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## Real and Complex Analysis by Rudin

Code:
 Preface
Prologue: The Exponential Function
Abstract Integration 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

Positive Borel Measures Vector Spaces
Topological preliminaries
The Riesz representation theorem
Regularity properties of Borel measures
Lebesgue measure
Continuity properties of measurable functions
Exercises

L^p-Spaces Convex functions and inequalities
The L^p-spaces
Approximation by continuous functions
Exercises

Elementary Hilbert Space Theory Inner products and linear functionals
Orthonormal sets
Trigonometric series
Exercises

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
Exercises

Complex Measures Total Variation
Absolute continuity
Bounded linear functionals on L^p
The Riesz representation theorem
Exercises

Differentiation Derivatives of measures
The fundamental theorem of Calculus
Differentiable transformations
Exercises

Integration on Product Spaces Measurability on cartesian products
Product measures
The Fubini theorem
Completion of product measures
Convolutions
Distribution functions
Exercises

Fourier Transforms Formal properties
The inversion theorem
The Plancherel theorem
The Banach algebra L^1
Exercises

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

Harmonic Functions The Cauchy-Riemann equations
The Poisson integral
The mean value property
Boundary behavior of Poisson integrals
Representation theorems
Exercises

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

Approximation by Rational Functions Preperation
Runge's theorem
The Mittag-Leffler theorem
Simply connected regions
Exercises

Conformal Mapping 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

Zeros of Holomorphic Functions Infinite products
The Weierstrass factorization theorem
An interpolation problem
Jensen's formula
Blaschke products
The Muntz-Szasz theorem
Exercises

Analytic Continuation Regular points and singular points
Continuation along curves
The monodromy theorem
Construction of a modular function
The Picard theorem
Exercises

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

Elementary Theory of Banach Algebras Introduction
The invertible elements
Ideals and homomorphisms
Applications
Exercises

Holomorphic Fourier Transforms Introduction
Two theorems of Paley and Wiener
Quasi-analytic classes
The Denjoy-Carleman theorem
Exercises

Uniform Approximation by Polynomials Introduction
Some lemmas
Mergelyan's theorem
Exercises

Appendix: Hausdorff's Maximality Theorem
Bibliography
List of Special Symbols
Index

 Would Spivak's Calculus and Pugh's Real Math Analysis be sufficient grounding for this text, at least for the "real" part of it? Also, what books and in what order would I need to work my way up from absolute nothing to this book's level in complex analysis? Order of growth towards this book's rigor seems straightforward when it comes to real analysis: regular calculus -> (maybe some intro to proofs) -> Calculus by Spivak like text -> (maybe some intro to proofs) -> baby Rudin... How does this work with complex variables? Does one need some sort of complex pre/calculus before attempting complex analysis? Thanks.
 Recognitions: Homework Help Science Advisor I own this book, have studied it from time to time in the past as a grad student, but have learned little from it. Experts consider it a classic, but as with other books by this author i find it unilluminating. I did find it an excellent source of practice problems for PhD prelims in analysis.

## Real and Complex Analysis by Rudin

Same reaction as mathwonk. I learned from Folland's book instead.

 Quote by Vargo Same reaction as mathwonk. I learned from Folland's book instead.
OK, so I have very mixed feelings about Rudin and Folland. People say that Rudin makes more effort to impress the reader with his cleverness and that the proofs aren't enlightening (I once heard a professor describe it as being "pathelogically elegant".) But, Folland is nothing if not terse and there are a lot of steps that are skipped in proofs that the reader is assumed to a) know they are there and b)fill them in. Now, if you are learning, this isn't the worst thing that can happen, but it gets irritating.

My biggest gripe with Rudin is that he doesn't start abstract enough, whereas Folland starts out with premeasures on arbitrary sets and builds the theory from there and then says "hey, look at that, the 'dx' that you already know about is actually Lebesgue measure." Rudin abstracts a little at a time and it gets annoying.

However, I still enjoyed Rudin's book and learned a lot from it.