental Theorems for Normed
and Banach Spaces. . . . . . . . . . . 209
4.1 Zorn's Lemma 210
4.2 Hahn-Banach Theorem 213
4.3 Hahn-Banach Theorem for Complex Vector Spaces and
Normed Spaces 218
4.4 Application to Bounded Linear ~unctionals on
C[a, b] 225
4.6 Reflexive Spaces 239
4.7 Category Theorem. Uniform Boundedness Theorem 246
4.8 Strong and Weak Convergence 256
4.9 Convergence of Sequences of Operators and
Functionals 263
4.10 Application to Summability of Sequences 269
4.11 Numerical Integration and Weak* Convergence 276
4.12 Open Mapping Theorem 285
4.13 Closed Linear Operators. Closed Graph Theorem 291
Chapter 5. Further Applications: Banach Fixed
Point Theorem . . . . . . . . . . . . 299
5.1 Banach Fixed Point Theorem 299
5.2 Application of Banach's Theorem to Linear Equations 307
5.3 Applications of Banach's Theorem to Differential
Equations 314
5.4 Application of Banach's Theorem to Integral
Equations 319
Chapter 6. Further Applications: Approximation
Theory ..... . . . . . . . 327
6.1 Approximation in Normed Spaces 327
6.2 Uniqueness, Strict Convexity 330
6.3 Uniform Approximation 336
6.4 Chebyshev Polynomials 345
6.5 Approximation in Hilbert Space 352
6.6 Splines 356
Chapter 7. Spectral Theory of Linear Operators
in Normed, Spaces . . . . . . . . . . . 363
7.1 Spectral Theory in Finite Dimensional Normed Spaces 364
7.2 Basic Concepts 370
Contents
7.3 Spectral Properties of Bounded Linear Operators 374
7.4 Further Properties of Resolvent and Spectrum 379
7.5 Use of Complex Analysis in Spectral Theory 386
7.6 Banach Algebras 394
7.7 Further Properties of Banach Algebras 398
Chapter 8. Compact Linear Operators on Normed
xi
Spaces and Their Spectrum . 405
8.1 Compact Linear Operators on Normed Spaces 405
8.2 Further Properties of Compact Linear Operators 412
8.3 Spectral Properties of Compact Linear Operators on
Normed Spaces 419
8.4 Further Spectral Properties of Compact Linear
Operators 428
8.5 Operator Equations Involving Compact Linear
Operators 436
8.6 Further Theorems of Fredholm Type 442
8.7 Fredholm Alternative 451
Chapter 9. Spectral Theory of Bounded
9.1 Spectral Properties of Bounded Self-Adjoint Linear
Operators 460
9.2 Further Spectral Properties of Bounded Self-Adjoint
Linear Operators 465
9.3 Positive Operators 469
9.4 Square Roots of a Positive Operator 476
9.5 Projection Operators 480
9.6 Further Properties of Projections 486
9.7 Spectral Family 492
9.8 Spectral Family of a Bounded Self-Adjoint Linear
Operator 497
. .. 459
9.9 Spectral Representation of Bounded Self-Adjoint Linear
Operators 505
9.10 Extension of the Spectral Theorem to Continuous
Functions 512
9.11 Properties of the Spectral Family of a Bounded Self-Ad,ioint Linear Operator 516
xII ( 'onlellis
Chapter 10. Unbounded Linear Operators in
Hilbert Space . . . . . . . . . . . . 523
10.1 Unbounded Linear Operators and their
Linear Operators 530
10.3 Closed Linear Operators and Cldsures 535
10.4 Spectral Properties of Self-Adjoint Linear Operators 541
10.5 Spectral Representation of Unitary Operators 546
10.6 Spectral Representation of Self-Adjoint Linear Operators
556
10.7 Multiplication Operator and Differentiation Operator
562
Chapter 11. Unbounded Linear Operators in
Quantum Mechanics . . . . . . 571
11.1 Basic Ideas. States, Observables, Position Operator 572
11.2 Momentum Operator. Heisenberg Uncertainty Principle
576
11.3 Time-Independent Schrodinger Equation 583
11.4 Hamilton Operator 590
11.5 Time-Dependent Schrodinger Equation 598
Appendix 1. Some Material for Review and
Reference . . . . . . . . . . . . . . 609
A1.1 Sets 609
A1.2 Mappings 613
A1.3 Families 617
A1.4 Equivalence Relations 618
A1.5 Compactness 618
A1.6 Supremum and Infimum 619
A1.7 Cauchy Convergence Criterion 620
A1.8 Groups 622
Appendix 2. Answers to Odd-Numbered Problems. 623
Appendix 3. References. .675
Index . . . . . . . . . . .681