You are here

A Concrete Approach to Classical Analysis

Number of Pages: 

This book has a lot of weaknesses, and it’s hard to recommend it for anything. The most serious problem is that it is a hodge-podge and lacks focus. A large part of the book is a traditional course in real (not classical) analysis, similar to but less comprehensive than Apostol’s Mathematical Analysis (Addison-Wesley, 2nd ed. 1974) or Rudin’s Principles of Mathematical Analysis. Another large part of the book is a collection of specific results about particular series, functions, and constants; this is probably what the “classical analysis” in the title refers to.

The book starts out with two chapters on topology and abstract spaces (including Hilbert spaces but not Banach spaces), but most of this material is not used later. Metric spaces are used in many places, but not consistently; the exposition jumps back and forth between metric spaces and the real line without any explanation. The title term “Concrete” is explained in the Preface but I did not understand the explanation; it might mean “applied”, but there is not any applied math in the book.

Some stated results are simply wrong. The book claims on p. 17 that completeness of the real numbers is equivalent to the Archimedes principle. It claims on p. 209 that if the kth derivative of a Ck function has at least k+1 zeroes in an interval, then the function has a zero in the interval.

The exposition needs work too, although most of this could be fixed by better copyediting. The English is often unidiomatic (many theorems start with “It holds” or “It holds that”). Often terms are used many pages before they are defined. A number of theorems are stated without proof, while others are stated with proof but the proof is delayed for several or many pages, and there’s usually no indication of whether a proof will follow or not.

Classical analysis (which I define as “the kind that has numbers in it”) seems to be out of fashion today. My favorite classical analysis text, which has been out of print for many years, is Karl R. Stromberg’s An Introduction to Classical Real Analysis (Wadsworth, 1981). You can also find good coverage of this topic in many older books with “Theory of Functions” in the title, such as Titchmarsh’s Theory of Functions.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Date Received: 
Monday, November 17, 2008
Include In BLL Rating: 
Marian Mureşan
CMS Books in Mathematics
Publication Date: 
Allen Stenger

1 Sets and Numbers
1.1 Sets
1.1.1 The concept of a set
1.1.2 Operations on sets
1.1.3 Relations and functions
1.2 Sets of numbers
1.2.1 Two examples
1.2.2 The real number system
1.2.3 Elements of algebra
1.2.4 Elements of topology on R 
1.2.5 The extended real number system
1.2.6 The complex number system
1.3 Exercises
1.4 References and comments

2 Vector Spaces and Metric Spaces
2.1 Vector spaces
2.1.1 Finite-dimensional vector spaces
2.1.2 Vector spaces
2.1.3 Normed spaces
2.1.4 Hilbert spaces
2.1.5 Inequalities
2.2 Metric spaces
2.3 Compact spaces
2.4 Exercises
2.5 References and comments

3 Sequences and Series
3.1 Numerical sequences
3.1.1 Convergent sequences
3.1.2 Subsequences
3.1.3 Cauchy sequences
3.1.4 Monotonic sequences
3.1.5 Upper limits and lower limits
3.1.6 The big Oh and small oh notations
3.1.7 Stolz-Cesaro theorem and some of its consequences
3.1.8 Certain combinatorial numbers
3.1.9 Unimodal, log-convex, and Pó1ya-frequency sequences
3.1.10 Some special sequences
3.2 Sequences of functions
3.3 Numerical series
3.3.1 Series of nonnegative terms
3.3.2 The root and the ratio tests
3.3.3 Partial summation
3.3.4 Absolutely and conditionally convergent series
3.3.5 The W – Z method
3.4 Series of functions
3.4.1 Power series
3.4.2 Hypergeometric series
3.5 The Riemann Zeta function ζ(p)
3.6 Exercises
3.7 References and comments

4 Limits and Continuity
4.1 Limits
4.1.1 The limit of a function
4.1.2 Right-hand side and left-hand side limits
4.2 Continuity
4.2.1 Continuity and compactness
4.2.2 Uniform continuous mappings
4.2.3 Continuity and connectedness
4.2.4 Discontinuities
4.2.5 Monotonic functions
4.3 Periodic functions
4.4 Darboux functions
4.5 Lipschitz functions
4.6 Convex functions
4.6.1 Convex functions
4.6.2 Jensen convex functions
4.7 Functions of bounded variations
4.8 Continuity of sequences of functions
4.9 Continuity of series of functions
4.10 Exercises
4.11 References and comments

5 Differential Calculus on R
5.1 The derivative of a real function
5.2 Mean value theorems
5.2.1 Consequences of the mean value theorems
5.3 The continuity and the surjectivity of derivatives
5.4 L’Hospital theorem
5.5 Higher-order derivatives and the Taylor formula
5.6 Convex functions and differentiability
5.6.1 Inequalities
5.7 Differentiability of sequences and series of functions
5.8 Power series and Taylor series
5.8.1 Operations with power series
5.8.2 The Taylor expansion of some elementary functions
5.8.3 Bernoulli numbers and polynomials
5.9 Some elementary functions introduced by recurrences
5.9.1 The square root function
5.9.2 The logarithm function
5.9.3 The exponential function
5.9.4 The arctangent function
5.10 Functions with primitives
5.10.1 The concept of a primitive function
5.10.2 The existence of primitives for continuous functions
5.11 Exercises
5.12 References and comments

6 Integral Calculus on R
6.1 The Darboux-Stieltjes integral
6.1.1 The Darboux integral
6.1.2 The Darboux-Stieltjes integral
6.2 Integrability of sequences and series of functions
6.3 Improper integrals
6.4 Euler integrals
6.4.1 Gamma function
6.4.2 Beta function
6.5 Polylogarithms
6.6 e and π are transcendental
6.7 The Gronwall inequality
6.8 Exercises
6.9 References and comments

7 Differential Calculus on Rn 
7.1 Linear and bounded mappings
7.1.1 Multilinear mappings
7.1.2 Quadratic mappings
7.2 Differentiable functions
7.2.1 Variations
7.2.2 Gateaux differential
7.2.3 Frechet differential
7.2.4 Properties of the Frechet differentiable functions
7.3 Partial derivatives
7.3.1 The inverse function theorem and the implicit function theorem
7.3.2 Directional derivatives and gradients
7.4 Higher-order differentials and partial derivatives
7.4.1 The case X = Rn
7.5 Taylor formula
7.6 Problems of local extremes
7.6.1 First-order conditions
7.6.2 Second-order conditions
7.6.3 Constraint local extremes
7.7 Exercises
7.8 References and comments

8 Double Integrals, Triple Integrals, and Line Integrals
8.1 Double integrals
8.1.1 Double integrals on rectangles
8.1.2 Double integrals on simple domains
8.2 Triple integrals
8.2.1 Triple integrals on parallelepipeds
8.2.2 Triple integrals on simple domains
8.3 n-fold integrals
8.3.1 n-fold integrals on hyperrectangles
8.3.2 n-fold integrals on simple domains
8.4 Line integrals
8.4.1 Line integrals with respect to arc length
8.4.2 Line integrals with respect to axis
8.4.3 Green formula
8.5 Integrals depending on parameters
8.6 Exercises
8.7 References and comments

9 Constants
9.1 Pythagoras's constant
9.1.1 Sequences approaching √2
9.2 Archimedes' constant
9.2.1 Recurrence relation
9.2.2 Buffon needle problem
9.3 Arithmetic-geometric mean
9.4 BBP formulas
9.4.1 Computing the nth binary or hexadecimal digit of π
9.4.2 BBP formulas by binomial sums
9.5 Ramanujan formulas
9.6 Several natural ways to introduce number e
9.7 Optimal stopping problem
9.8 References and comments

10 Asymptotic and Combinatorial Estimates
10.1 Asymptotic estimates
10.2 Algorithm analysis
10.3 Combinatorial estimates
10.3.1 Counting relations, topologies, and partial orders
10.3.2 Generalized Fubini numbers
10.3.3 The Catalan numbers and binary trees
10.4 References and comments


List of Symbols

Author Index

Subject Index

Publish Book: 
Modify Date: 
Monday, March 16, 2009