دانلود کتاب A passage to modern analysis

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Cover......Page 1
Title page......Page 4
List of Figures......Page 18
Preface......Page 20
1.1. Set Notation and Operations......Page 30
Exercises......Page 33
1.2. Functions......Page 34
Exercises......Page 35
1.3. The Natural Numbers and Induction......Page 36
Exercises......Page 40
1.4. Equivalence of Sets and Cardinality......Page 41
Exercises......Page 44
1.5. Notes and References......Page 45
Chapter 2. The Complete Ordered Field of Real Numbers......Page 46
2.1.1. The Field Axioms......Page 47
2.1.2. The Order Axiom and Ordered Fields......Page 49
Exercises......Page 52
2.2. The Complete Ordered Field of Real Numbers......Page 53
2.3. The Archimedean Property and Consequences......Page 57
Exercises......Page 64
2.4. Sequences......Page 65
Exercises......Page 71
2.5. Nested Intervals and Decimal Representations......Page 72
Exercises......Page 76
2.6. The Bolzano-Weierstrass Theorem......Page 77
2.7. Convergence of Cauchy Sequences......Page 79
2.8.1. Properties that Characterize Completeness......Page 81
2.8.2. Why Calculus Does Not Work in ��......Page 82
2.8.3. The Existence of a Complete Ordered Field......Page 83
Exercise......Page 84
3.1. Some Special Sequences......Page 86
Exercises......Page 89
3.2. Introduction to Series......Page 90
3.3. The Geometric Series......Page 93
Exercises......Page 94
3.4. The Cantor Set......Page 95
Exercises......Page 97
3.5. A Series for the Euler Number......Page 98
3.6. Alternating Series......Page 100
3.7. Absolute Convergence and Conditional Convergence......Page 101
Exercise......Page 102
3.8. Convergence Tests for Series with Positive Terms......Page 103
3.9. Geometric Comparisons: The Ratio and Root Tests......Page 104
Exercises......Page 105
3.10. Limit Superior and Limit Inferior......Page 106
Exercises......Page 108
3.11.1. Absolute Convergence: The Root and Ratio Tests......Page 109
3.11.2. Conditional Convergence: Abel’s and Dirichlet’s Tests......Page 112
3.12. Rearrangements and Riemann’s Theorem......Page 115
3.13. Notes and References......Page 119
4.1. Open Sets and Closed Sets......Page 120
Exercises......Page 127
4.2. Compact Sets......Page 128
4.3. Connected Sets......Page 131
4.4. Limit of a Function......Page 132
4.5. Continuity at a Point......Page 138
4.6. Continuous Functions on an Interval......Page 140
Exercises......Page 141
4.7. Uniform Continuity......Page 142
4.8. Continuous Image of a Compact Set......Page 144
Exercises......Page 145
4.9. Classification of Discontinuities......Page 146
Exercises......Page 148
5.1. The Derivative: Definition and Properties......Page 150
5.2. The Mean Value Theorem......Page 156
5.3. The One-Dimensional Inverse Function Theorem......Page 160
5.4. Darboux’s Theorem......Page 162
5.5. Approximations by Contraction Mapping......Page 163
5.6. Cauchy’s Mean Value Theorem......Page 168
5.6.1. Limits of Indeterminate Forms......Page 170
Exercises......Page 171
5.7. Taylor’s Theorem with Lagrange Remainder......Page 172
5.8. Extreme Points and Extreme Values......Page 174
5.9. Notes and References......Page 176
6.1. Partitions and Riemann-Darboux Sums......Page 178
Exercises......Page 179
6.2. The Integral of a Bounded Function......Page 180
6.3. Continuous and Monotone Functions......Page 183
6.4. Lebesgue Measure Zero and Integrability......Page 186
6.5. Properties of the Integral......Page 188
6.6. Integral Mean Value Theorems......Page 192
6.7. The Fundamental Theorem of Calculus......Page 194
Exercises......Page 199
6.8. Taylor’s Theorem with Integral Remainder......Page 200
Exercises......Page 202
6.9.1. Functions on [��,∞) or (-∞,��]......Page 203
6.9.2. Functions on (��,��] or [��,��)......Page 204
6.9.3. Functions on (��,∞), (-∞,��) or (-∞,∞)......Page 205
6.9.4. Cauchy Principal Value......Page 206
Exercises......Page 207
6.10. Notes and References......Page 208
7.1.1. Pointwise Convergence......Page 210
7.1.2. Uniform Convergence......Page 212
Exercises......Page 218
7.2. Series of Functions......Page 220
7.2.1. Integration and Differentiation of Series......Page 221
7.2.2. Weierstrass’s Test: Uniform Convergence of Series......Page 222
7.3. A Continuous Nowhere Differentiable Function......Page 223
7.4. Power Series; Taylor Series......Page 225
Exercises......Page 230
7.5. Exponentials, Logarithms, Sine and Cosine......Page 231
7.5.1. Exponentials and Logarithms......Page 232
7.5.2. Power Functions......Page 237
7.5.3. Sine and Cosine Functions......Page 238
7.5.5. The Elementary Transcendental Functions......Page 241
Exercises......Page 242
7.6. The Weierstrass Approximation Theorem......Page 244
7.7. Notes and References......Page 247
8.1. The Vector Space ��ⁿ......Page 248
8.2. The Euclidean Inner Product......Page 253
8.3. Norms......Page 256
Exercises......Page 265
8.4. Fourier Expansion in ��ⁿ......Page 267
Exercises......Page 270
8.5.1. Definitions and Preliminary Results......Page 271
8.5.2. The Spectral Theorem for Real Symmetric Matrices......Page 274
Exercises......Page 276
8.6. The Euclidean Metric Space ��ⁿ......Page 277
Exercise......Page 279
8.7. Sequences and the Completeness of ��ⁿ......Page 280
Exercises......Page 281
8.8.1. Topology of ��ⁿ......Page 282
8.8.2. Relative Topology of a Subset......Page 283
Exercises......Page 284
8.9. Nested Intervals and the Bolzano-Weierstrass Theorem......Page 285
8.10.1. Limits of Functions and Continuity......Page 286
Exercises......Page 288
8.10.2. Continuity on a Domain......Page 289
8.10.4. Continuous Images of Compact Sets......Page 291
Exercises......Page 293
8.10.5. Differentiation under the Integral......Page 294
Exercises......Page 296
8.10.6. Continuous Images of Connected Sets......Page 297
8.11. Notes and References......Page 299
9.1. Basic Topology in Metric Spaces......Page 300
Exercises......Page 306
9.2. The Contraction Mapping Theorem......Page 307
9.3. The Completeness of ��[��,��] and ��²......Page 309
Exercises......Page 311
9.4. The ��^{��} Sequence Spaces......Page 312
9.5.1. Matrix Norms......Page 316
9.5.2. Completeness of ��^{����}......Page 321
Exercises......Page 322
9.6. Notes and References......Page 324
10.1. Partial Derivatives......Page 326
Exercises......Page 332
10.2. Differentiability: Real Functions and Vector Functions......Page 334
Exercises......Page 335
10.3. Matrix Representation of the Derivative......Page 336
Exercise......Page 337
10.4. Existence of the Derivative......Page 338
10.5. The Chain Rule......Page 341
10.6. The Mean Value Theorem: Real Functions......Page 344
Exercises......Page 347
10.7. The Two-Dimensional Implicit Function Theorem......Page 348
10.8. The Mean Value Theorem: Vector Functions......Page 351
Exercises......Page 356
10.9. Taylor’s Theorem......Page 357
10.10. Relative Extrema without Constraints......Page 360
Exercises......Page 363
10.11. Notes and References......Page 364
11.1. Matrix Geometric Series and Inversion......Page 366
11.2. The Inverse Function Theorem......Page 370
Exercises......Page 375
11.3. The Implicit Function Theorem......Page 376
Exercises......Page 379
11.4. Constrained Extrema and Lagrange Multipliers......Page 380
Exercises......Page 383
11.5. The Morse Lemma......Page 384
11.6. Notes and References......Page 389
12.1. Bounded Functions on Closed Intervals......Page 390
12.2. Bounded Functions on Bounded Sets......Page 394
12.3. Jordan Measurable Sets; Sets with Volume......Page 396
12.4. Lebesgue Measure Zero......Page 398
12.5. A Criterion for Riemann Integrability......Page 402
12.6. Properties of Volume and Integrals......Page 406
Exercises......Page 412
12.7. Multiple Integrals......Page 413
Exercises......Page 417
Chapter 13. Transformation of Integrals......Page 418
13.1. A Space-Filling Curve......Page 419
13.2. Volume and Integrability under ��¹ Maps......Page 420
Exercises......Page 423
13.3. Linear Images of Sets with Volume......Page 424
13.4. The Change of Variables Formula......Page 431
Exercises......Page 441
13.5. The Definition of Surface Integrals......Page 443
13.6. Notes and References......Page 449
14.1. Scalar Differential Equations......Page 450
14.2. Systems of Ordinary Differential Equations......Page 454
14.2.1. Definition of Solution and the Integral Equation......Page 455
14.2.2. Completeness of ��_{��}[��,��]......Page 456
14.2.3. The Local Lipchitz Condition......Page 458
14.2.4. Existence and Uniqueness of Solutions......Page 461
Exercises......Page 463
14.3.1. The Maximal Interval of Definition......Page 464
14.3.2. An Example of a Newtonian System......Page 467
14.4.1. Continuous Dependence on Initial Conditions, Parameters, and Vector Fields......Page 468
Exercises......Page 471
14.4.2. Newtonian Equations and Examples of Stability......Page 472
Exercises......Page 473
14.5. Matrix Exponentials and Linear Autonomous Systems......Page 475
14.6. Notes and References......Page 479
Chapter 15. The Dirichlet Problem and Fourier Series......Page 480
15.1. Introduction to Laplace’s Equation......Page 481
15.2. Orthogonality of the Trigonometric Set......Page 482
Exercises......Page 484
15.3. The Dirichlet Problem for the Disk......Page 485
Exercises......Page 494
Exercises......Page 496
Exercise......Page 499
15.5. The Best Mean Square Approximation......Page 500
Exercises......Page 504
15.6. Convergence of Fourier Series......Page 505
Exercises......Page 514
15.7. Fejér’s Theorem......Page 515
Exercises......Page 519
15.8. Notes and References......Page 520
Chapter 16. Measure Theory and Lebesgue Measure......Page 522
16.1. Algebras and ��-Algebras......Page 523
16.2. Arithmetic in the Extended Real Numbers......Page 527
16.3. Measures......Page 528
16.4. Measure from Outer Measure......Page 534
16.5.1. Lebesgue Measure on the Real Line......Page 539
16.5.2. Metric Outer Measure; Lebesgue Measure on Euclidean Space......Page 543
Exercises......Page 553
16.6. Notes and References......Page 554
Chapter 17. The Lebesgue Integral......Page 556
17.1. Measurable Functions......Page 557
Exercises......Page 563
17.2. Simple Functions and the Integral......Page 564
17.3. Definition of the Lebesgue Integral......Page 566
17.4. The Limit Theorems......Page 568
Exercises......Page 576
17.5. Comparison with the Riemann Integral......Page 578
17.6. Banach Spaces of Integrable Functions......Page 581
17.7. Notes and References......Page 584
18.1. Examples of Orthonormal Sets......Page 586
Exercises......Page 587
18.2.1. Basic Results for Inner Product Spaces......Page 588
18.2.2. Complete Spaces and Complete Orthonormal Sets......Page 592
Exercises......Page 596
18.3. Mean Square Convergence......Page 598
18.3.1. Comparison of Pointwise, Uniform, and ��² Norm Convergence......Page 599
Exercises......Page 600
18.3.2. Mean Square Convergence for ����[-��,��]......Page 601
18.3.3. Mean Square Convergence for ℛ[-��,��]......Page 602
18.4. Hilbert Spaces of Integrable Functions......Page 605
Exercises......Page 613
18.5. Notes and References......Page 614
A.1. Proof of the Schroeder-Bernstein Theorem......Page 616
Exercise......Page 617
B.1. Symbols and Notations Reference List......Page 618
B.2. The Greek Alphabet......Page 620
Bibliography......Page 622
Index......Page 626
Back Cover......Page 638