# Book:Walter Rudin/Real and Complex Analysis

## Walter Rudin: *Real and Complex Analysis*

Published $\text {1966}$, **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