Real and Complex Analysis 3 9780070542341 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.
Real and Complex Analysis

Real and Complex Analysis

3rd Edition
By Walter Rudin
ISBN10: 0070542341
ISBN13: 9780070542341
Copyright: 1987
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ISBN10: 0070542341 | ISBN13: 9780070542341



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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


About the Author

Walter Rudin