Book:Karl R. Stromberg/An Introduction to Classical Real Analysis

Subject Matter

 * Real Analysis

Contents

 * 0 PRELIMINARIES
 * Sets and Subsets
 * Operations on Sets
 * Ordered Pairs and Relations
 * Equivalence Relations
 * Functions
 * Products of Sets


 * 1 NUMBERS
 * Axioms for $\R$
 * The Supremum Principle
 * The Natural Numbers
 * Integers
 * Decimal Representation of Natural Numbers
 * Roots
 * Rational and Irrational Numbers
 * Complex Numbers
 * Some Inequalities
 * Extended Real Numbers
 * Finite and Infinite Sets
 * Newton's Binomial Theorem
 * Exercises


 * 2 SEQUENCES AND SERIES
 * Sequences in $\C$
 * Sequences in $\R^\#$
 * Cauchy Sequences
 * Subsequences
 * Series of Complex Terms
 * Series of Nonnegative Terms
 * Decimal Expansions
 * The Number $e$
 * The Root and Ratio Tests
 * Power Series
 * Multiplication of Series
 * Lebesgue Outer Measure
 * Cantor Sets
 * Exercises


 * 3 LIMITS AND CONTINUITY
 * Metric Spaces
 * Topological Spaces
 * Compactness
 * Connectedness
 * Completeness
 * Baire Category
 * Exercises
 * Limits of Functions at a Point
 * Exercises
 * Compactness, Connectedness, and Continuity
 * Exercises
 * Simple Discontinuities and Monotone Functions
 * Exercises
 * Exp and Log
 * Powers
 * Exercises
 * Uniform Convergence
 * Exercises
 * Stone-Weierstrass Theorems
 * Exercises
 * Total Variation
 * Absolute Continuity
 * Exercises
 * Equicontinuity
 * Exercises


 * 4 DIFFERENTIATION
 * Dini Derivates
 * ** A Nowhere Differentiable, Everywhere Continuous, Function
 * Some Elementary Formulas
 * Local Extrema
 * Mean Value Theorems
 * L'Hospital's Rule
 * Exercises
 * Higher Order Derivatives
 * Taylor Polynomials
 * Exercises
 * * Convex Functions
 * * Exercises
 * Differentiability Almost Everywhere
 * Exercises
 * * Termwise Differentiation of Sequences
 * * Exercises
 * * Complex Derivatives
 * * Exercises


 * 5 THE ELEMENTARY TRANSCENDENTAL FUNCTIONS
 * The Exponential Function
 * The Trigonometric Functions
 * The Argument
 * Exercises
 * * Complex Logarithms and Powers
 * * Exercises
 * ** $\pi$ is Irrational
 * ** Exercises
 * * Log Series and the Inverse Tangent
 * ** Rational Approximation to $\pi$
 * * Exercises
 * ** The Sine Product and Related pansions
 * ** Stirling's Formula
 * ** Exercises


 * 6 INTEGRATION
 * Step Functions
 * The First Extension
 * Integrable Functions
 * Two Limit Theorems
 * The Riemann Integral
 * Exercises
 * Measureable Functions
 * Complex-Valued Functions
 * Measurable Sets
 * Structure of Measurable Functions
 * Integration Over Measurable Sets
 * Exercises
 * The Fundamental Theorem of Calculus
 * Integration by Parts
 * Integration Substitution
 * Two Mean Value Theorems
 * * Arc Length
 * Exercises
 * Hölder's and Minkowski's Inequalities
 * The $L_p$ Spaces
 * Exercises
 * Integration on $\R^n$
 * Iteration of Integrals
 * Exercises
 * Some Differential Calculus in Higher Dimensions
 * Exercises
 * Transformations of Integrals on $\R^n$
 * Exercises


 * 7 INFINITE SERIES AND INFINITE PRODUCTS
 * Series Having Monotone Terms
 * Limit Comparison Tests
 * ** Two Log Tests
 * ** Other Ratio Tests
 * * Exercises
 * ** Infinite Products
 * ** Exercises
 * Some Theorems of Abel
 * Exercises
 * ** Another Ratio Test and the Binomial Series
 * ** Exercises
 * Rearrangements and Double Series
 * Exercises
 * ** The Gamma Function
 * ** Exercises
 * Divergent Series
 * Exercises
 * Tauberian Theorems
 * Exercises


 * 8 TRIGONOMETRIC SERIES
 * Trigonometric Series and Fourier Series
 * Which Trigonometric Series are Fourier Series?
 * Exercises
 * * Divergent Fourier Series
 * * Exercises
 * Summability of Fourier Series
 * Riemann Localization and Convergence Criteria
 * Growth Rate of Partial Sums
 * Exercises


 * BIBLIOGRAPHY
 * INDEX