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Main Question or Discussion Point
- Author: Walter Rudin
- Title: Real and Complex Analysis
- Amazon Link: https://www.amazon.com/dp/0070542341/?tag=pfamazon01-20
- Prerequisities: Baby Rudin
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Prologue: The Exponential Function
[*] Abstract Integration
[LIST]
[*] Set-theoretic notatons and terminology
[*] The concept of measurability
[*] 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
[*] Exercises
[/LIST]
[*] Positive Borel Measures
[LIST]
[*] Vector Spaces
[*] Topological preliminaries
[*] The Riesz representation theorem
[*] Regularity properties of Borel measures
[*] Lebesgue measure
[*] Continuity properties of measurable functions
[*] Exercises
[/LIST]
[*] L^p-Spaces
[LIST]
[*] Convex functions and inequalities
[*] The L^p-spaces
[*] Approximation by continuous functions
[*] Exercises
[/LIST]
[*] Elementary Hilbert Space Theory
[LIST]
[*] Inner products and linear functionals
[*] Orthonormal sets
[*] Trigonometric series
[*] Exercises
[/LIST]
[*] Examples of Banach Space Techniques
[LIST]
[*] 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
[/LIST]
[*] Complex Measures
[LIST]
[*] Total Variation
[*] Absolute continuity
[*] Consequences of the Radon-Nikodym theorem
[*] Bounded linear functionals on L^p
[*] The Riesz representation theorem
[*] Exercises
[/LIST]
[*] Differentiation
[LIST]
[*] Derivatives of measures
[*] The fundamental theorem of Calculus
[*] Differentiable transformations
[*] Exercises
[/LIST]
[*] Integration on Product Spaces
[LIST]
[*] Measurability on cartesian products
[*] Product measures
[*] The Fubini theorem
[*] Completion of product measures
[*] Convolutions
[*] Distribution functions
[*] Exercises
[/LIST]
[*] Fourier Transforms
[LIST]
[*] Formal properties
[*] The inversion theorem
[*] The Plancherel theorem
[*] The Banach algebra L^1
[*] Exercises
[/LIST]
[*] Elementary Properties of Holomorphic Functions
[LIST]
[*] 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
[/LIST]
[*] Harmonic Functions
[LIST]
[*] The Cauchy-Riemann equations
[*] The Poisson integral
[*] The mean value property
[*] Boundary behavior of Poisson integrals
[*] Representation theorems
[*] Exercises
[/LIST]
[*] The Maximum Modulus Principle
[LIST]
[*] Introduction
[*] The Schwarz lemma
[*] The Phragmen-Lindelof method
[*] An interpolation theorem
[*] A converse of the maximum modulus theorem
[*] Exercises
[/LIST]
[*] Approximation by Rational Functions
[LIST]
[*] Preperation
[*] Runge's theorem
[*] The Mittag-Leffler theorem
[*] Simply connected regions
[*] Exercises
[/LIST]
[*] Conformal Mapping
[LIST]
[*] Preservation of angles
[*] Linear fractional transformations
[*] Normal families
[*] The Riemann mapping theorem
[*] The class \mathcal{S}
[*] Continuity and the boundary
[*] Conformal mapping of an annulus
[*] Exercises
[/LIST]
[*] Zeros of Holomorphic Functions
[LIST]
[*] Infinite products
[*] The Weierstrass factorization theorem
[*] An interpolation problem
[*] Jensen's formula
[*] Blaschke products
[*] The Muntz-Szasz theorem
[*] Exercises
[/LIST]
[*] Analytic Continuation
[LIST]
[*] Regular points and singular points
[*] Continuation along curves
[*] The monodromy theorem
[*] Construction of a modular function
[*] The Picard theorem
[*] Exercises
[/LIST]
[*] H^p-Spaces
[LIST]
[*] Subharmonic functons
[*] The spaces H^p and N
[*] The theorem of F. and M. Riesz
[*] Factorization theorems
[*] The shift operator
[*] Conjugate functions
[*] Exercises
[/LIST]
[*] Elementary Theory of Banach Algebras
[LIST]
[*] Introduction
[*] The invertible elements
[*] Ideals and homomorphisms
[*] Applications
[*] Exercises
[/LIST]
[*] Holomorphic Fourier Transforms
[LIST]
[*] Introduction
[*] Two theorems of Paley and Wiener
[*] Quasi-analytic classes
[*] The Denjoy-Carleman theorem
[*] Exercises
[/LIST]
[*] Uniform Approximation by Polynomials
[LIST]
[*] Introduction
[*] Some lemmas
[*] Mergelyan's theorem
[*] Exercises
[/LIST]
[*] Appendix: Hausdorff's Maximality Theorem
[*] Notes and Comments
[*] Bibliography
[*] List of Special Symbols
[*] Index
[/LIST]
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