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Elementary Classical Analysis

Jerrold E. Marsden and Michael J. Hoffman
W. H. Freeman
Publication Date: 
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BLL Rating: 

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

[Reviewed by
Allen Stenger
, on

This is an introductory text in real analysis, aimed at upper-division undergraduates. The coverage is similar to that in Rudin’s Principles of Mathematical Analysis and Apostol’s Mathematical Analysis. This book differs from these earlier books primarily in being more talkative: explanations are written out at greater length, there are more worked examples, and there is a much larger number of exercises at all levels of difficulty.

One unusual feature of the present book is that in the body of each chapter the theorems are motivated, stated in detail, and illustrated through examples — but not proven. Complete proofs are given in a lengthy appendix to each chapter. The chapters are like survey articles, with supplements. I wasn’t completely satisfied with this approach, because at this level of course we are primarily interested in the proofs, but it is workable.

Some topics are omitted. Functions of several variables are included, but not vector calculus (vector-valued functions). The integration is strictly Riemannian, although sets of measure zero are included, as are limited versions of the Lebesgue convergence theorems.

Apart from the segregation of the proofs, I had just a couple of minor gripes. The book uses the reversed bracket notation for open intervals, for example ]0, 1[ is the open interval from 0 to 1. It also uses a mixed reference style, where references cited in the body have their bibliographic information given there, while there is a separate, disjoint list of References and Suggestions for Further Study.

Although the material is classical, it is not what most people call “classical analysis,”. The book does not deal with classical topics such as inequalities, approximation of functions, and special numbers and functions (as does, for example, Duren’s recent Invitation to Classical Analysis).

Bottom line: A good choice for students who find Rudin too austere.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

1. Introduction: Sets and Functions
    Supplement on the Axioms of Set Theory
  2. The Real Line and Euclidean Space
    Ordered Fields and the Number Systems
    Completeness and the Real Number System
    Least Upper Bounds
    Cauchy Sequences
    Cluster Points: lim inf and lim sup
    Euclidean Space
    Norms, Inner Products, and Metrics
    The Complex Numbers
  3. Topology of Euclidean Space
    Open Sets
    Interior of a Set
    Closed Sets
    Accumulation Points
    Closure of a Set
    Boundary of a Set
    Series of Real Numbers and Vectors
  4. Compact and Connected Sets
    The Heine-Borel Theorem
    Nested Set Property
    Path-Connected Sets
    Connected Sets
  5. Continuous Mappings
    Images of Compact and Connected Sets
    Operations on Continuous Mappings
    The Boundedness of Continuous Functions of Compact Sets
    The Intermediate Value Theorem
    Uniform Continuity
    Differentiation of Functions of One Variable
    Integration of Functions of One Variable
  6. Uniform Convergence
    Pointwise and Uniform Convergence
    The Weierstrass M Test
    Integration and Differentiation of Series
    The Elementary Functions
    The Space of Continuous Functions
    The Arzela-Ascoli Theorem
    The Contraction Mapping Principle and Its Applications
    The Stone-Weierstrass Theorem
    The Dirichlet and Abel Tests
    Power Series and Cesaro and Abel Summability
  7. Differentiable Mappings
    Definition of the Derivative
    Matrix Representation
    Continuity of Differentiable Mappings; Differentiable Paths
    Conditions for Differentiability
    The Chain Rule
    Product Rule and Gradients
    The Mean Value Theorem
    Taylor's Theorem and Higher Derivatives
    Maxima and Minima
  8. The Inverse and Implicit Function Theorems and Related Topics
    Inverse Function Theorem
    Implicit Function Theorem
    The Domain-Straightening Theorem
    Further Consequences of the
    Implicit Function Theorem
    An Existence Theorem for Ordinary Differential Equations
    The Morse Lemma
    Constrained Extrema and Lagrange Multipliers
  9. Integration
    Integrable Functions
    Volume and Sets of Measure Zero
    Lebesgue's Theorem
    Properties of the Integral
    Improper Integrals
    Some Convergence Theorems
    Introduction to Distributions
  10. Fubini's Theorem and the Change of Variables Formula
    Fubini's Theorem
    Change of Variables Theorem
    Polar Coordinates
    Spherical Coordinates and Cylindrical Coordinates
    A Note on the Lebesgue Integral
    Interchange of Limiting Operations
  11. Fourier Analysis
    Inner Product Spaces
    Orthogonal Families of Functions
    Completeness and Convergence Theorems
    Functions of Bounded Variation and Fejér Theory (Optional)
    Computation of Fourier Series
    Further Convergence Theorems
    Fourier Integrals
    Quantum Mechanical Formalism
  Miscellaneous Exercises
  Answers to Selected Odd-Numbered Exercises