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A Passage to Modern Analysis
 
William J. Terrell Virginia Commonwealth University, Richmond, VA
A Passage to Modern Analysis
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
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
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
A Passage to Modern Analysis
Click above image for expanded view
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
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
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
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 412019; 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:

    Readership

    Undergraduate and graduate students interested in real analysis.

  • Table of Contents
     
     
    • 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
    • Back Cover
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Volume: 412019; 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:

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
  • Back Cover
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