A Passage to Modern Analysis (Pure and Applied Undergraduate Texts)

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. 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