# Real and Complex Analysis

3^{rd}Edition

ISBN10: 0070542341

ISBN13: 9780070542341

Copyright: 1987

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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.

### 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

### 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

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