| SoftcoverISBN: | 978-1-4704-6454-7 |
| Product Code: | TEXT/70 |
| List Price: | $69.00 |
| MAA Member Price: | $51.75 |
| AMS Member Price: | $51.75 |
| eBookISBN: | 978-1-4704-6847-7 |
| Product Code: | TEXT/70.E |
| List Price: | $69.00 |
| MAA Member Price: | $51.75 |
| AMS Member Price: | $51.75 |
| SoftcoverISBN: | 978-1-4704-6454-7 |
| eBookISBN: | 978-1-4704-6847-7 |
| Product Code: | TEXT/70.B |
| List Price: | $138.00$103.50 |
| MAA Member Price: | $103.50$77.63 |
| AMS Member Price: | $103.50$77.63 |
| Softcover ISBN: | 978-1-4704-6454-7 |
| Product Code: | TEXT/70 |
| List Price: | $69.00 |
| MAA Member Price: | $51.75 |
| AMS Member Price: | $51.75 |
| eBook ISBN: | 978-1-4704-6847-7 |
| Product Code: | TEXT/70.E |
| List Price: | $69.00 |
| MAA Member Price: | $51.75 |
| AMS Member Price: | $51.75 |
| Softcover ISBN: | 978-1-4704-6454-7 |
| eBookISBN: | 978-1-4704-6847-7 |
| Product Code: | TEXT/70.B |
| List Price: | $138.00$103.50 |
| MAA Member Price: | $103.50$77.63 |
| AMS Member Price: | $103.50$77.63 |
-
Book DetailsAMS/MAA TextbooksVolume: 70; 2022; 360 ppMSC: Primary 26; 37; 39;
Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and \(n\)-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series.
The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.ReadershipUndergraduate students interested in learning real analysis.
-
Table of Contents
-
Cover
-
Title page
-
Copyright
-
Contents
-
Preface
-
Chapter 0. Where We’re Starting and Where We’re Going
-
Chapter 1. Essential Tools
-
1.1. Sets and statements
-
1.2. Functions
-
1.3. Countability and uncountability
-
1.4. Induction
-
1.5. Order in the real line
-
1.6. Some vital inequalities
-
1.7. Exercises
-
Chapter 2. Metric Spaces
-
2.1. The definition of a metric space
-
2.2. Important metrics in Rⁿ
-
2.3. Open balls and open sets in metric spaces
-
2.4. Closed sets and limit points
-
2.5. Interior, closure, and boundary
-
2.6. Dense subsets
-
2.7. Equivalent metrics
-
2.8. Normed vector spaces
-
2.9. A brief note about conventions
-
2.10. Exercises
-
Chapter 3. Sequences
-
3.1. Convergence
-
3.2. Discrete dynamical systems
-
3.3. Sequences and limit points
-
3.4. Algebraic theorems for sequences
-
3.5. Subsequences
-
3.6. Completeness
-
3.7. The contraction mapping principle
-
3.8. Sets of sequences as metric spaces
-
3.9. Exercises
-
Chapter 4. Continuity
-
4.1. The definition of continuity
-
4.2. Equivalent formulations of continuity
-
4.3. Continuity and limit theorems for scalar- valued functions
-
4.4. Continuity and products of metric spaces
-
4.5. Uniform continuity
-
4.6. The metric space 𝐶([𝑎,𝑏],R )
-
4.7. An application to functional equations
-
4.8. Exercises
-
Chapter 5. Compactness and Connectedness
-
5.1. Basic definitions and results on compactness
-
5.2. The nested set property for compact sets
-
5.3. Compactness and continuity
-
5.4. Other facts about compactness
-
5.5. Connectedness
-
5.6. Periodic points of maps on intervals
-
5.7. Injective continuous functions defined on intervals
-
5.8. Exercises
-
Chapter 6. The Derivative
-
6.1. The definition of the derivative
-
6.2. Differentiation rules
-
6.3. Applications of the derivative
-
6.4. Exercises
-
Chapter 7. The Riemann Integral
-
7.1. Partitions and the definition of the integral
-
7.2. Basic properties of the integral
-
7.3. The fundamental theorem of calculus
-
7.4. Ordinary differential equations
-
7.5. Exercises
-
Chapter 8. Sequences of Functions
-
8.1. Infinite series
-
8.2. Power series
-
8.3. Higher derivatives and Taylor polynomials
-
8.4. Differentiation and integration of sequences of functions
-
8.5. The exponential function
-
8.6. Compact subsets in 𝐶[𝑎,𝑏]
-
8.7. Exercises
-
Chapter 9. Chaos in Discrete Dynamical Systems
-
9.1. The definition of chaos
-
9.2. Semiconjugacy
-
9.3. Subshifts of finite type
-
9.4. Itineraries and piecewise expanding maps
-
9.5. A dynamical system with a dense orbit but no periodic points
-
9.6. Exercises
-
Chapter 10. The Hausdorff Metric and Fractals
-
10.1. Definition of the Hausdorff metric
-
10.2. Properties of the Hausdorff metric
-
10.3. Fractals in the plane
-
10.4. Exercises
-
Bibliography
-
Index
-
Back Cover
-
-
Additional Material
-
Reviews
-
In terms of writing: this book is clear, challenging, and rewarding in its exposition. The ordering of topics is well-vetted; examples and proof details are well-chosen; and the exposition is well-written and imminently readable. The text offers reading questions after each section, as well as a variety of exercises at the end of each chapter. I appreciated both forms of questions; the 'exercises' are varied and completely appropriate, while the 'reading questions' allow for students to check their comprehension (and model the kinds of questions we want students to ask themselves as they read any mathematical text). Additionally, the text benefits from 'analytical advice' text boxes that offer just-in-time discussions of proof styles and big ideas in proof-writing.
...This seems to be an excellent text which truly highlights how the core elements of analysis can be applied in a number of powerful ways beyond calculus. I would argue against using this text in a course that uses Real Analysis as an 'introduction' to proof-writing, as such students might find this text a bit of a struggle. However, a well-prepared undergraduate will benefit immensely from the exposition here, and I would recommend the text to such a student without reservation.
John Ross, Southwestern University
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-
RequestsReview Copy – for reviewers who would like to review an AMS bookDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and \(n\)-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series.
The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.
Undergraduate students interested in learning real analysis.
-
Cover
-
Title page
-
Copyright
-
Contents
-
Preface
-
Chapter 0. Where We’re Starting and Where We’re Going
-
Chapter 1. Essential Tools
-
1.1. Sets and statements
-
1.2. Functions
-
1.3. Countability and uncountability
-
1.4. Induction
-
1.5. Order in the real line
-
1.6. Some vital inequalities
-
1.7. Exercises
-
Chapter 2. Metric Spaces
-
2.1. The definition of a metric space
-
2.2. Important metrics in Rⁿ
-
2.3. Open balls and open sets in metric spaces
-
2.4. Closed sets and limit points
-
2.5. Interior, closure, and boundary
-
2.6. Dense subsets
-
2.7. Equivalent metrics
-
2.8. Normed vector spaces
-
2.9. A brief note about conventions
-
2.10. Exercises
-
Chapter 3. Sequences
-
3.1. Convergence
-
3.2. Discrete dynamical systems
-
3.3. Sequences and limit points
-
3.4. Algebraic theorems for sequences
-
3.5. Subsequences
-
3.6. Completeness
-
3.7. The contraction mapping principle
-
3.8. Sets of sequences as metric spaces
-
3.9. Exercises
-
Chapter 4. Continuity
-
4.1. The definition of continuity
-
4.2. Equivalent formulations of continuity
-
4.3. Continuity and limit theorems for scalar- valued functions
-
4.4. Continuity and products of metric spaces
-
4.5. Uniform continuity
-
4.6. The metric space 𝐶([𝑎,𝑏],R )
-
4.7. An application to functional equations
-
4.8. Exercises
-
Chapter 5. Compactness and Connectedness
-
5.1. Basic definitions and results on compactness
-
5.2. The nested set property for compact sets
-
5.3. Compactness and continuity
-
5.4. Other facts about compactness
-
5.5. Connectedness
-
5.6. Periodic points of maps on intervals
-
5.7. Injective continuous functions defined on intervals
-
5.8. Exercises
-
Chapter 6. The Derivative
-
6.1. The definition of the derivative
-
6.2. Differentiation rules
-
6.3. Applications of the derivative
-
6.4. Exercises
-
Chapter 7. The Riemann Integral
-
7.1. Partitions and the definition of the integral
-
7.2. Basic properties of the integral
-
7.3. The fundamental theorem of calculus
-
7.4. Ordinary differential equations
-
7.5. Exercises
-
Chapter 8. Sequences of Functions
-
8.1. Infinite series
-
8.2. Power series
-
8.3. Higher derivatives and Taylor polynomials
-
8.4. Differentiation and integration of sequences of functions
-
8.5. The exponential function
-
8.6. Compact subsets in 𝐶[𝑎,𝑏]
-
8.7. Exercises
-
Chapter 9. Chaos in Discrete Dynamical Systems
-
9.1. The definition of chaos
-
9.2. Semiconjugacy
-
9.3. Subshifts of finite type
-
9.4. Itineraries and piecewise expanding maps
-
9.5. A dynamical system with a dense orbit but no periodic points
-
9.6. Exercises
-
Chapter 10. The Hausdorff Metric and Fractals
-
10.1. Definition of the Hausdorff metric
-
10.2. Properties of the Hausdorff metric
-
10.3. Fractals in the plane
-
10.4. Exercises
-
Bibliography
-
Index
-
Back Cover
-
In terms of writing: this book is clear, challenging, and rewarding in its exposition. The ordering of topics is well-vetted; examples and proof details are well-chosen; and the exposition is well-written and imminently readable. The text offers reading questions after each section, as well as a variety of exercises at the end of each chapter. I appreciated both forms of questions; the 'exercises' are varied and completely appropriate, while the 'reading questions' allow for students to check their comprehension (and model the kinds of questions we want students to ask themselves as they read any mathematical text). Additionally, the text benefits from 'analytical advice' text boxes that offer just-in-time discussions of proof styles and big ideas in proof-writing.
...This seems to be an excellent text which truly highlights how the core elements of analysis can be applied in a number of powerful ways beyond calculus. I would argue against using this text in a course that uses Real Analysis as an 'introduction' to proof-writing, as such students might find this text a bit of a struggle. However, a well-prepared undergraduate will benefit immensely from the exposition here, and I would recommend the text to such a student without reservation.
John Ross, Southwestern University
