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Real and complex analysis
Real and complex analysis by walter rudins
This book contains a first-year graduate course in which the basic techniques and
theorems of analysis are presented in such a way that the intimate connections
between its various branches are strongly emphasized. The traditionally separate
subjects of "real analysis" and "complex analysis" are thus united; some of the
basic ideas from functional analysis are also included.
Here are some examples of the way in which these connections are demon-
strated and exploited. The Riesz representation theorem and the Hahn-Banach
theorem allow one to " guess" the Poisson integral formula. They team up in the
proof of Runge's theorem. They combine with Blaschke's theorem on the zeros of
bounded holomorphic functions to give a proof of the Miintz-Szasz theorem,
which concerns approximation on an interval. The fact that 13 is a Hilbert space
is used in the proof of the Radon-Nikodym theorem, which leads to the theorem
about differentiation of indefinite integrals, which in turn yields the existence of
radial limits of bounded harmonic functions. The theorems of Plancherel and
Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in
the Denjoy-Carleman theorem about infinitely differentiable functions on the real
line. The maximum modulus theorem gives information about linear transform-
ations on fl'-spaces.
Since most of the results presented here are quite classical (the novelty lies in
the arrangement, and some of the proofs are new), I have not attempted to docu-
ment the source of every item. References are gathered at the end, in Notes and
Comments. They are not always to the original sources, but more often to more
recent works where further references can be found. In no case does the absence
of a reference imply any claim to originality on my part.
The prerequisite for this book is a good course in advanced calculus
(settheoretic manipulations, metric spaces, uniform continuity, and uniform
convergence). The first seven chapters of my earlier book" Principles of Mathe-
matical Analysis" furnish sufficient preparation. Experience with the first edition shows that first-year graduate students can
study the first 15 chapters in two semesters, plus some topics from 1 or 2 of the
remaining 5. These latter are quite independent of each other. The first 15 should
be taken up in the order in which they are presented, except for Chapter 9, which
can be postponed.
The most important difference between this third edition and the previous
ones is the entirely new chapter on differentiation. The basic facts about differen-
tiation are now derived fro~ the existence of Lebesgue points, which in turn is an
easy consequence of the so-called "weak type" inequality that is satisfied by the
maximal functions of measures on euclidean spaces. This approach yields strong
theorems with minimal effort. Even more important is that it familiarizes stu-
dents with maximal functions,' since these have become increasingly useful in
several areas of analysis.
One of these is the study of the boundary behavior of Poisson integrals. A
related one concerns HP-spaces. Accordingly, large parts of Chapters 11 and 17
were rewritten
This book contains a first-year graduate course in which the basic techniques and
theorems of analysis are presented in such a way that the intimate connections
between its various branches are strongly emphasized. The traditionally separate
subjects of "real analysis" and "complex analysis" are thus united; some of the
basic ideas from functional analysis are also included.
Here are some examples of the way in which these connections are demon-
strated and exploited. The Riesz representation theorem and the Hahn-Banach
theorem allow one to " guess" the Poisson integral formula. They team up in the
proof of Runge's theorem. They combine with Blaschke's theorem on the zeros of
bounded holomorphic functions to give a proof of the Miintz-Szasz theorem,
which concerns approximation on an interval. The fact that 13 is a Hilbert space
is used in the proof of the Radon-Nikodym theorem, which leads to the theorem
about differentiation of indefinite integrals, which in turn yields the existence of
radial limits of bounded harmonic functions. The theorems of Plancherel and
Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in
the Denjoy-Carleman theorem about infinitely differentiable functions on the real
line. The maximum modulus theorem gives information about linear transform-
ations on fl'-spaces.
Since most of the results presented here are quite classical (the novelty lies in
the arrangement, and some of the proofs are new), I have not attempted to docu-
ment the source of every item. References are gathered at the end, in Notes and
Comments. They are not always to the original sources, but more often to more
recent works where further references can be found. In no case does the absence
of a reference imply any claim to originality on my part.
The prerequisite for this book is a good course in advanced calculus
(settheoretic manipulations, metric spaces, uniform continuity, and uniform
convergence). The first seven chapters of my earlier book" Principles of Mathe-
matical Analysis" furnish sufficient preparation. Experience with the first edition shows that first-year graduate students can
study the first 15 chapters in two semesters, plus some topics from 1 or 2 of the
remaining 5. These latter are quite independent of each other. The first 15 should
be taken up in the order in which they are presented, except for Chapter 9, which
can be postponed.
The most important difference between this third edition and the previous
ones is the entirely new chapter on differentiation. The basic facts about differen-
tiation are now derived fro~ the existence of Lebesgue points, which in turn is an
easy consequence of the so-called "weak type" inequality that is satisfied by the
maximal functions of measures on euclidean spaces. This approach yields strong
theorems with minimal effort. Even more important is that it familiarizes stu-
dents with maximal functions,' since these have become increasingly useful in
several areas of analysis.
One of these is the study of the boundary behavior of Poisson integrals. A
related one concerns HP-spaces. Accordingly, large parts of Chapters 11 and 17
were rewritten
858 pages, Kindle Edition
Published November 18, 2020
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