A Passage to Modern Analysis

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A Passage to Modern Analysis

William J. Terrell : Virginia Commonwealth University, Richmond, VA

Hardcover ISBN:	978-1-4704-5135-6
Product Code:	AMSTEXT/41

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eBook ISBN:	978-1-4704-5520-0
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Hardcover ISBN:	978-1-4704-5135-6
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A Passage to Modern Analysis

William J. Terrell : Virginia Commonwealth University, Richmond, VA

Hardcover ISBN:	978-1-4704-5135-6
Product Code:	AMSTEXT/41

List Price:	$99.00
MAA Member Price:	$89.10
AMS Member Price:	$79.20
Sale Price:	$64.35

eBook ISBN:	978-1-4704-5520-0
Product Code:	AMSTEXT/41.E

List Price:	$85.00
MAA Member Price:	$76.50
AMS Member Price:	$68.00
Sale Price:	$55.25

Hardcover ISBN:	978-1-4704-5135-6
eBook ISBN:	978-1-4704-5520-0
Product Code:	AMSTEXT/41.B

List Price:	$184.00 $141.50
MAA Member Price:	$165.60 $127.35
AMS Member Price:	$147.20 $113.20
Sale Price:	$119.60 $91.98

Add to cart

Book Details
Table of Contents
Additional Material
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Pure and Applied Undergraduate Texts

Volume: 41; 2019; 607 pp

MSC: Primary 26; 54; 28

A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level.

A Passage to Modern Analysis is grounded solidly in the analysis of $\mathbf{R}$ and $\mathbf{R}^{n}$, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments.

Ancillaries:

Instructor's Manual

Readership

Undergraduate and graduate students interested in real analysis.

Cover
Title page
List of Figures
Preface
Chapter 1. Sets and Functions
1.1. Set Notation and Operations
Exercises
1.2. Functions
Exercises
1.3. The Natural Numbers and Induction
Exercises
1.4. Equivalence of Sets and Cardinality
Exercises
1.5. Notes and References
Chapter 2. The Complete Ordered Field of Real Numbers
2.1. Algebra in Ordered Fields
2.1.1. The Field Axioms
2.1.2. The Order Axiom and Ordered Fields
Exercises
2.2. The Complete Ordered Field of Real Numbers
Exercises
2.3. The Archimedean Property and Consequences
Exercises
2.4. Sequences
Exercises
2.5. Nested Intervals and Decimal Representations
Exercises
2.6. The Bolzano-Weierstrass Theorem
Exercises
2.7. Convergence of Cauchy Sequences
Exercises
2.8. Summary: A Complete Ordered Field
2.8.1. Properties that Characterize Completeness
2.8.2. Why Calculus Does Not Work in 𝑄
2.8.3. The Existence of a Complete Ordered Field
2.8.4. The Uniqueness of a Complete Ordered Field
Exercise
Chapter 3. Basic Theory of Series
3.1. Some Special Sequences
Exercises
3.2. Introduction to Series
Exercises
3.3. The Geometric Series
Exercises
3.4. The Cantor Set
Exercises
3.5. A Series for the Euler Number
3.6. Alternating Series
Exercises
3.7. Absolute Convergence and Conditional Convergence
Exercise
3.8. Convergence Tests for Series with Positive Terms
Exercises
3.9. Geometric Comparisons: The Ratio and Root Tests
Exercises
3.10. Limit Superior and Limit Inferior
Exercises
3.11. Additional Convergence Tests
3.11.1. Absolute Convergence: The Root and Ratio Tests
3.11.2. Conditional Convergence: Abel’s and Dirichlet’s Tests
Exercises
3.12. Rearrangements and Riemann’s Theorem
Exercise
3.13. Notes and References
Chapter 4. Basic Topology, Limits, and Continuity
4.1. Open Sets and Closed Sets
Exercises
4.2. Compact Sets
Exercises
4.3. Connected Sets
Exercise
4.4. Limit of a Function
Exercises
4.5. Continuity at a Point
Exercises
4.6. Continuous Functions on an Interval
Exercises
4.7. Uniform Continuity
Exercises
4.8. Continuous Image of a Compact Set
Exercises
4.9. Classification of Discontinuities
Exercises
Chapter 5. The Derivative
5.1. The Derivative: Definition and Properties
Exercises
5.2. The Mean Value Theorem
Exercises
5.3. The One-Dimensional Inverse Function Theorem
Exercises
5.4. Darboux’s Theorem
Exercise
5.5. Approximations by Contraction Mapping
Exercises
5.6. Cauchy’s Mean Value Theorem
5.6.1. Limits of Indeterminate Forms
Exercises
5.7. Taylor’s Theorem with Lagrange Remainder
Exercises
5.8. Extreme Points and Extreme Values
Exercises
5.9. Notes and References
Chapter 6. The Riemann Integral
6.1. Partitions and Riemann-Darboux Sums
Exercises
6.2. The Integral of a Bounded Function
Exercises
6.3. Continuous and Monotone Functions
Exercises
6.4. Lebesgue Measure Zero and Integrability
Exercises
6.5. Properties of the Integral
Exercises
6.6. Integral Mean Value Theorems
Exercises
6.7. The Fundamental Theorem of Calculus
Exercises
6.8. Taylor’s Theorem with Integral Remainder
Exercises
6.9. Improper Integrals
6.9.1. Functions on [𝑎,∞) or (-∞,𝑏]
6.9.2. Functions on (𝑎,𝑏] or [𝑎,𝑏)
6.9.3. Functions on (𝑎,∞), (-∞,𝑏) or (-∞,∞)
6.9.4. Cauchy Principal Value
Exercises
6.10. Notes and References
Chapter 7. Sequences and Series of Functions
7.1. Sequences of Functions: Pointwise and Uniform Convergence
7.1.1. Pointwise Convergence
7.1.2. Uniform Convergence
Exercises
7.2. Series of Functions
7.2.1. Integration and Differentiation of Series
7.2.2. Weierstrass’s Test: Uniform Convergence of Series
Exercises
7.3. A Continuous Nowhere Differentiable Function
Exercises
7.4. Power Series; Taylor Series
Exercises
7.5. Exponentials, Logarithms, Sine and Cosine
7.5.1. Exponentials and Logarithms
7.5.2. Power Functions
7.5.3. Sine and Cosine Functions
7.5.4. Some Inverse Trigonometric Functions
7.5.5. The Elementary Transcendental Functions
Exercises
7.6. The Weierstrass Approximation Theorem
Exercise
7.7. Notes and References
Chapter 8. The Metric Space 𝑅ⁿ
8.1. The Vector Space 𝑅ⁿ
Exercises
8.2. The Euclidean Inner Product
Exercises
8.3. Norms
Exercises
8.4. Fourier Expansion in 𝑅ⁿ
Exercises
8.5. Real Symmetric Matrices
8.5.1. Definitions and Preliminary Results
8.5.2. The Spectral Theorem for Real Symmetric Matrices
Exercises
8.6. The Euclidean Metric Space 𝑅ⁿ
Exercise
8.7. Sequences and the Completeness of 𝑅ⁿ
Exercises
8.8. Topological Concepts for 𝑅ⁿ
8.8.1. Topology of 𝑅ⁿ
8.8.2. Relative Topology of a Subset
Exercises
8.9. Nested Intervals and the Bolzano-Weierstrass Theorem
Exercises
8.10. Mappings of Euclidean Spaces
8.10.1. Limits of Functions and Continuity
Exercises
8.10.2. Continuity on a Domain
8.10.3. Open Mappings
Exercises
8.10.4. Continuous Images of Compact Sets
Exercises
8.10.5. Differentiation under the Integral
Exercises
8.10.6. Continuous Images of Connected Sets
Exercises
8.11. Notes and References
Chapter 9. Metric Spaces and Completeness
9.1. Basic Topology in Metric Spaces
Exercises
9.2. The Contraction Mapping Theorem
Exercises
9.3. The Completeness of 𝐶[𝑎,𝑏] and 𝑙²
Exercises
9.4. The 𝑙^{𝑝} Sequence Spaces
Exercises
9.5. Matrix Norms and Completeness
9.5.1. Matrix Norms
9.5.2. Completeness of 𝑅^{𝑛×𝑛}
Exercises
9.6. Notes and References
Chapter 10. Differentiation in 𝑅ⁿ
10.1. Partial Derivatives
Exercises
10.2. Differentiability: Real Functions and Vector Functions
Exercises
10.3. Matrix Representation of the Derivative
Exercise
10.4. Existence of the Derivative
Exercises
10.5. The Chain Rule
Exercises
10.6. The Mean Value Theorem: Real Functions
Exercises
10.7. The Two-Dimensional Implicit Function Theorem
Exercises
10.8. The Mean Value Theorem: Vector Functions
Exercises
10.9. Taylor’s Theorem
Exercises
10.10. Relative Extrema without Constraints
Exercises
10.11. Notes and References
Chapter 11. The Inverse and Implicit Function Theorems
11.1. Matrix Geometric Series and Inversion
Exercises
11.2. The Inverse Function Theorem
Exercises
11.3. The Implicit Function Theorem
Exercises
11.4. Constrained Extrema and Lagrange Multipliers
Exercises
11.5. The Morse Lemma
Exercises
11.6. Notes and References
Chapter 12. The Riemann Integral in Euclidean Space
12.1. Bounded Functions on Closed Intervals
Exercises
12.2. Bounded Functions on Bounded Sets
Exercise
12.3. Jordan Measurable Sets; Sets with Volume
Exercises
12.4. Lebesgue Measure Zero
Exercises
12.5. A Criterion for Riemann Integrability
Exercise
12.6. Properties of Volume and Integrals
Exercises
12.7. Multiple Integrals
Exercises
Chapter 13. Transformation of Integrals
13.1. A Space-Filling Curve
13.2. Volume and Integrability under 𝐶¹ Maps
Exercises
13.3. Linear Images of Sets with Volume
Exercises
13.4. The Change of Variables Formula
Exercises
13.5. The Definition of Surface Integrals
Exercises
13.6. Notes and References
Chapter 14. Ordinary Differential Equations
14.1. Scalar Differential Equations
Exercises
14.2. Systems of Ordinary Differential Equations
14.2.1. Definition of Solution and the Integral Equation
Exercise
14.2.2. Completeness of 𝐶_{𝑛}[𝑎,𝑏]
Exercises
14.2.3. The Local Lipchitz Condition
Exercises
14.2.4. Existence and Uniqueness of Solutions
Exercises
14.3. Extension of Solutions
14.3.1. The Maximal Interval of Definition
Exercise
14.3.2. An Example of a Newtonian System
Exercise
14.4. Continuous Dependence
14.4.1. Continuous Dependence on Initial Conditions, Parameters, and Vector Fields
Exercises
14.4.2. Newtonian Equations and Examples of Stability
Exercises
14.5. Matrix Exponentials and Linear Autonomous Systems
Exercises
14.6. Notes and References
Chapter 15. The Dirichlet Problem and Fourier Series
15.1. Introduction to Laplace’s Equation
15.2. Orthogonality of the Trigonometric Set
Exercises
15.3. The Dirichlet Problem for the Disk
Exercises
15.4. More Separation of Variables
15.4.1. The Heat Equation: Two Basic Problems
Exercises
15.4.2. The Wave Equation with Fixed Ends
Exercise
15.5. The Best Mean Square Approximation
Exercises
15.6. Convergence of Fourier Series
Exercises
15.7. Fejér’s Theorem
Exercises
15.8. Notes and References
Chapter 16. Measure Theory and Lebesgue Measure
16.1. Algebras and 𝜎-Algebras
Exercise
16.2. Arithmetic in the Extended Real Numbers
16.3. Measures
Exercises
16.4. Measure from Outer Measure
Exercises
16.5. Lebesgue Measure in Euclidean Space
16.5.1. Lebesgue Measure on the Real Line
Exercises
16.5.2. Metric Outer Measure; Lebesgue Measure on Euclidean Space
Exercises
16.6. Notes and References
Chapter 17. The Lebesgue Integral
17.1. Measurable Functions
Exercises
17.2. Simple Functions and the Integral
Exercises
17.3. Definition of the Lebesgue Integral
Exercises
17.4. The Limit Theorems
Exercises
17.5. Comparison with the Riemann Integral
Exercises
17.6. Banach Spaces of Integrable Functions
Exercises
17.7. Notes and References
Chapter 18. Inner Product Spaces and Fourier Series
18.1. Examples of Orthonormal Sets
Exercises
18.2. Orthonormal Expansions
18.2.1. Basic Results for Inner Product Spaces
18.2.2. Complete Spaces and Complete Orthonormal Sets
Exercises
18.3. Mean Square Convergence
18.3.1. Comparison of Pointwise, Uniform, and 𝐿² Norm Convergence
Exercises
18.3.2. Mean Square Convergence for 𝐶𝑃[-𝜋,𝜋]
18.3.3. Mean Square Convergence for ℛ[-𝜋,𝜋]
18.4. Hilbert Spaces of Integrable Functions
Exercises
18.5. Notes and References
Appendix A. The Schroeder-Bernstein Theorem
A.1. Proof of the Schroeder-Bernstein Theorem
Exercise
Appendix B. Symbols and Notations
B.1. Symbols and Notations Reference List
B.2. The Greek Alphabet
Bibliography
Index
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