Real analysis and foundations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2014
|
| Edition: | 3. ed. |
| Series: | Textbooks in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVIII, 412 S. graph. Darst. |
| ISBN: | 9781466587311 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Table
of
Contents
Preface to the Third Edition
iii
Preface to the Second Edition
v
Preface to the First Edition
vii
1
Number Systems
1
1.1
The Real Numbers
.......................... 1
EXERCISES
............................. 8
1.2
The Complex Numbers
........................ 9
EXERCISES
............................. 14
2
Sequences
15
2.1
Convergence of Sequences
...................... 15
EXERCISES
............................. 21
2.2
Subsequences
............................. 22
EXERCISES
............................. 25
2.3
Lim
sup and
Lim inf.........................
26
EXERCISES
............................. 28
2.4
Some Special Sequences
....................... 29
EXERCISES
............................. 31
3
Series of Numbers
33
3.1
Convergence of Series
........................ 33
EXERCISES
............................. 37
3.2
Elementary Convergence Tests
................... 39
EXERCISES
............................. 45
3.3
Advanced Convergence Tests
.................... 46
EXERCISES
............................. 51
3.4
Some Special Series
.......................... 52
EXERCISES
............................. 57
xv
XVI
3.5
Operations on Series
......................... 59
EXERCISES
............................. 62
4
Basic Topology
63
4.1
Open and Closed Sets
........................ 63
EXERCISES
............................. 68
4.2
Further Properties of Open and Closed Sets
............ 69
EXERCISES
............................. 72
4.3
Compact Sets
............................. 73
EXERCISES
............................. 76
4.4
The Cantor Set
............................ 76
EXERCISES
............................. 79
4.5
Connected and Disconnected Sets
.................. 80
EXERCISES
............................. 82
4.6
Perfect Sets
.............................. 83
EXERCISES
............................. 84
5
Limits and Continuity of Functions
85
5.1
Basic Properties of the Limit of a Function
............ 85
EXERCISES
............................. 90
5.2
Continuous Functions
........................ 91
EXERCISES
............................. 96
5.3
Topological Properties and Continuity
............... 96
EXERCISES
............................. 102
5.4
Classifying Discontinuities and
Monotonicity
........... 104
EXERCISES
............................. 107
6
Differentiation of Functions 111
6.1
The Concept of Derivative
......................
Ill
EXERCISES
............................. 119
6.2
The Mean Value Theorem and Applications
............ 120
EXERCISES
............................. 126
6.3
More on the Theory of Differentiation
............... 127
EXERCISES
. . . .......................... 130
7
The Integral
133
7.1
Partitions and the Concept of Integral
............... 133
EXERCISES
............................. 138
7.2
Properties of the Riemann Integral
................. 140
EXERCISES
............................. 147
7.3
Another Look at the Integral
.................... 149
EXERCISES
............................. 153
7.4
Advanced Results on Integration Theory
.............. 153
EXERCISES
.............. .............. 160
XVII
8
Sequences and Series of Functions
163
8.1
Partial Sums and Pointwise Convergence
.............. 163
EXERCISES
............................. 167
8.2
More on Uniform Convergence
................... 168
EXERCISES
............................. 171
8.3
Series of Functions
.......................... 172
EXERCISES
............................. 175
8.4
The
Weierstrass
Approximation Theorem
............. 176
EXERCISES
............................. 180
9
Elementary Transcendental Functions
183
9.1
Power Series
.............................. 183
EXERCISES
............................. 188
9.2
More on Power Series: Convergence Issues
............. 189
EXERCISES
............................. 193
9.3
The Exponential and Trigonometric Functions
.......... 194
EXERCISES
............................. 199
9.4
Logarithms and Powers of Real Numbers
............. 201
EXERCISES
............................. 203
10
Differential Equations
205
10.1
Picard s Existence and Uniqueness Theorem
............ 205
10.1.1
The Form of a Differential Equation
............ 205
10.1.2
Picard s Iteration Technique
................. 206
10.1.3
Some Illustrative Examples
................. 207
10.1.4
Estimation of the
Picard
Iterates
.............. 209
EXERCISES
............................. 210
10.2
Power Series Methods
........................ 212
EXERCISES
............................. 220
11
Introduction to Harmonic Analysis
223
11.1
The Idea of Harmonic Analysis
................... 223
EXERCISES
............................. 224
11.2
The Elements of Fourier Series
................... 225
EXERCISES
............................. 231
11.3
An Introduction to the Fourier Transform
............. 235
11.3.1
APPENDIX: Approximation by Smooth Functions
.... 238
EXERCISES
............................. 240
11.4
Fourier Methods and Differential Equations
............ 243
11.4.1
Remarks on Different Fourier Notations
.......... 243
11.4.2
The Dirichlet Problem on the Disc
............. 244
EXERCISES
............................. 248
XVIII
12
Functions of Several Variables
253
12.1
A New Look at the Basic Concepts of Analysis
.......... 253
EXERCISES
............................. 257
12.2
Properties of the Derivative
..................... 258
EXERCISES
............................. 263
12.3
The Inverse and Implicit Function Theorems
........... 264
EXERCISES
............................. 269
13
Advanced Topics
271
13.1
Metric Spaces
............................. 271
EXERCISES
............................. 275
13.2
Topology in a Metric Space
..................... 276
EXERCISES
............................. 279
13.3
The Bake Category Theorem
.................... 280
EXERCISES
............................. 284
13.4
The
Ascołi-Arzela
Theorem
..................... 284
EXERCISES
............................. 287
14
Normed Linear Spaces
289
14.1
What Is This Subject About?
.................... 289
EXERCISES
............................. 290
14.2
What Is a Normed Linear Space?
.................. 290
EXERCISES
............................. 293
14.3
Finite-Dimensional Spaces
...................... 294
EXERCISES
............................. 295
14.4
Linear Operators
........................... 296
EXERCISES
............................. 298
14.5
The Three Big Results
........................ 299
EXERCISES
............................. 304
14.6
Applications of the Big Three
.................... 305
EXERCISES
............................. 315
APPENDIX I: Elementary Number Systems
317
APPENDIX II: Logic and Set Theory
335
APPENDIX III: Review of Linear Algebra
369
Table of Notation
377
Glossary
383
Bibliography
403
Index
407
Real Analysis
and Foundations
THIRD EDITION
Back by popular demand, Real Analysis and Foundations, Third Edition
bridges the gap between classic theoretical texts and less rigorous ones,
providing a smooth transition from logic and proofs to real analysis. Along
with the basic material, the text covers Riemann-Stieltjes integrals, Fourier
analysis, metric spaces and applications, and differential equations.
Offering a more streamlined presentation, this edition moves elementary
number systems and set theory and logic to appendices and removes
the material on wavelet theory, measure theory, differential forms, and the
method of characteristics. It also adds a chapter on normed linear spaces
and includes more examples and varying levels of exercises.
Features
•
Presents a clear, thorough treatment of the theorems and concepts of
real analysis
•
Includes a new chapter on normed linear spaces
•
Provides more examples throughout the text and additional exercises
at the end of each section
•
Designates challenging exercises with an asterisk
With extensive examples and thorough explanations, this best-selling book
continues to give you a solid foundation in mathematical analysis and its
applications. It prepares you for further exploration of measure theory,
functional analysis, harmonic analysis, and beyond.
ISBN:
CRC
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an
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|
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 399 SK 400 SK 420 ST 110 |
| classification_tum | MAT 260f |
| ctrlnum | (OCoLC)931116502 (DE-599)BVBBV041154997 |
| dewey-full | 515.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.8 |
| dewey-search | 515.8 |
| dewey-sort | 3515.8 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 3. ed. |
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| indexdate | 2024-07-10T00:40:50Z |
| institution | BVB |
| isbn | 9781466587311 |
| language | English |
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| physical | XVIII, 412 S. graph. Darst. |
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| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Real analysis and foundations Steven G. Krantz 3. ed. Boca Raton [u.a.] CRC Press 2014 XVIII, 412 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd rswk-swf Reelle Analysis (DE-588)4627581-2 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s Reelle Variable (DE-588)4202614-3 s DE-604 Reelle Analysis (DE-588)4627581-2 s 1\p DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026130389&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026130389&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Krantz, Steven G. 1951- Real analysis and foundations Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4202614-3 (DE-588)4627581-2 (DE-588)4001865-9 |
| title | Real analysis and foundations |
| title_auth | Real analysis and foundations |
| title_exact_search | Real analysis and foundations |
| title_full | Real analysis and foundations Steven G. Krantz |
| title_fullStr | Real analysis and foundations Steven G. Krantz |
| title_full_unstemmed | Real analysis and foundations Steven G. Krantz |
| title_short | Real analysis and foundations |
| title_sort | real analysis and foundations |
| topic | Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable (DE-588)4202614-3 gnd Reelle Analysis (DE-588)4627581-2 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis - Reelle Variable Functions of real variables Mathematical analysis Reelle Variable Reelle Analysis Analysis |
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