Table of Contents

Foundations of Mathematical Analysis

Richard Johnsonbaugh

W. E. Pfaffenberger

Preface

Preface to the Dover Edition
  1. Sets and Functions
    2.
    1. Sets
    2. Functions
  2. The Real Number System
    3.
    1. The Algebraic Axioms of the Real Numbers
    2. The Order Axiom of the Real Numbers
    3. The Least-Upper-Bound Axiom
    4. The Set of Positive Integers
    5. Integers, Rationals, and Exponents
  3. Set Equivalence
    4.
    1. Definitions and Examples
    2. Countable and Uncountable Sets
  4. Sequences of Real Numbers
    5.
    1. Limit of a Sequence
    2. Subsequences
    3. The Algebra of Limits
    4. Bounded Sequences
    5. Further Limit Theorems
    6. Divergent Sequences
    7. Monotone Sequences and the Number e
    8. Real Exponents
    9. The Bolzano-Weierstrass Theorem
    10. The Cauchy Condition
    11. The lim sup and lim inf of Bounded Sequences
    12. The lim sup and lim inf of Unbounded Sequences
  5. Infinite Series
    6.
    1. The Sum of an Infinite Series
    2. Algebraic Operations on Series
    3. Series with Nonnegative Terms
    4. The Alternating Series Test
    5. Absolute Convergence
    6. Power Series
    7. Conditional Convergence
    8. Double Series and Applications
  6. Limits of Real-Valued Functions and Continuous Functions on the Real Line
    7.
    1. Definition of the Limit of a Function
    2. Limit Theorems for Functions
    3. One-Sided and Infinite Limits
    4. Continuity
    5. The Heine-Borel Theorem and a Consequence for Continuous Functions
  7. Metric Spaces
    8.
    1. The Distance Function
    2. Rn, l2, and the Cauchy-Schwarz Inequality
    3. Sequences in Metric Spaces
    4. Closed Sets
    5. Open Sets
    6. Continuous Functions on Metric Spaces
    7. The Relative Metric
    8. Compact Metric Spaces
    9. The Bolzano-Weierstrass Characterization of a Compact Metric Space
    10. Continuous Functions on Compact Metric Spaces
    11. Connected Metric Spaces
    12. Complete Metric Spaces
    13. Baire Category Theorem
  8. Differential Calculus of the Real Line
    9.
    1. Basic Definitions and Theorems
    2. Mean-Value Theorems and L'Hospital's Rule
    3. Taylor's Theorem
  9. The Riemann-Stieltjes Integral
    10.
    1. Riemann-Stieltjes Integration with Respect to an Increasing Integrator
    2. Riemann-Stieltjes Sums
    3. Riemann-Stieltjes Integration with Respect to an Arbitrary Integrator
    4. Functions of Bounded Variation
    5. Riemann-Stieltjes Integration with Respect to Functions of Bounded Variation
    6. The Riemann Integral
    7. Measure Zero
    8. A Necessary and Sufficient Condition for the Existence of the Riemann Integral
    9. Improper Riemann-Stieltjes Integrals
  10. Sequences and Series of Functions
    11.
    1. Pointwise Convergence and Uniform Convergence
    2. Integration and Differentiation of Uniformly Convergent Sequences
    3. Series of Functions
    4. Applications to Power Series
    5. Abel's Limit Theorems
    6. Summability Methods and Tauberian Theorems
  11. Transcendental Functions
    12.
    1. The Exponential Function
    2. The Natural Logarithm Function
    3. The Trigonometric Functions
  12. Inner Product Spaces and Fourier Series
    13.
    1. Normed Linear Spaces
    2. The Inner Product Space R3
    3. Inner Product Spaces
    4. Orthogonal Sets in Inner Product Spaces
    5. Periodic Functions
    6. Fourier Series: Definition and Examples
    7. Orthonormal Expansions in Inner Product Spaces
    8. Pointwise Convergence of Fourier Series in R[a,a + 2π]
    9. Cesàro Summability of Fourier Series
    10. Fourier Series in R[a,a + 2π]
    11. A Tauberian Theorem and an Application to Fourier Series
  13. Normed Linear Spaces and the Riesz Representation Theorem
    14.
    1. Normed Linear Spaces and Continuous Linear Transformations
    2. The Normed Linear Space of Continuous Linear Transformations
    3. The Dual Space of a Normed Linear Space
    4. Introduction to the Riesz Representation Theorem
    5. Proof of the Riesz Representation Theorem
  14. The Lebesgue Integral
    15.
    1. The Extended Real Line
    2. σ-Algebras and Positive Measures
    3. Measurable Functions
    4. Integration on Positive Measure Spaces
    5. Lebesgue Measure on R
    6. Lebesgue Measure on [a,b]
    7. The Hilbert Spaces L2(X,M,μ)

Appendix: Vector Spaces

References

Hints to Selected Exercises

Index

Errata