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Product Code:  AMSTEXT/25 
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Electronic ISBN:  9781470434946 
Product Code:  AMSTEXT/25.E 
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MAA Member Price:  $80.10 
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Book DetailsPure and Applied Undergraduate TextsVolume: 25; 2016; 348 ppMSC: Primary 26; 54;
Mathematical analysis is often referred to as generalized calculus. But it is much more than that. This book has been written in the belief that emphasizing the inherent nature of a mathematical discipline helps students to understand it better. With this in mind, and focusing on the essence of analysis, the text is divided into two parts based on the way they are related to calculus: completion and abstraction. The first part describes those aspects of analysis which complete a corresponding area of calculus theoretically, while the second part concentrates on the way analysis generalizes some aspects of calculus to a more general framework. Presenting the contents in this way has an important advantage: students first learn the most important aspects of analysis on the classical space \(\mathbb{R}\) and fill in the gaps of their calculusbased knowledge. Then they proceed to a stepbystep development of an abstract theory, namely, the theory of metric spaces which studies such crucial notions as limit, continuity, and convergence in a wider context.
The readers are assumed to have passed courses in one and severalvariable calculus and an elementary course on the foundations of mathematics. A large variety of exercises and the inclusion of informal interpretations of many results and examples will greatly facilitate the reader's study of the subject.ReadershipUndergraduate students interested in mathematical analysis.

Table of Contents

Cover

Title page

Contents

To the Instructor

To the Student

Introduction and Outline of the Book

Acknowledgments

Part 1 . Rebuilding the Calculus Building

Chapter 1. The Real Number System Revisited

1.1. The Algebraic Axioms

1.2. The Order Axioms

1.3. Absolute Value, Distance, and Neighborhoods

1.4. Natural Numbers and Mathematical Induction

1.5. The Axiom of Completeness and Its Uses

1.6. The Complex Number System

Notes on Essence and Generalizability

Exercises

Chapter 2. Sequences and Series of Real Numbers

2.1. Real Sequences, Their Convergence, and Boundedness

2.2. Subsequences, Limit Superior and Limit Inferior

2.3. Cauchy Sequences

2.4. Sequences in Closed and Bounded Intervals

2.5. Series: Revisiting Some Convergence Tests

2.6. Rearrangements of Series

2.7. Power Series

Notes on Essence and Generalizability

Exercises

Chapter 3. Limit and Continuity of Real Functions

3.1. Limit Points and Some Other Classes of Points in ℝ

3.2. A More General Definition of Limit

3.3. Limit at Infinity

3.4. OneSided Limits\index{onesided!limit}

3.5. Continuity and Two Kinds of Discontinuity

3.6. Continuity on [𝑎,𝑏]: Results and Applications

3.7. Uniform Continuity

Notes on Essence and Generalizability

Exercises

Chapter 4. Derivative and Differentiation

4.1. The Why and What of the Concept of Derivative

4.2. The Basic Properties of Derivative

4.3. Local Extrema and Derivative

4.4. The Mean Value Theorem: More Applications of Derivative

4.5. Taylor Series: A First Glance

4.6. Taylor’s Theorem and the Convergence of Taylor Series

Notes on Essence and Generalizability

Exercises

Chapter 5. The Riemann Integral

5.1. Motivation: The Area Problem

5.2. The Riemann Integral: Definition and Basic Results

5.3. Some Integrability Theorems

5.4. Antiderivatives and the Fundamental Theorem of Calculus

Notes on Essence and Generalizability

Exercises

Part 2 . Abstraction and Generalization

Chapter 6. Basic Theory of Metric Spaces

6.1. A First Generalization: The Definition of Metric Space

6.2. Neighborhoods and Some Classes of Points

6.3. Open and Closed Sets

6.4. Metric Subspaces

6.5. Boundedness and Total Boundedness

Notes on Essence and Generalizability

Exercises

Chapter 7. Sequences in General Metric Spaces

7.1. Convergence and Divergence in Metric Spaces

7.2. Cauchy Sequences and Complete Metric Spaces

7.3. Compactness: Definition and Some Basic Results

7.4. Compactness: Some Equivalent Forms

7.5. Perfect Sets and Cantor’s Set

Notes on Essence and Generalizability

Exercises

Chapter 8. Limit and Continuity of Functions in Metric Spaces

8.1. The Definition of Limit in General Metric Spaces

8.2. Continuity and Uniform Continuity

8.3. Continuity and Compactness

8.4. Connectedness and Its Relation to Continuity

8.5. Banach’s Fixed Point Theorem

Notes on Essence and Generalizability

Exercises

Chapter 9. Sequences and Series of Functions

9.1. Sequences of Functions and Their Pointwise Convergence

9.2. Uniform Convergence

9.3. Weierstrass’s Approximation Theorem

9.4. Series of Functions and Their Convergence

Notes on Essence and Generalizability

Exercises

Appendix

Real Sequences and Series

Limit and Continuity of Functions

The Concepts of Derivative and Differentiability

The Riemann Integral

Bibliography

Index

Back Cover


Additional Material

Reviews

[T]his is an attractive text, one that certainly merits a look by anybody trying to find a text for a course in undergraduate analysis.
Mark Hunacek, MAA Reviews


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Mathematical analysis is often referred to as generalized calculus. But it is much more than that. This book has been written in the belief that emphasizing the inherent nature of a mathematical discipline helps students to understand it better. With this in mind, and focusing on the essence of analysis, the text is divided into two parts based on the way they are related to calculus: completion and abstraction. The first part describes those aspects of analysis which complete a corresponding area of calculus theoretically, while the second part concentrates on the way analysis generalizes some aspects of calculus to a more general framework. Presenting the contents in this way has an important advantage: students first learn the most important aspects of analysis on the classical space \(\mathbb{R}\) and fill in the gaps of their calculusbased knowledge. Then they proceed to a stepbystep development of an abstract theory, namely, the theory of metric spaces which studies such crucial notions as limit, continuity, and convergence in a wider context.
The readers are assumed to have passed courses in one and severalvariable calculus and an elementary course on the foundations of mathematics. A large variety of exercises and the inclusion of informal interpretations of many results and examples will greatly facilitate the reader's study of the subject.
Undergraduate students interested in mathematical analysis.

Cover

Title page

Contents

To the Instructor

To the Student

Introduction and Outline of the Book

Acknowledgments

Part 1 . Rebuilding the Calculus Building

Chapter 1. The Real Number System Revisited

1.1. The Algebraic Axioms

1.2. The Order Axioms

1.3. Absolute Value, Distance, and Neighborhoods

1.4. Natural Numbers and Mathematical Induction

1.5. The Axiom of Completeness and Its Uses

1.6. The Complex Number System

Notes on Essence and Generalizability

Exercises

Chapter 2. Sequences and Series of Real Numbers

2.1. Real Sequences, Their Convergence, and Boundedness

2.2. Subsequences, Limit Superior and Limit Inferior

2.3. Cauchy Sequences

2.4. Sequences in Closed and Bounded Intervals

2.5. Series: Revisiting Some Convergence Tests

2.6. Rearrangements of Series

2.7. Power Series

Notes on Essence and Generalizability

Exercises

Chapter 3. Limit and Continuity of Real Functions

3.1. Limit Points and Some Other Classes of Points in ℝ

3.2. A More General Definition of Limit

3.3. Limit at Infinity

3.4. OneSided Limits\index{onesided!limit}

3.5. Continuity and Two Kinds of Discontinuity

3.6. Continuity on [𝑎,𝑏]: Results and Applications

3.7. Uniform Continuity

Notes on Essence and Generalizability

Exercises

Chapter 4. Derivative and Differentiation

4.1. The Why and What of the Concept of Derivative

4.2. The Basic Properties of Derivative

4.3. Local Extrema and Derivative

4.4. The Mean Value Theorem: More Applications of Derivative

4.5. Taylor Series: A First Glance

4.6. Taylor’s Theorem and the Convergence of Taylor Series

Notes on Essence and Generalizability

Exercises

Chapter 5. The Riemann Integral

5.1. Motivation: The Area Problem

5.2. The Riemann Integral: Definition and Basic Results

5.3. Some Integrability Theorems

5.4. Antiderivatives and the Fundamental Theorem of Calculus

Notes on Essence and Generalizability

Exercises

Part 2 . Abstraction and Generalization

Chapter 6. Basic Theory of Metric Spaces

6.1. A First Generalization: The Definition of Metric Space

6.2. Neighborhoods and Some Classes of Points

6.3. Open and Closed Sets

6.4. Metric Subspaces

6.5. Boundedness and Total Boundedness

Notes on Essence and Generalizability

Exercises

Chapter 7. Sequences in General Metric Spaces

7.1. Convergence and Divergence in Metric Spaces

7.2. Cauchy Sequences and Complete Metric Spaces

7.3. Compactness: Definition and Some Basic Results

7.4. Compactness: Some Equivalent Forms

7.5. Perfect Sets and Cantor’s Set

Notes on Essence and Generalizability

Exercises

Chapter 8. Limit and Continuity of Functions in Metric Spaces

8.1. The Definition of Limit in General Metric Spaces

8.2. Continuity and Uniform Continuity

8.3. Continuity and Compactness

8.4. Connectedness and Its Relation to Continuity

8.5. Banach’s Fixed Point Theorem

Notes on Essence and Generalizability

Exercises

Chapter 9. Sequences and Series of Functions

9.1. Sequences of Functions and Their Pointwise Convergence

9.2. Uniform Convergence

9.3. Weierstrass’s Approximation Theorem

9.4. Series of Functions and Their Convergence

Notes on Essence and Generalizability

Exercises

Appendix

Real Sequences and Series

Limit and Continuity of Functions

The Concepts of Derivative and Differentiability

The Riemann Integral

Bibliography

Index

Back Cover

[T]his is an attractive text, one that certainly merits a look by anybody trying to find a text for a course in undergraduate analysis.
Mark Hunacek, MAA Reviews