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