Book:Walter Rudin/Real and Complex Analysis

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Walter Rudin: Real and Complex Analysis

Published $1966$, McGraw-Hill

ISBN 0-070-54234-1.


Subject Matter


Contents

Preface

Prologue: The Exponential Function

Chapter 1 Abstract Integration

Set-theoretic notations and terminology
The concept of mesurability
Simple functions
Elementary properties of measures
Arithmetic in $[0,\infty]$
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 $L^p$-Spaces

Convex functions and inequalities
The $L^p$-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 $L^1$-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 $L^p$
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
Convolutions
Distribution functions

Chapter 9 Fourier Transforms

Formal properties
The inversion theorem
The Plancherel theorem
The Banach algebra $L^1$

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

Introduction
The Schwarz lemma
The Phragmen-Lindelof method
An interpolation theorem
A converse of the maximum modulus theorem

Chapter 13 Approximation by Rational Functions

Preparation
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 $\mathscr{S}$
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 Muntz-Szasz 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 $H^p$-Spaces

Subharmonic functions
The spaces $H^p$ and $N$
The theorem of F. and M. Riesz
Factorization theorems
The shift operator
Conjugate functions

Chapter 18 Elementary Theory of Banach Algebras

Introduction
The invertible elements
Ideals and homomorphisms
Applications

Chapter 19 Holomorphic Fourier Transforms

Introduction
Two theorems of Paley and Wiener
Quasi-analytic classes
The Denjoy-Carleman theorem

Chapter 20 Uniform Approximation by Polynomials

Introduction
Some lemmas
Mergelyan's theorem

Appendix: Hausdorff's Maximality Theorem