Real analysis through modern infinitesimals:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Encyclopedia of mathematics and its applications
140 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 565 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9781107002029 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804145722033963008 |
|---|---|
| adam_text | Titel: Real analysis through modern infinitesimals
Autor: Vakil, Nader
Jahr: 2011
CONTENTS
Preface page xiii
Introduction 1
0.1 Infinite sets and the continuum 1
0.2 An analytic model of the straight line 2
0.3 The rise and fall of infinitesimals 4
0.4 The return of infinitesimals 5
0.5 Ultrafilters and ultrapowers 7
0.6 What is internal set theory? 8
0.7 Internal, extemal, and Standard sets 10
PART I ELEMENTS OF REAL ANALYSIS
1 Internal set theory 17
1.1 The basic language of IST 18
1.2 Exercises 20
1.3 Classes 21
1.4 Basic concepts and axioms of IST 24
1.5 Exercises 27
1.6 Relations and functions 28
1.7 Exercises 31
1.8 The replacement axiom 32
1.9 The regularity and infinity axioms 33
1.10 The transfer axiom 35
1.11 Exercises 38
2 The real number System 40
2.1 Ordered field properties of R 41
viii Contents
2.2 Integers, rationals, and irrationals 44
2.3 Exercises 53
2.4 The supremum principle 54
2.5 Exercises 59
2.6 Ordering properties of the integers 60
2.7 Exercises 62
2.8 Absolute value and intervals 63
2.9 Exercises 66
2.10 Finite and infinite sets 66
2.11 Exercises 70
2.12 Idealization axiom 71
2.13 Nonstandard numbers 71
2.14 Exercises 77
2.15 Standardization axiom 78
2.16 Exercises 83
2.17 Standard finite sets 83
2.18 Exercises 87
3 Sequences and series 88
3.1 Convergence of sequences 88
3.2 Exercises 95
3.3 Monotone sequences 97
3.4 Exercises 101
3.5 Subsequences 102
3.6 Exercises 105
3.7 Cauchy convergence criterion 106
3.8 Exercises 107
3.9 Infinite series 109
3.10 Exercises 120
3.11 Power and logarithmic functions 122
3.12 More on the exponential function 129
3.13 Exercises 132
4 ThetopologyofE 134
4.1 Open and closed sets 134
4.2 Exercises 140
4.3 The Heine-Borel theorem 141
4.4 Exercises 145
5 Limits and continuity 146
5.1 Limit of a function at a point 147
5.2 Exercises 154
Contents ix
156
160
161
165
166
169
170
174
176
182
185
187
187
196
199
202
203
206
208
212
213
215
219
220
226
226
229
231
233
241
244
245
255
257
262
263
268
8 Sequences and series of functions 270
8.1 Pointwise and uniform convergence 271
8.2 Exercises 278
5.3 One-sided limits and infinite limits
5.4 Exercises
5.5 Continuous functions
5.6 Exercises
5.7 Properties of continuous functions
5.8 Exercises
5.9 Uniform continuity
5.10 Exercises
5.11 Monotone functions
5.12 Oscillation of a function
5.13 Exercises
6 Differentiation
6.1 Definitions and basic properties
6.2 Exercises
6.3 Differentiability and local properties
6.4 Exercises
6.5 The mean value theorem
6.6 Exercises
6.7 L Hospital s rule
6.8 Exercises
6.9 Higher-order derivatives
6.10 Taylor polynomials
6.11 Exercises
6.12 Convex functions
6.13 Exercises
6.14 Darboux s theorem
6.15 Exercises
7 Integration
7.1 Definition of the Riemann integral
7.2 Basic properties of the Riemann integral
7.3 Exercises
7.4 The Lebesgue-Riemann theorem
7.5 Exercises
7.6 The fundamental theorem of calculus
7.7 Exercises
7.8 Improper Riemann integration
7.9 Exercises
x Contents
8.3 Applications of uniform convergence 280
8.4 Exercises 291
9 Infinite series 293
9.1 Upper and lower limits of sequences 294
9.2 Exercises 300
9.3 Absolute and conditional convergence 302
9.4 Exercises 314
9.5 Power series 315
9.6 Functions defined by power series 318
9.7 Exercises 328
PART II ELEMENTS OF ABSTRACT ANALYSIS
10 Point set topology 333
334
341
352
354
360
361
367
371
373
376
11 Metrie Spaces 379
379
383
385
395
396
400
401
12 Complete metric Spaces 404
12.1 Completeness 404
12.2 Total boundedness 409
12.3 Compactness in metric spaces 414
12.4 Uniform and Lipschitz continuity 416
12.5 Products of metric Spaces 420
12.6 The completion of a metric space 424
12.7 Exercises 430
Point set topology
10.1 Topological spaces
10.2 Monads in topological Spaces
10.3 Exercises
10.4 Continuous functions
10.5 Exercises
10.6 Compactness
10.7 Local compactness
10.8 Connectedness
10.9 Monads of Alters
10.10 Exercises
Metric Spaces
11.1 The metric topology
11.2 Normed vector spaces
11.3 Metric space properties of W
11.4 Exercises
11.5 Standard hulls of classes and functions
11.6 Exercises
11.7 The Peano existence theorem
Contents xi
13 Some applications of completeness 433
13.1 Baire category theorem 433
13.2 Two extension theorems 438
13.3 Banach fixed point theorem 440
14 Linear Operators 443
14.1 The p-norm of a linear Operator 443
14.2 The Operator norm 453
14.3 Invertible Operators 455
14.4 Integral Operators 456
14.5 Exercises 459
15 Differential calculus on W 461
15.1 First-order differentials 461
15.2 Directional and partial derivatives 468
15.3 Exercises 475
15.4 Higher-order differentials 477
15.5 Inverse and implicit function theorems 489
15.6 Exercises 501
16 Function space topologies 503
16.1 The JC-convergence topology 503
16.2 Metrization of/C-convergence topology 506
16.3 The /C-open topology 507
16.4 The Ascoli theorem 510
16.5 The Stone-Weierstrass theorem 514
16.6 Exercises 518
Appendix A Vector spaces 521
Appendix B The 6-adic representation of numbers 523
B.l ib-adic representation of integers 524
B.2 6-adic representation of real numbers 527
B.3 Exercises 531
B.4 Some examples of proofs by induction 532
B.5 Existence of roots 534
Appendix C Finite, denumerable, and uncountable sets 536
C.l Finite sets 536
C.2 Exercises 538
C.3 Denumerable sets 538
C.4 Uncountable sets 542
C.5 Exercises 543
xii Contents
Appendix D The syntax of mathematical languages 544
D. 1 Constituents of mathematical Statements 544
D.2 Logical and non-logical symbols 546
D.3 Terms and formulas 548
D.4 Exercises 552
References 554
Index 557
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T23:23:58Z |
| institution | BVB |
| isbn | 9781107002029 |
| language | English |
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| spelling | Vakil, Nader 1948- Verfasser (DE-588)1012842223 aut Real analysis through modern infinitesimals Nader Vakil 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2011 XIX, 565 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 140 Infinitesimalanalysis (DE-588)4161657-1 gnd rswk-swf Mathematical analysis Transformations, Infinitesimal Infinitesimalanalysis (DE-588)4161657-1 s DE-604 Encyclopedia of mathematics and its applications 140 (DE-604)BV000903719 140 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022572735&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vakil, Nader 1948- Real analysis through modern infinitesimals Encyclopedia of mathematics and its applications Infinitesimalanalysis (DE-588)4161657-1 gnd |
| subject_GND | (DE-588)4161657-1 |
| title | Real analysis through modern infinitesimals |
| title_auth | Real analysis through modern infinitesimals |
| title_exact_search | Real analysis through modern infinitesimals |
| title_full | Real analysis through modern infinitesimals Nader Vakil |
| title_fullStr | Real analysis through modern infinitesimals Nader Vakil |
| title_full_unstemmed | Real analysis through modern infinitesimals Nader Vakil |
| title_short | Real analysis through modern infinitesimals |
| title_sort | real analysis through modern infinitesimals |
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