# Introduction to Classical Real Analysis

###### Karl R. Stromberg
Publisher:
Publication Date:
1981
Number of Pages:
576
Format:
Hardcover
Series:
Price:
0.00
ISBN:
out-of-print
Category:
General
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
09/2/2015
]

The great strength of this book is its exercises. They are lengthy, difficult, and many (described as “projects”) are broken into manageable pieces. The author says (p. iv) about the exercises that “I spent at least three items as much effort in preparing them as I did on the main text itself,” and it shows. The level of the exercises is somewhere around that of the first sections of Pólya and Szego’s Problems and Theorems in Analysis, although that book is slanted heavily towards complex function theory and the present book sticks to real functions.

The narrative part of the book is well-done too, and contains many interesting things, but it is not such a standout as the exercises. The term “classical” in the title indicates that the book is slanted towards the concrete and has quite a lot on properties of particular series and integrals. In olden days it might have been titled Advanced Calculus, although it doesn’t go very far into multi-variable calculus. In modern terms it is a text for a first rigorous course in mathematical analysis. This is a very competitive field, and as the present book was written in 1981, you would expect some better texts to have come out since then. Considering only the narrative part and not the exercises, I think Ross’s Elementary Analysis would be a better choice for most courses. The present book is more advanced in some aspects, and in particular it develops the Lebesgue integral rather than the Riemann integral (through step functions rather than measure). Despite the slant towards concreteness, it does prove results in more generality when it’s not much harder to do so. For example, it proves the Stone-Weierstrass theorem in its full generality rather than the Weierstrass approximation theorem.

If your interest is primarily in the classical analysis aspects rather than rigorous analysis, there are some recent samplers that are valuable and interesting: Duren’s Invitation to Classical Analysis and Chen’s Excursions in Classical Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

• 0 PRELIMINARIES
• Sets and Subsets
• Operations on Sets
• Ordered Pairs and Relations
• Equivalence Relations
• Functions
• Products of Sets
• 1 NUMBERS
• Axioms for $\mathbb{R}$
• 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 $\mathbb{C}$
• Sequences in $\mathbb{R}^\#$
• 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 $\mathbb{R}^n$
• Iteration of Integrals
• Exercises
• Some Differential Calculus in Higher Dimensions
• Exercises
• Transformations of Integrals on $\mathbb{R}^n$
• 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
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