# Real and Complex Analysis

3^{rd}Edition

ISBN10: 0070542341

ISBN13: 9780070542341

Copyright: 1987

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ISBN10: 0070542341 | ISBN13: 9780070542341

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The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

### Program Details

### Preface

### 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

#### Exercises

### 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

#### Exercises

### Chapter 3: *L*^{p}-Spaces

^{p}

#### Convex functions and inequalities

#### The *L*^{p}-spaces

^{p}

#### Approximation by continuous functions

#### Exercises

### Chapter 4: Elementary Hilbert Space Theory

#### Inner products and linear functionals

#### Orthonormal sets

#### Trigonometric series

#### Exercises

### Chapter 5: Examples of Banach Space Techniques

#### Banach spaces

#### Consequences of Baire's theorem

#### Fourier series of continuous functions

#### Fourier coefficients of *L*^{1}-functions

#### The Hahn-Banach theorem

#### An abstract approach to the Poisson integral

#### Exercises

### Chapter 6: Complex Measures

#### Total variation

#### Absolute continuity

#### Consequences of the Radon-Nikodym theorem

#### Bounded linear functionals on *L*^{p}

^{p}

#### The Riesz representation theorem

#### Exercises

### Chapter 7: Differentiation

#### Derivatives of measures

#### The fundamental theorem of Calculus

#### Differentiable transformations

#### Exercises

### Chapter 8: Integration on Product Spaces

#### Measurability on cartesian products

#### Product measures

#### The Fubini theorem

#### Completion of product measures

#### Convolutions

#### Distribution functions

#### Exercises

### Chapter 9: Fourier Transforms

#### Formal properties

#### The inversion theorem

#### The Plancherel theorem

#### The Banach algebra *L*^{1}

#### Exercises

### 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

#### Exercises

### Chapter 11: Harmonic Functions

#### The Cauchy-Riemann equations

#### The Poisson integral

#### The mean value property

#### Boundary behavior of Poisson integrals

#### Representation theorems

#### Exercises

### Chapter 12: The Maximum Modulus Principle

#### Introduction

#### The Schwarz lemma

#### The Phragmen-Lindelöf method

#### An interpolation theorem

#### A converse of the maximum modulus theorem

#### Exercises

### Chapter 13: Approximation by Rational Functions

#### Preparation

#### Runge's theorem

#### The Mittag-Leffler theorem

#### Simply connected regions

#### Exercises

### 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

#### Exercises

### Chapter 15: Zeros of Holomorphic Functions

#### Infinite Products

#### The Weierstrass factorization theorem

#### An interpolation problem

#### Jensen's formula

#### Blaschke products

#### The Müntz-Szas theorem

#### Exercises

### Chapter 16: Analytic Continuation

#### Regular points and singular points

#### Continuation along curves

#### The monodromy theorem

#### Construction of a modular function

#### The Picard theorem

#### Exercises

### Chapter 17: *H*^{p}-Spaces

^{p}

#### Subharmonic functions

#### The spaces *H*^{p} and N

^{p}

#### The theorem of F. and M. Riesz

#### Factorization theorems

#### The shift operator

#### Conjugate functions

#### Exercises

### Chapter 18: Elementary Theory of Banach Algebras

#### Introduction

#### The invertible elements

#### Ideals and homomorphisms

#### Applications

#### Exercises

### Chapter 19: Holomorphic Fourier Transforms

#### Introduction

#### Two theorems of Paley and Wiener

#### Quasi-analytic classes

#### The Denjoy-Carleman theorem

#### Exercises

### Chapter 20: Uniform Approximation by Polynomials

#### Introduction

#### Some lemmas

#### Mergelyan's theorem

#### Exercises

### Appendix: Hausdorff's Maximality Theorem

### Notes and Comments

### Bibliography

### List of Special Symbols

### Index

# About the Author

**Walter Rudin**

### Preface

### 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

#### Exercises

### 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

#### Exercises

### Chapter 3: *L*^{p}-Spaces

^{p}

#### Convex functions and inequalities

#### The *L*^{p}-spaces

^{p}

#### Approximation by continuous functions

#### Exercises

### Chapter 4: Elementary Hilbert Space Theory

#### Inner products and linear functionals

#### Orthonormal sets

#### Trigonometric series

#### Exercises

### Chapter 5: Examples of Banach Space Techniques

#### Banach spaces

#### Consequences of Baire's theorem

#### Fourier series of continuous functions

#### Fourier coefficients of *L*^{1}-functions

#### The Hahn-Banach theorem

#### An abstract approach to the Poisson integral

#### Exercises

### Chapter 6: Complex Measures

#### Total variation

#### Absolute continuity

#### Consequences of the Radon-Nikodym theorem

#### Bounded linear functionals on *L*^{p}

^{p}

#### The Riesz representation theorem

#### Exercises

### Chapter 7: Differentiation

#### Derivatives of measures

#### The fundamental theorem of Calculus

#### Differentiable transformations

#### Exercises

### Chapter 8: Integration on Product Spaces

#### Measurability on cartesian products

#### Product measures

#### The Fubini theorem

#### Completion of product measures

#### Convolutions

#### Distribution functions

#### Exercises

### Chapter 9: Fourier Transforms

#### Formal properties

#### The inversion theorem

#### The Plancherel theorem

#### The Banach algebra *L*^{1}

#### Exercises

### 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

#### Exercises

### Chapter 11: Harmonic Functions

#### The Cauchy-Riemann equations

#### The Poisson integral

#### The mean value property

#### Boundary behavior of Poisson integrals

#### Representation theorems

#### Exercises

### Chapter 12: The Maximum Modulus Principle

#### Introduction

#### The Schwarz lemma

#### The Phragmen-Lindelöf method

#### An interpolation theorem

#### A converse of the maximum modulus theorem

#### Exercises

### Chapter 13: Approximation by Rational Functions

#### Preparation

#### Runge's theorem

#### The Mittag-Leffler theorem

#### Simply connected regions

#### Exercises

### 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

#### Exercises

### Chapter 15: Zeros of Holomorphic Functions

#### Infinite Products

#### The Weierstrass factorization theorem

#### An interpolation problem

#### Jensen's formula

#### Blaschke products

#### The Müntz-Szas theorem

#### Exercises

### Chapter 16: Analytic Continuation

#### Regular points and singular points

#### Continuation along curves

#### The monodromy theorem

#### Construction of a modular function

#### The Picard theorem

#### Exercises

### Chapter 17: *H*^{p}-Spaces

^{p}

#### Subharmonic functions

#### The spaces *H*^{p} and N

^{p}

#### The theorem of F. and M. Riesz

#### Factorization theorems

#### The shift operator

#### Conjugate functions

#### Exercises

### Chapter 18: Elementary Theory of Banach Algebras

#### Introduction

#### The invertible elements

#### Ideals and homomorphisms

#### Applications

#### Exercises

### Chapter 19: Holomorphic Fourier Transforms

#### Introduction

#### Two theorems of Paley and Wiener

#### Quasi-analytic classes

#### The Denjoy-Carleman theorem

#### Exercises

### Chapter 20: Uniform Approximation by Polynomials

#### Introduction

#### Some lemmas

#### Mergelyan's theorem

#### Exercises

### Appendix: Hausdorff's Maximality Theorem

### Notes and Comments

### Bibliography

### List of Special Symbols

### Index

# About the Author

**Walter Rudin**

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