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

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