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

## Karl R. Stromberg: An Introduction to Classical Real Analysis

Published $\text {1981}$, Kluwer Academic Publishers

ISBN 978 0534980122

### 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