Spaces: an introduction to real analysis
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2017]
|
| Series: | Pure and applied undergraduate texts
29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 369 Seiten |
| ISBN: | 9781470440626 |
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Record in the Search Index
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| adam_text | Contents Preface ix Introduction - Mainly to the Students 1 Chapter 1. Preliminaries: Proofs, Sets, and Functions 1.1. Proofs 1.2. Sets and Boolean operations 1.3. Families of sets 1.4. Functions 1.5. Relations and partitions 1.6. Countability Notes and references for Chapter 1 5 5 8 11 13 17 20 22 Chapter 2. The Foundation of Calculus 2.1. Epsilon-delta and all that 2.2. Completeness 2.3. Four important theorems Notes and references for Chapter 2 23 24 29 37 42 j Chapter 3. Metric Spaces 3.1. Definitions and examples 3.2. Convergence and continuity 3.3. Open and closed sets 3.4. Complete spaces 3.5. Compact sets 3.6. An alternative description of compactness 3.7. The completion of a metric space 43 43 48 52 59 63 68 71
Contents VI Notes and references for Chapter 3 Chapter 4. 76 Spaces of Continuous Functions 79 4.1. 4.2. 4.3. Modes of continuity Modes of convergence Integrating and differentiating sequences 79 81 86 4.4. 4.5. Applications to power series Spaces of bounded functions 92 99 4.6. 4.7. Spaces of bounded, continuous functions Applications to differential equations 4.8. Compact sets of continuous functions 4.9. Differential equations revisited 4.10. Polynomials are dense in the continuous function 4.11. The Stone-Weierstrass Theorem Notes and references for Chapter 4 Chapter 5. Normed Spaces and Linear Operators 5.1. Normed spaces 101 103 107 112 116 123 131 133 133 5.2. 5.3. 5.4. 5.5. 5.6. Infinite sums and bases Inner product spaces Linear operators Inverse operators and Neumann series Baire’s Category Theorem 140 142 150 155 161 5.7. A group of famous theorems 167 Notes and references for Chapter 5 171 Chapter 6. Differential Calculus in Normed Spaces 173 6.1. 6.2. 6.3. 6.4. The derivative Finding derivatives The Mean Value Theorem The Riemann Integral 174 182 187 190 6.5. 6.6. Taylor’s Formula Partial derivatives 194 201 6.7. 6.8. 6.9. The Inverse Function Theorem The Implicit Function Theorem Differential equations yet again 206 212 216 6.10. Multilinear maps 6.11. Higher order derivatives Notes and references for Chapter 6 226 230 238
Сол ten էտ Chaptei■ 7. Measure and Integration 7.1. Measure spaces 7.2. Complete measures 7.3. 7.4. Measurable functions Integration of simple functions 7.5. Integrals of nonnegative functions 7.6. Integrable functions 7.7. Spaces of integrable functions 7.8. Ways to converge 7.9. Integration of complex functions Notes and references for Chapter 7 Chapter 8. Constructing Measures 8.1. Outer measure 8.2. 8.3. 8.4. 8.5. Measurable sets Carathéodory’s Theorem Lebesgue measure on the real line Approximation results The coin tossing measure 8.6. 8.7. Product measures 8.8. Fubini’s Theorem Notes and references for Chapter 8 Chapter 9. Fourier Series 9.1. Fourier coefficients and Fourier series 9.2. Convergence in mean square vii 239 240 248 252 257 262 271 276 285 288 290 291 292 294 297 304 307 311 313 316 324 9.3. 9.4. The Dirichlet kernel The Fejér kernel 325 327 333 336 341 9.5. The Riemann-Lebesgue Lemma 347 9.6. 9.7. Dini’s Test Pointwise divergence of Fourier series 350 354 9.8. Termwise operations Notes and references for Chapter 9 356 359 Bibliography 361 Index 363
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| isbn | 9781470440626 |
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| spelling | Lindstrøm, Tom 1954- Verfasser (DE-588)1145652689 aut Spaces an introduction to real analysis Tom L. Lindstrøm Providence, Rhode Island American Mathematical Society [2017] © 2017 xii, 369 Seiten txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 29 The Sally series Includes bibliographical references and index Topologischer Raum (DE-588)4137586-5 gnd rswk-swf Reelle Analysis (DE-588)4627581-2 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Topologischer Raum (DE-588)4137586-5 s Reelle Analysis (DE-588)4627581-2 s Maßtheorie (DE-588)4074626-4 s Funktionalanalysis (DE-588)4018916-8 s DE-604 b DE-604 Pure and applied undergraduate texts 29 (DE-604)BV035489189 29 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=030039426&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lindstrøm, Tom 1954- Spaces an introduction to real analysis Pure and applied undergraduate texts Includes bibliographical references and index Topologischer Raum (DE-588)4137586-5 gnd Reelle Analysis (DE-588)4627581-2 gnd Maßtheorie (DE-588)4074626-4 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4137586-5 (DE-588)4627581-2 (DE-588)4074626-4 (DE-588)4018916-8 (DE-588)4123623-3 |
| title | Spaces an introduction to real analysis |
| title_auth | Spaces an introduction to real analysis |
| title_exact_search | Spaces an introduction to real analysis |
| title_full | Spaces an introduction to real analysis Tom L. Lindstrøm |
| title_fullStr | Spaces an introduction to real analysis Tom L. Lindstrøm |
| title_full_unstemmed | Spaces an introduction to real analysis Tom L. Lindstrøm |
| title_short | Spaces |
| title_sort | spaces an introduction to real analysis |
| title_sub | an introduction to real analysis |
| topic | Topologischer Raum (DE-588)4137586-5 gnd Reelle Analysis (DE-588)4627581-2 gnd Maßtheorie (DE-588)4074626-4 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Topologischer Raum Reelle Analysis Maßtheorie Funktionalanalysis Lehrbuch |
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