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Real Analysis and Foundations, Third Edition 3rd New edition [Hardback]

4.38/5 (14 ratings by Goodreads)
(Washington University, St. Louis, Missouri, USA)
  • Formāts: Hardback, 430 pages, height x width: 235x156 mm, weight: 748 g, 55 Illustrations, black and white
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 24-Jun-2013
  • Izdevniecība: CRC Press Inc
  • ISBN-10: 1466587318
  • ISBN-13: 9781466587311
Citas grāmatas par šo tēmu:
Real Analysis and Foundations, Third Edition 3rd New editionPalielināt
  • Formāts: Hardback, 430 pages, height x width: 235x156 mm, weight: 748 g, 55 Illustrations, black and white
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 24-Jun-2013
  • Izdevniecība: CRC Press Inc
  • ISBN-10: 1466587318
  • ISBN-13: 9781466587311
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781466587311.html
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A Readable yet Rigorous Approach to an Essential Part of Mathematical Thinking
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.

New to the Third Edition
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.

Extensive Examples and Thorough Explanations Cultivate an In-Depth Understanding
This best-selling book continues to give students a solid foundation in mathematical analysis and its applications. It prepares them for further exploration of measure theory, functional analysis, harmonic analysis, and beyond.

Recenzijas

Praise for the Second Edition: "The book is recommended as a source for middle-level mathematical courses. It can be used not only in mathematical departments, but also by physicists, engineers, economists, and other experts in applied sciences who want to understand the main ideas of analysis in order to use them to study mathematical models of different type processes." Zentralblatt MATH









"The book contains many well-chosen examples and each of the fifteen chapters is followed by almost 500 exercises. Illustrative pictures are instructive and the design of the book makes reading it a real pleasure. The book can be recommended for university libraries, teachers, and students." EMS Newsletter

Preface to the Third Edition iii
Preface to the Second Edition v
Preface to the First Edition vii
1 Number Systems
1(14)
1.1 The Real Numbers
1(8)
Exercises
8(1)
1.2 The Complex Numbers
9(6)
Exercises
14(1)
2 Sequences
15(18)
2.1 Convergence of Sequences
15(7)
Exercises
21(1)
2.2 Subsequences
22(4)
Exercises
25(1)
2.3 Lim sup and Lim inf
26(3)
Exercises
28(1)
2.4 Some Special Sequences
29(4)
Exercises
31(2)
3 Series of Numbers
33(30)
3.1 Convergence of Series
33(6)
Exercises
37(2)
3.2 Elementary Convergence Tests
39(7)
Exercises
45(1)
3.3 Advanced Convergence Tests
46(6)
Exercises
51(1)
3.4 Some Special Series
52(7)
Exercises
57(2)
3.5 Operations on Series
59(4)
Exercises
62(1)
4 Basic Topology
63(22)
4.1 Open and Closed Sets
63(6)
Exercises
68(1)
4.2 Further Properties of Open and Closed Sets
69(4)
Exercises
72(1)
4.3 Compact Sets
73(3)
Exercises
76(1)
4.4 The Cantor Set
76(4)
Exercises
79(1)
4.5 Connected and Disconnected Sets
80(3)
Exercises
82(1)
4.6 Perfect Sets
83(2)
Exercises
84(1)
5 Limits and Continuity of Functions
85(26)
5.1 Basic Properties of the Limit of a Function
85(6)
Exercises
90(1)
5.2 Continuous Functions
91(5)
Exercises
96(1)
5.3 Topological Properties and Continuity
96(8)
Exercises
102(2)
5.4 Classifying Discontinuities and Monotonicity
104(7)
Exercises
107(4)
6 Differentiation of Functions
111(22)
6.1 The Concept of Derivative
111(9)
Exercises
119(1)
6.2 The Mean Value Theorem and Applications
120(7)
Exercises
126(1)
6.3 More on the Theory of Differentiation
127(6)
Exercises
130(3)
7 The Integral
133(30)
7.1 Partitions and the Concept of Integral
133(7)
Exercises
138(2)
7.2 Properties of the Riemann Integral
140(9)
Exercises
147(2)
7.3 Another Look at the Integral
149(4)
Exercises
153(1)
7.4 Advanced Results on Integration Theory
153(10)
Exercises
160(3)
8 Sequences and Series of Functions
163(20)
8.1 Partial Sums and Pointwise Convergence
163(5)
Exercises
167(1)
8.2 More on Uniform Convergence
168(4)
Exercises
171(1)
8.3 Series of Functions
172(4)
Exercises
175(1)
8.4 The Weierstrass Approximation Theorem
176(7)
Exercises
180(3)
9 Elementary Transcendental Functions
183(22)
9.1 Power Series
183(6)
Exercises
188(1)
9.2 More on Power Series: Convergence Issues
189(5)
Exercises
193(1)
9.3 The Exponential and Trigonometric Functions
194(7)
Exercises
199(2)
9.4 Logarithms and Powers of Real Numbers
201(4)
Exercises
203(2)
10 Differential Equations
205(18)
10.1 Picard's Existence and Uniqueness Theorem
205(7)
10.1.1 The Form of a Differential Equation
205(1)
10.1.2 Picard's Iteration Technique
206(1)
10.1.3 Some Illustrative Examples
207(2)
10.1.4 Estimation of the Picard Iterates
209(1)
Exercises
210(2)
10.2 Power Series Methods
212(11)
Exercises
220(3)
11 Introduction to Harmonic Analysis
223(30)
11.1 The Idea of Harmonic Analysis
223(2)
Exercises
224(1)
11.2 The Elements of Fourier Series
225(10)
Exercises
231(4)
11.3 An Introduction to the Fourier Transform
235(8)
11.3.1 APPENDIX: Approximation by Smooth Functions
238(2)
Exercises
240(3)
11.4 Fourier Methods and Differential Equations
243(10)
11.4.1 Remarks on Different Fourier Notations
243(1)
11.4.2 The Dirichlet Problem on the Disc
244(4)
Exercises
248(5)
12 Functions of Several Variables
253(18)
12.1 A New Look at the Basic Concepts of Analysis
253(5)
Exercises
257(1)
12.2 Properties of the Derivative
258(6)
Exercises
263(1)
12.3 The Inverse and Implicit Function Theorems
264(7)
Exercises
269(2)
13 Advanced Topics
271(18)
13.1 Metric Spaces
271(5)
Exercises
275(1)
13.2 Topology in a Metric Space
276(4)
Exercises
279(1)
13.3 The Baire Category Theorem
280(4)
Exercises
284(1)
13.4 The Ascoli-Arzela Theorem
284(5)
Exercises
287(2)
14 Normed Linear Spaces
289(28)
14.1 What Is This Subject About?
289(1)
Exercises
290(1)
14.2 What Is a Normed Linear Space?
290(4)
Exercises
293(1)
14.3 Finite-Dimensional Spaces
294(2)
Exercises
295(1)
14.4 Linear Operators
296(3)
Exercises
298(1)
14.5 The Three Big Results
299(6)
Exercises
304(1)
14.6 Applications of the Big Three
305(12)
Exercises
315(2)
Appendix I Elementary Number Systems 317(18)
Appendix II Logic and Set Theory 335(34)
Appendix III Review of Linear Algebra 369(8)
Table of Notation 377(6)
Glossary 383(20)
Bibliography 403(4)
Index 407
Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.