A companion to analysis: a second first and first second course in analysis
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2004
|
| Series: | Graduate studies in mathematics
62 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 590 S. graph. Darst. |
| ISBN: | 9780821834473 0821834479 |
Staff View
MARC
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| 100 | 1 | |a Körner, T. W. |d 1946- |e Verfasser |0 (DE-588)120225379 |4 aut | |
| 245 | 1 | 0 | |a A companion to analysis |b a second first and first second course in analysis |c T. W. Körner |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2004 | |
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Record in the Search Index
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| adam_text | Contents
Introduction xi
Chapter 1. The Real Line 1
§1.1. Why do we bother? 1
§1.2. Limits 3
§1.3. Continuity 7
§1.4. The fundamental axiom 9
§1.5. The axiom of Archimedes 10
§1.6. Lion hunting 15
§1.7. The mean value inequality 18
§1.8. Full circle 21
§1.9. Are the real numbers unique? 22
Chapter 2. A First Philosophical Interlude W 25
§2.1. Is the intermediate value theorem obvious? v ? 25
Chapter 3. Other Versions of the Fundamental Axiom 31
§3.1. The supremum 31
§3.2. The Bolzano Weierstrass theorem 37
§3.3. Some general remarks 42
vii
viii Contents
Chapter 4. Higher Dimensions 43
§4.1. Bolzano Weierstrass in Higher Dimensions 43
§4.2. Open and closed sets 48
§4.3. A central theorem of analysis 56
§4.4. The mean value theorem 59
§4.5. Uniform continuity 64
§4.6. The general principle of convergence 66
Chapter 5. Sums and Suchlike ^? 73
§5.1. Comparison tests 9? 73
§5.2. Conditional convergence C 75
§5.3. Interchanging limits ? 80
§5.4. The exponential function s? 88
§5.5. The trigonometric functions s? 95
§5.6. The logarithm V 99
§5.7. Powers 9 105
§5.8. The fundamental theorem of algebra C 109
Chapter 6. Differentiation 117
§6.1. Preliminaries 117
§6.2. The operator norm and the chain rule 123
§6.3. The mean value inequality in higher dimensions 130
Chapter 7. Local Taylor Theorems 135
§7.1. Some one dimensional Taylor theorems 135
§7.2. Some many dimensional local Taylor theorems 139
§7.3. Critical points 147
Chapter 8. The Riemann Integral 161
§8.1. Where is the problem ? 161
§8.2. Riemann integration 164
§8.3. Integrals of continuous functions 173
§8.4. First steps in the calculus of variations 9? 181
§8.5. Vector valued integrals 192
Contents ix
Chapter 9. Developments and Limitations of the Riemann Integral v 195
§9.1. Why go further? 195
§9.2. Improper integrals 9 197
§9.3. Integrals over areas V 201
§9.4. The Riemann Stieltjes integral V 206
§9.5. How long is a piece of string? 9 213
Chapter 10. Metric Spaces 221
§10.1. Sphere packing 9 221
§10.2. Shannon s theorem 9 224
§10.3. Metric spaces 229
§10.4. Norms and the interaction of algebra and analysis 234
§10.5. Geodesies 9 241
Chapter 11. Complete Metric Spaces 249
§11.1. Completeness 249
§11.2. The Bolzano Weierstrass property 257
§11.3. The uniform norm 261
§11.4. Uniform convergence 265
§11.5. Power series 273
§11.6. Fourier series 9 282
Chapter 12. Contraction Mappings and Differential Equations 287
§12.1. Banach s contraction mapping theorem 287
§12.2. Existence of solutions of differential equations 289
§12.3. Local to global 9 294
§12.4. Green s function solutions C 301
Chapter 13. Inverse and Implicit Functions 311
§13.1. The inverse function theorem 311
§13.2. The implicit function theorem 9 320
§13.3. Lagrange multipliers ? 328
x Contents
Chapter 14. Completion 335
§14.1. What is the correct question? 335
§14.2. The solution 341
§14.3. Why do we construct the reals? 9 344
§14.4. How do we construct the reals? C 348
§14.5. Paradise lost? W 354
Appendix A. Ordered Fields 357
Appendix B. Countability 361
Appendix C. The Care and Treatment of Counterexamples 365
Appendix D. A More General View of Limits 371
Appendix E. Traditional Partial Derivatives 377
Appendix F. Another Approach to the Inverse Function Theorem 383
Appendix G. Completing Ordered Fields 387
Appendix H. Constructive Analysis 391
Appendix I. Miscellany 395
Appendix J. Executive Summary 401
Appendix K. Exercises 405
Bibliography 583
Index 585
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T20:00:33Z |
| institution | BVB |
| isbn | 9780821834473 0821834479 |
| language | English |
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| spelling | Körner, T. W. 1946- Verfasser (DE-588)120225379 aut A companion to analysis a second first and first second course in analysis T. W. Körner Providence, RI American Math. Soc. 2004 XIV, 590 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 62 Analyse mathématique Análise matemática (textos intermediários) larpcal Reële functies gtt aMathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2105-2 Graduate studies in mathematics 62 (DE-604)BV009739289 62 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012908450&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Körner, T. W. 1946- A companion to analysis a second first and first second course in analysis Graduate studies in mathematics Analyse mathématique Análise matemática (textos intermediários) larpcal Reële functies gtt aMathematical analysis Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4151278-9 |
| title | A companion to analysis a second first and first second course in analysis |
| title_auth | A companion to analysis a second first and first second course in analysis |
| title_exact_search | A companion to analysis a second first and first second course in analysis |
| title_full | A companion to analysis a second first and first second course in analysis T. W. Körner |
| title_fullStr | A companion to analysis a second first and first second course in analysis T. W. Körner |
| title_full_unstemmed | A companion to analysis a second first and first second course in analysis T. W. Körner |
| title_short | A companion to analysis |
| title_sort | a companion to analysis a second first and first second course in analysis |
| title_sub | a second first and first second course in analysis |
| topic | Analyse mathématique Análise matemática (textos intermediários) larpcal Reële functies gtt aMathematical analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analyse mathématique Análise matemática (textos intermediários) Reële functies aMathematical analysis Analysis Einführung |
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