A passage to modern analysis:
A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage...
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2019]
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| Schriftenreihe: | Pure and applied undergraduate texts
41 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level. A Passage to Modern Analysis is grounded solidly in the analysis of \mathbf{R} and \mathbf{R} {n}, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments |
| Beschreibung: | xxvii, 607 Seiten Illustrationen |
| ISBN: | 9781470451356 |
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Datensatz im Suchindex
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| adam_text | Contents List of Figures xvii Preface xix Chapter 1. Sets and Functions 1 1.1. Set Notation and Operations Exercises 1 4 1.2. Functions Exercises 5 6 1.3. The Natural Numbers and Induction Exercises 7 11 1.4. Equivalence of Sets and Cardinality Exercises 12 15 1.5. 16 Notes and References Chapter 2. The Complete Ordered Field of Real Numbers 17 2.1. Algebra in Ordered Fields 2.1.1. The Field Axioms 2.1.2. The Order Axiom and Ordered Fields Exercises 18 18 20 23 2.2. The Complete Ordered Field of Real Numbers Exercises 24 28 2.3. The Archimedean Property and Consequences Exercises 28 35 2.4. Sequences Exercises 36 42 2.5. Nested Intervals and Decimal Representations Exercises 43 47 vii
Contents viii 2.6. The Bolzano-Weierstrass Theorem Exercises 2.7. Convergence of Cauchy Sequences Exercises 2.8. Summary: A Complete Ordered Field 2.8.1. Properties that Characterize Completeness 2.8.2. Why Calculus Does Not Work in Q 2.8.3. The Existence of a Complete Ordered Field 2.8.4. The Uniqueness of a Complete Ordered Field Exercise Chapter 3. Basic Theory of Series 48 50 50 52 52 52 53 54 55 55 57 3.1. Some Special Sequences Exercises 3.2. Introduction to Series Exercises 3.3. The Geometric Series Exercises 3.4. The Cantor Set Exercises 3.5. A Series for the Euler Number 3.6. Alternating Series Exercises 57 60 61 64 64 65 66 68 69 71 72 3.7. Absolute Convergence and Conditional Convergence Exercise 3.8. Convergence Tests for Series with Positive Terms Exercises 3.9. Geometric Comparisons: The Ratio and Root Tests Exercises 3.10. Limit Superior and Limit Inferior Exercises 3.11. Additional Convergence Tests 3.11.1. Absolute Convergence: The Root and Ratio Tests 3.11.2. Conditional Convergence: Abel’s and Dirichlet’s Tests Exercises 72 73 74 75 75 76 77 79 80 80 83 86 3.12. Rearrangements and Riemann’s Theorem Exercise 3.13. Notes and References 86 90 90 Chapter 4. Basic Topology, Limits, and Continuity 4.1. Open Sets and Closed Sets Exercises 4.2. Compact Sets Exercises 91 91 98 99 102
Contents ix 4.3. Connected Sets Exercise 4.4. Limit of a Function Exercises 102 103 103 109 4.5. Continuity at a Point Exercises 4.6. Continuous Functions on an Interval Exercises 4.7. Uniform Continuity Exercises 4.8. Continuous Image of a Compact Set Exercises 4.9. Classification of Discontinuities Exercises 109 111 111 112 113 115 115 116 117 119 Chapter 5. The Derivative 5.1. The Derivative: Definition and Properties Exercises 5.2. The Mean Value Theorem Exercises 5.3. The One-Dimensional Inverse Function Theorem Exercises 5.4. Darboux’s Theorem Exercise 5.5. Approximations by Contraction Mapping Exercises 5.6. Cauchy’s Mean Value Theorem 5.6.1. Limits of Indeterminate Forms Exercises 5.7. Taylor’s Theorem with Lagrange Remainder Exercises 5.8. Extreme Points and Extreme Values Exercises 5.9. Notes and References 121 121 127 127 131 131 133 133 134 134 139 139 141 142 143 145 145 147 147 Chapter 6. The Riemann Integral 6.1. Partitions and Riemann-Darboux Sums Exercises 6.2. The Integral of a Bounded Function Exercises 6.3. Continuous and Monotone Functions Exercises 6.4. Lebesgue Measure Zero and Integrability Exercises 149 149 150 151 154 154 157 157 159
Contents x 6.5. Properties of the Integral Exercises 6.6. Integral Mean Value Theorems Exercises 6.7. The Fundamental Theorem of Calculus Exercises 159 163 163 165 165 170 6.8. Taylor’s Theorem with Integral Remainder Exercises 6.9. Improper Integrals 6.9.1. Functions on [a, oo) or (—οο,ό] 6.9.2. Functions on (a, b] or [a, ծ) 6.9.3. Functions on (a,oo), (—οο,ό) or (—00,00) 6.9.4. Cauchy Principal Value Exercises 6.10. Notes and References 171 173 174 174 175 176 177 178 179 Chapter 7. Sequences and Series of Functions 7.1. Sequences of Functions: Pointwise and Uniform Convergence 7.1.1. Pointwise Convergence 7.1.2. Uniform Convergence Exercises 7.2. Series of Functions 7.2.1. Integration and Differentiation of Series 7.2.2. Weierstrass’s Test: Uniform Convergence of Series Exercises 7.3. A Continuous Nowhere Differentiable Function Exercises 7.4. Power Series; Taylor Series Exercises 7.5. Exponentials, Logarithms, Sine and Cosine 7.5.1. Exponentials and Logarithms 7.5.2. Power Functions 7.5.3. Sine and Cosine Functions 7.5.4. Some Inverse Trigonometric Functions 7.5.5. The Elementary Transcendental Functions Exercises 7.6. The Weierstrass Approximation Theorem Exercise 7.7. Notes and References 181 181 181 183 189 191 192 193 194 194 196 196 201 202 203 208 209 212 212 213 215 218 218 Chapter 8. The Metric Space Rn 8.1. The Vector Space Rn Exercises 219 219 224 8.2. The Euclidean Inner Product Exercises 224 227
Contents xi 8.3. Norms Exercises 8.4. Fourier Expansion in R Exercises 227 236 238 241 8.5. Real Symmetric Matrices 8.5.1. Definitions and Preliminary Results 8.5.2. The Spectral Theorem for Real Symmetric Matrices Exercises 8.6. The Euclidean Metric Space Rn Exercise 8.7. Sequences and the Completeness of R Exercises 242 242 245 247 248 250 251 252 8.8. Topological Concepts for R 8.8.1. Topology of Rn 8.8.2. Relative Topology of a Subset Exercises 8.9. Nested Intervals and the Bolzano-Weierstrass Theorem Exercises 8.10. Mappings of Euclidean Spaces 8.10.1. Limits of Functions and Continuity Exercises 8.10.2. Continuity on a Domain 8.10.3. Open Mappings Exercises 8.10.4. Continuous Images of Compact Sets Exercises 8.10.5. Differentiation under the Integral Exercises 8.10.6. Continuous Images of Connected Sets Exercises 8.11. Notes and References 253 253 254 255 256 257 257 257 259 260 262 262 262 264 265 267 268 270 270 Chapter 9. Metric Spaces and Completeness 9.1. Basic Topology in Metric Spaces Exercises 271 271 277 9.2. The Contraction Mapping Theorem Exercises 9.3. The Completeness of C[a, b and l2 Exercises 9.4. The lp Sequence Spaces Exercises 9.5. Matrix Norms and Completeness 9.5.1. Matrix Norms 9.5.2. Completeness of Rnxn 278 280 280 282 283 287 287 287 292
Contents xii Exercises 9.6. Notes and References 293 295 Chapter 10. Differentiation in R 10.1. Partial Derivatives Exercises 10.2. Differentiability: Real Functions and Vector Functions Exercises 10.3. Matrix Representation of the Derivative Exercise 10.4. Existence of the Derivative Exercises 10.5. The Chain Rule Exercises 10.6. The Mean Value Theorem: Real Functions Exercises 10.7. The Two-Dimensional Implicit Function Theorem Exercises 10.8. The Mean Value Theorem: Vector Functions Exercises 10.9. Taylor’s Theorem Exercises 10.10. Relative Extrema without Constraints Exercises 10.11. Notes and References 297 297 303 305 306 307 308 309 312 312 315 315 318 319 322 322 327 328 331 331 334 335 Chapter 11. The Inverse and Implicit Function Theorems 11.1. Matrix Geometric Series and Inversion Exercises 11.2. The Inverse Function Theorem Exercises 11.3. The Implicit Function Theorem Exercises 11.4. Constrained Extrema and Lagrange Multipliers Exercises 11.5. The Morse Lemma Exercises 11.6. Notes and References 337 337 341 341 346 347 350 351 354 355 360 360 Chapter 12. The Riemann Integral in Euclidean Space 12.1. Bounded Functions on Closed Intervals Exercises 361 361 365 12.2. Bounded Functions on Bounded Sets Exercise 365 367
Contents xiii 12.3. Jordan Measurable Sets; Sets with Volume Exercises 12.4. Lebesgue Measure Zero Exercises 12.5. A Criterion for Riemann Integrability Exercise 367 369 369 373 373 377 12.6. Properties of Volume and Integrals Exercises 12.7. Multiple Integrals Exercises 377 383 384 388 Chapter 13. 13.1. Transformation of Integrals A Space-Filling Curve 13.2. Volume and Integrability under C1 Maps Exercises 13.3. Linear Images of Sets with Volume Exercises 13.4. The Change of Variables Formula Exercises 13.5. The Definition of Surface Integrals Exercises 13.6. Notes and References Chapter 14. Ordinary Differential Equations 14.1. Scalar Differential Equations Exercises 14.2. Systems of Ordinary Differential Equations 14.2.1. Definition of Solution and the Integral Equation Exercise 14.2.2. Completeness of Cn[a, b] Exercises 14.2.3. The Local Lipchitz Condition Exercises 14.2.4. Existence and Uniqueness of Solutions Exercises 389 390 391 394 395 402 402 412 414 420 420 421 421 425 425 426 427 427 429 429 432 432 434 14.3. Extension of Solutions 14.3.1. The Maximal Interval of Definition Exercise 14.3.2. An Example of a Newtonian System Exercise 435 435 438 438 439 14.4. Continuous Dependence 14.4.1. Continuous Dependence on Initial Conditions, Parameters, and Vector Fields Exercises 439 439 442
Contents XIV 14.4.2. Newtonian Equations and Examples of Stability Exercises 14.5. Matrix Exponentials and Linear Autonomous Systems Exercises 443 444 446 450 14.6. 450 Notes and References Chapter 15. The Dirichlet Problem and Fourier Series 451 15.1. Introduction to Laplace’s Equation 15.2. Orthogonality of the Trigonometric Set Exercises 15.3. The Dirichlet Problem for the Disk Exercises 15.4. More Separation of Variables 15.4.1. The Heat Equation: Two Basic Problems Exercises 15.4.2. The Wave Equation with Fixed Ends Exercise 452 453 455 456 465 467 467 467 470 470 15.5. The Best Mean Square Approximation Exercises 471 475 15.6. Convergence of Fourier Series Exercises 476 485 15.7. Fejér’s Theorem Exercises 15.8. Notes and References 486 490 491 Chapter 16. Measure Theory and Lebesgue Measure 493 16.1. Algebras and σ-Algebras Exercise 16.2. Arithmetic in the Extended Real Numbers 494 498 498 16.3. Measures Exercises 499 505 16.4. Measure from Outer Measure Exercises 505 510 16.5. Lebesgue Measure in Euclidean Space 16.5.1. Lebesgue Measure on the Real Line Exercises 16.5.2. Metric Outer Measure; Lebesgue Measure on Euclidean Space Exercises 16.6. Notes and References 510 510 514 514 524 525 Chapter 17. The Lebesgue Integral 17.1. Measurable Functions Exercises 527 528 534
Contents XV 17.2. Simple Functions and the Integral Exercises 17.3. Definition of the Lebesgue Integral Exercises 17.4. The Limit Theorems Exercises 535 537 537 539 539 547 17.5. Comparison with the Riemann Integral Exercises 17.6. Banach Spaces of Integrable Functions Exercises 17.7. Notes and References 549 552 552 555 555 Chapter 18. Inner Product Spaces andFourier Series 18.1. Examples of Orthonormal Sets Exercises 18.2. Orthonormal Expansions 18.2.1. Basic Results for Inner Product Spaces 18.2.2. Complete Spaces and Complete Orthonormal Sets Exercises 18.3. Mean Square Convergence 18.3.1. Comparison of Pointwise, Uniform, and L2 NormConvergence Exercises 18.3.2. Mean Square Convergence for CP[—π, π] 18.3.3. Mean Square Convergence for 77[—π, π] 18.4. Hilbert Spaces of Integrable Functions Exercises 18.5. Notes and References 557 557 558 559 559 563 567 569 570 571 572 573 576 584 585 Appendix A. The Scliroeder-BernsteinTheorem A.l. Proof of the Scliroeder-Bernstein Theorem Exercise 587 587 588 Appendix B. Symbols and Notations B.l. Symbols and Notations Reference List 589 589 B.2. The Greek Alphabet 591 Bibliography 593 Index 597
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| spelling | Terrell, William J. Verfasser (DE-588)120036614X aut A passage to modern analysis William J. Terrell Providence, Rhode Island American Mathematical Society [2019] xxvii, 607 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 41 A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level. A Passage to Modern Analysis is grounded solidly in the analysis of \mathbf{R} and \mathbf{R} {n}, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 9781470455200 Erscheint auch als Terrell, William J A Passage to Modern Analysis Providence : American Mathematical Society, 2019 1 online resource (638 pages) Online-Ausgabe 978-1-4704-5520-0 Pure and applied undergraduate texts 41 (DE-604)BV035489189 41 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031731617&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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