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

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

Published $1981$, **Kluwer Academic Publishers**

- ISBN 978 0534980122.

### Subject Matter

### 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