# Real and Complex Analysis by Rudin

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## Main Question or Discussion Point

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[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
[*] Bibliography
[*] List of Special Symbols
[*] Index
[/LIST]

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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.

mathwonk
Homework Helper
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.

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

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.