A passage to modern analysis

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