Real and Complex Analysis / Edition 3

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This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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Table of Contents


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

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  • Anonymous

    Posted September 21, 2004

    A start in analysis.

    I am a fan of Rudin's books. This one 'Real and Complex Analysis' has served as a standard textbook in the first graduate course in analysis at lots of universities in the US, and around the world. The book is divided in the two main parts, real and complex analysis. But in addition, it contains a good amount of functional and harmonic analysis; and a little operator theory. I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know. What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting. Some of the exercises are notoriously hard, but I think that is good: It simply means that they serve as work-projects when the students use the book. And this approach probably is more pedagogical as well. After surviving some of the hard exercises in Rudin's Real and Complex, I think we learn things that stay with us for life; you will be 'marked for life!' Review by Palle Jorgensen, September 2004.

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  • Anonymous

    Posted February 7, 2001

    A classic Real Analysis Book

    Truly one of the best in the field. The theory is almost complete; it is well presented, although the proofs sometimes seem 'too clever'. Exercises are excellent and challenging. Requires a high level of math maturity.

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