Contents

• 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: LP-Spaces

• Convex functions and inequalities
• The LP-spaces
• 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 L¹-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 LP
• 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 L¹
• 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 S
• 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-Szasz 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: HP-Spaces

• Subharmonic functions
• The spaces HP and N
• 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