Understanding analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English German |
| Veröffentlicht: |
New York [u.a.]
Springer
2001
|
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 257 S. graph. Darst. |
| ISBN: | 0387950605 |
Internformat
MARC
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Datensatz im Suchindex
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| adam_text | Contents
Preface v
1 The Real Numbers 1
1.1 Discussion: The Irrationality of y/2 1
1.2 Some Preliminaries 4
1.3 The Axiom of Completeness 13
1.4 Consequences of Completeness 18
1.5 Cantor s Theorem 29
1.6 Epilogue 33
2 Sequences and Series 35
2.1 Discussion: Rearrangements of Infinite Series 35
2.2 The Limit of a Sequence 38
2.3 The Algebraic and Order Limit Theorems 44
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series 50
2.5 Subsequences and the Bolzano Weierstrass Theorem 55
2.6 The Cauchy Criterion 58
2.7 Properties of Infinite Series 62
2.8 Double Summations and Products of Infinite Series 69
2.9 Epilogue 73
3 Basic Topology of R 75
3.1 Discussion: The Cantor Set 75
3.2 Open and Closed Sets 78
3.3 Compact Sets 84
3.4 Perfect Sets and Connected Sets 89
3.5 Baire s Theorem 94
3.6 Epilogue 96
4 Functional Limits and Continuity 99
4.1 Discussion: Examples of Dirichlet and Thomae 99
4.2 Functional Limits 103
4.3 Combinations of Continuous Functions 109
xi
xii Contents
4.4 Continuous Functions on Compact Sets 114
4.5 The Intermediate Value Theorem 120
4.6 Sets of Discontinuity 125
4.7 Epilogue 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous? 129
5.2 Derivatives and the Intermediate Value Property 131
5.3 The Mean Value Theorem 137
5.4 A Continuous Nowhere Differentiable Function 144
5.5 Epilogue 148
6 Sequences and Series of Functions 151
6.1 Discussion: Branching Processes 151
6.2 Uniform Convergence of a Sequence of Functions 154
6.3 Uniform Convergence and Differentiation 164
6.4 Series of Functions 167
6.5 Power Series 169
6.6 Taylor Series 176
6.7 Epilogue 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration be Denned? 183
7.2 The Definition of the Riemann Integral 186
7.3 Integrating Functions with Discontinuities 191
7.4 Properties of the Integral 195
7.5 The Fundamental Theorem of Calculus 199
7.6 Lebesgue s Criterion for Riemann Integrability 203
7.7 Epilogue 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral 213
8.2 Metric Spaces and the Baire Category Theorem 222
8.3 Fourier Series 228
8.4 A Construction of R From Q 243
Bibliography 251
Index 253
|
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| indexdate | 2024-07-09T18:49:40Z |
| institution | BVB |
| isbn | 0387950605 |
| language | English German |
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| physical | XII, 257 S. graph. Darst. |
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| series2 | Undergraduate texts in mathematics |
| spelling | Abbott, Stephen 1964- Verfasser (DE-588)122743083 aut Understanding analysis Stephen Abbott New York [u.a.] Springer 2001 XII, 257 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Reelle Analysis - Lehrbuch Mathematical analysis Reelle Analysis (DE-588)4627581-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Reelle Analysis (DE-588)4627581-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009329231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4627581-2 (DE-588)4123623-3 |
| title | Understanding analysis |
| title_auth | Understanding analysis |
| title_exact_search | Understanding analysis |
| title_full | Understanding analysis Stephen Abbott |
| title_fullStr | Understanding analysis Stephen Abbott |
| title_full_unstemmed | Understanding analysis Stephen Abbott |
| title_short | Understanding analysis |
| title_sort | understanding analysis |
| topic | Reelle Analysis - Lehrbuch Mathematical analysis Reelle Analysis (DE-588)4627581-2 gnd |
| topic_facet | Reelle Analysis - Lehrbuch Mathematical analysis Reelle Analysis Lehrbuch |
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