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Real and Complex Analysis

Walter Rudin
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book is full of interesting things, mostly proofs. The chapter on Banach algebras is a gem; this subject combines algebra, analysis, and topology, and the exposition shows clearly how the three areas work together. Walter Rudin (1921–2010) wrote the book in 1966 to show that real and complex analysis should be studied together rather than as two subjects, and to give a a modern treatment. Fifty years later it is still modern.

The first third of the book is devoted to measure and integration. The presentation is based on measures on abstract spaces with \(\sigma\)-algebras. It includes brief introductions to Hilbert space and Banach spaces, with material that will be used in the complex-variables proofs later. This beginning section is the only part of the book that deals with spaces more general than the real line and the complex plane, however it’s not any harder than it would be if we stuck to the real line. This includes a chapter on differentiation (of measures) and a chapter on product spaces (i.e., the Fubini theorem). The rest of the book is about analysis on the complex plane. It starts with a short chapter on Fourier transforms, then presents a course in complex variables that is traditional in terms of the theorems proved, but has very slick proofs using what has gone before. The traditional part ends with the little Picard theorem. The last quarter of the book consists of several short chapters on advanced topics in complex analysis; these include \(H^p\) spaces, Banach algebras, holomorphic Fourier transforms, and a characterization of functions that are the uniform limit of polynomials (Mergelyan’s theorem).

The approach is not very concrete; there are very few worked examples (many of the exercises do deal with specific functions). The book does not have the detailed chapters that we are used to on evaluating series and integrals and on special functions. But it is also not very abstract; it truly is mostly complex analysis, not general spaces. The proofs are informed by the more general viewpoint, and there is a strong functional-analysis flavor. For example, much use is made of the Hahn-Banach Theorem and some use of the Urysohn lemma and Tietze extension theorem.

The book has been widely criticized for lacking motivation, and this criticism is accurate. You don’t absolutely need a lot of background to read the book, but it is a collection of beautiful proofs without much context. For example, the discussion of spectra comes out of nowhere and is very mysterious unless you are well-acquainted with linear algebra and eigenvalues.

The book was aimed at first-year graduates and has been used successfully in many first-year graduate courses, and I think that is still about the right level for it. Undergraduates interested in the subject matter would be better served, before they tackle this book, by a more traditional complex analysis book such as Bak & Newman’s Complex Analysis or Ahlfors’s more advanced Complex Analysis, and by one of the many good introductions to Lebesgue integration (I like Boas’s A Primer of Real Functions).

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.


Prologue: The Exponential Function

Chapter 1: Abstract Integration
Set-theoretic notations and terminology
The concept of measurability
Simple functions
Elementary properties of measures
Arithmetic in [0, ∞]
Integration of positive functions
Integration of complex functions
The role played by sets of measure zero

Chapter 2: Positive Borel Measures
Vector spaces
Topological preliminaries
The Riesz representation theorem
Regularity properties of Borel measures
Lebesgue measure
Continuity properties of measurable functions

Chapter 3: Lp-Spaces
Convex functions and inequalities
The Lp-spaces
Approximation by continuous functions

Chapter 4: Elementary Hilbert Space Theory
Inner products and linear functionals
Orthonormal sets
Trigonometric series

Chapter 5: Examples of Banach Space Techniques
Banach spaces
Consequences of Baire's theorem
Fourier series of continuous functions
Fourier coefficients of L1-functions
The Hahn-Banach theorem
An abstract approach to the Poisson integral

Chapter 6: Complex Measures
Total variation
Absolute continuity
Consequences of the Radon-Nikodym theorem
Bounded linear functionals on Lp
The Riesz representation theorem

Chapter 7: Differentiation
Derivatives of measures
The fundamental theorem of Calculus
Differentiable transformations

Chapter 8: Integration on Product Spaces
Measurability on cartesian products
Product measures
The Fubini theorem
Completion of product measures
Distribution functions

Chapter 9: Fourier Transforms
Formal properties
The inversion theorem
The Plancherel theorem
The Banach algebra L1

Chapter 10: Elementary Properties of Holomorphic Functions
Complex differentiation
Integration over paths
The local Cauchy theorem
The power series representation
The open mapping theorem
The global Cauchy theorem
The calculus of residues

Chapter 11: Harmonic Functions
The Cauchy-Riemann equations
The Poisson integral
The mean value property
Boundary behavior of Poisson integrals
Representation theorems

Chapter 12: The Maximum Modulus Principle
The Schwarz lemma
The Phragmen-Lindelöf method
An interpolation theorem
A converse of the maximum modulus theorem

Chapter 13: Approximation by Rational Functions
Runge's theorem
The Mittag-Leffler theorem
Simply connected regions

Chapter 14: Conformal Mapping
Preservation of angles
Linear fractional transformations
Normal families
The Riemann mapping theorem
The class L
Continuity at the boundary
Conformal mapping of an annulus

Chapter 15: Zeros of Holomorphic Functions
Infinite Products
The Weierstrass factorization theorem
An interpolation problem
Jensen's formula
Blaschke products
The Müntz-Szas theorem

Chapter 16: Analytic Continuation
Regular points and singular points
Continuation along curves
The monodromy theorem
Construction of a modular function
The Picard theorem

Chapter 17: Hp-Spaces
Subharmonic functions
The spaces Hp and N
The theorem of F. and M. Riesz
Factorization theorems
The shift operator
Conjugate functions

Chapter 18: Elementary Theory of Banach Algebras
The invertible elements
Ideals and homomorphisms

Chapter 19: Holomorphic Fourier Transforms
Two theorems of Paley and Wiener
Quasi-analytic classes
The Denjoy-Carleman theorem

Chapter 20: Uniform Approximation by Polynomials
Some lemmas
Mergelyan's theorem

Appendix: Hausdorff's Maximality Theorem

Notes and Comments


List of Special Symbols