ABOUT THE AUTHOR. Walter Rudin is the author of three textbooks, Principles of Mathematical. Analysis, Real and Complex Analysis, and Functional Analysis, . REAL AND. COMPLEX. ANALYSIS. Third Edition. Walter Rudin. Professor of Mathematics. University of Wisconsin, Madison. Version No rights reserved. Chapter 1 The Real and Complex Number Systems. Introduction. Ordered Sets. Fields. The Real Field. The Extended Real Number System. The Complex Field.

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Problems and Solutions in REAL AND COMPLEX ANALYSIS As a lim sup of a sequence of measurable functions, gis measurable (Rudin [8], theorem I bought Rudin's Real and Complex Analysis at the age of 21 at the recommendation of a fellow student, some years my senior, who not only knew much more. This is the second book in the Rudin series suitable for the first year graduate student who has completed Rudin's first book, "Mathematical Analysis" (Chapters .

In many cases, no citation of a source is given for a definition or result, especially when a result is very basic or a definition evolved in such a way that it is difficult to decide who should receive credit for it, as is the case for the definition of the norm function. It must be emphasized that in no case does the lack of a citation imply any claim to priority on my part. The exercises, of which there are over , have several purposes. One obvious one is to provide the student with some practice in the use of the results developed in the text, and a few quite frankly have no reason-for Preface xiii their existence beyond that.

However, most do serve higher purposes. One is to extend the theory presented in the text. For example, Banach limits are defined and developed in Exercise 1. Another purpose of some of the exercises is to provide supplementary examples and counterexamples. Occasionally, an exercise presents an alternative development of a main result. For example, in Exercise 1. With the exception of a few extremely elementary facts presented in the first section of Chapter 1, none of the results stated and used in the body of the text have their proofs left as exercises.

Very rarely, a portion of an example begun in t. One final comment on the general approach involves the transliteration of Cyrillic names. I originally intended to use the modern scheme adopted by Mathematical Revie? S in However, in the end I decided to write these names as the authors themselves did in papers published in Western languages, or as the names have commonly appeared in other sources. For, example, the modern MR transliteration scheme would require that V.

Real Analysis Rudin

Smulian's last name be written as Shmul'yan. However, Smulian wrote many papers in Western languages, several of which are cited in this book, in which he gave his name the Czech diacritical transliteration that appears in this sentence. No doubt he was just following the custom of his time, but because of his own extensive use of the form Smulian I have presented his name as he wrote it and would have recognized it.

Synopsis Chapter 1 focuses on the metric theory of normed spaces. The first three sections present fundamental definitions and examples, as well as the most elementary properties of normed spaces such as the continuity of their vector space operations.

The fourth section contains a short development of the most basic properties of bounded linear operators between normed spaces, including properties of normed space isomorphisms, which are then used to show that every finite-dimensional normed space is a Banach space. The Baire category theorem for nonempty complete metric spaces is the subject of Section 1.

This section is, in a sense, optional, since none of the results outside of optional sections of this book depend directly on it, though some such results do depend on a weak form of the Baire category theorem that will be mentioned in the next paragraph. However, this section has not been marked optional, since a student far enough along in his or her mathematical career to be reading this book should become familiar with Baire category.

This section is placed just before the section on the open mapping theorem, closed graph theorem, and uniform boundedness xiv Preface principle for the benefit of the instructor wishing to substitute traditional Baire category proofs of those results for the ones given here. Since a course in functional analysis is not a prerequisite foJ this book, the reader may not have seen the open mapping theorem, closed graph theorem, and uniform boundedness principle for Banach spaces.

Section 1. All three are based on a very specific and easily proved form of the Baire category theorem, presented in Section 1. In Section 1. Following a section devoted to direct sums of normed spaces, Section 1. The same section contains a development of Minkowski functionals and gives an example of how they are used to prove versions of the Hahn-Banach separation theorem.

The next section discusses reflexivity and includes Pettis's theorem about the reflexivity of a closed subspace of a reflexive space and many of its consequences. This completes the basic material of Chapter 1. The last section of Chapter 1, Section 1. This section contains a number of useful characterizations of reflexivity, including James's theorem.

Some of the more basic of these are usually obtained as corollaries of the Eberlein-Smulian theorem, but are included here since they can be proved fairly easily without it. The most important of these basic characterizations are repeated in Section 2. The proof given here is a detailed version of James's proof []. While the development leading up to the proof could be abbreviated slightly by delaying this section until the Eberlein-Smulian theorem is available, there Preface xv are two reasons for my not doing so.

The second is due to the reputation that James's theorem has acquired as being formidably deep. The proof is admittedly a bit intricate, but it is entirely elementary, not all that long, and contains some very nice ideas. By placing the proof as early as possible in this book, I hope to stress its elementary nature and dispel a bit of the notion that it is inaccessible.

The first section includes some topological preliminaries, but is devoted primarily to a fairly extensive development of the theory of nets, including characterizations of topological properties in terms of the accumulation and convergence of certain nets. Even a student with a solid first course in general topology may never have dealt with nets, so several examples are given to illustrate both their similarities to and differences from sequences. A motivation of the somewhat nonintuitive definition of a subnet is given, along with examples.

The section includes a short discussion of topological groups, primarily to be able to obtain a characterization of relative compactness in topological groups in terms of the accumulation of nets that does not always hold in arbitrary topological spaces.

Ultranets are not discussed in this section, since they are not really needed in the rest of this book, but a brief discussion of ultranets is given in Appendix D for use by the instructor who wishes to show how ultranets can be used to simplify certain compactness arguments. Section 2. The section includes a brief introduction to the dual space of a topological vector space, and presents the versions of the Hahn-Banach separation theorem due to Mazur and Eidelheit as well as the consequences for locally convex spaces of Mazur's separation theorem that parallel the consequences for normed spaces of the normed space version of the Hahn-Banach extension theorem.

This is followed by a section on metrizable vector topologies.

This section is marked optional since the topologies of main interest in this book are either induced by a norm or not compatible with any metric whatever. An F -space is defined in this section to be a topological vector space whose topology is compatible with a complete metric, without the requirement that the metric be invariant.

Included is Victor Klee's result that every invariant metric inducing a topologically complete topology on a group is in fact a complete metric, which has the straightforward consequence that every F-space, as defined in this section, actually has its topology induced by a complete invariant metric, and thereby answers a question of Banach.

The study of the weak topology of a normed space begins in earnest in Section 2. This section is devoted primarily to summarizing and extending the fundamental properties of this topology already developed in more general settings earlier in this chapter, and exploring the connections between the weak and norm topologies. Included is Mazur's theorem that the closure and weak closure of a convex subset of a normed space are the same. Weak sequential completeness, Schur's property, and the RadonRiesz property are studied briefly.

The main results of this section are the Banach-Alaoglu theorem and Goldstine's theorem. Weak compactness is studied in Section 2. It was necessary to delay this section until after Sections 2. The Eberlein-Smulian theorem is obtained in this section, as is the result due to Krein and Smulian that the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact.

An Introduction to Banach Space Theory

The corresponding theorem by Mazur on norm compactness is also obtained, since it is an easy consequence of the same lemma that contains the heart of the proof of the Krein-Smulian result.

A brief look is taken at weakly compactly generated normed spaces. The goal of optional Section 2.

The section is relatively short since most of the work needed to obtain this result was done in the lemmas used to prove James's reflexivity theorem in Section 1. The topic of Section 2. The Krein-Milman theorem is obtained, as is Milman's partial converse of that result. Chapter 2 ends with an optional section on support points and subreflexivity.

Included are the Bishop-Phelps theorems on the density of support points in the boundaries of closed convex subsets of Banach spaces and on the subreflexivity of every Banach space. Chapter 3 contains a discussion of linear operators between normed spaces far more extensive than the brief introduction presented in Section 1. The first section of the chapter is devoted to adjoints of bounded Preface xvii linear operators between normed spaces. Section 3.

Riesz's analysis of the spectrum of a compact operator is obtained, and the method used yields the result for real Banach spaces as well as complex ones. The Fredholm alternative is then obtained from this analysis. Much of the rest of the section is devoted to the approximation property, especially to Grothendieck's result that shows that the classical definition of the approximation property in terms of the approximability of compact operators by finite-rank operators is equivalent to the common modern definition in terms of the uniform approximability of the identity operator on compact sets by finite-rank operators.

The section ends with a brief study of the relationship between Riesz's notion of operator compactness and Hilbert's property of complete continuity, and their equivalence for a linear operator whose domain is reflexive. The final section of Chapter 3 is devoted to weakly compact operators. The Dunford-Pettis property is examined briefly in this section. The purpose of Chapter 4 is to investigate Schauder bases for Banach spaces. Monotone bases and the existence of basic sequences are covered, and the relationship between Schauder bases and the approximation property is discussed.

Unconditional bases are investigated in Section 4. Results are presented about equivalently renorming Banach spaces with unconditional bases to be Banach algebras and Banach lattices. It is shown that neither the classical Schauder basis for C O, 1] nor the Haar basis for L 1 0, 1] is unconditional. Section 4. Characterizations of the standard unit vector bases for eo and l 1 arc. Weakly unconditionally Cauchy series are examined, and the Orlic:z:-Pettis theorem and Bessaga-Pelczynski selection principle are obtained.

The properties of the sequence of coordinate functionals for a Schauder basis are taken up in Section 4. The final section of Chapter 4 is optional and is devoted to an investigation of James's space J, which was the first example of a nonreflexive Banach space isometrically isomorphic to its second dual.

Chapter 5 focuses on various forms of rotundity, also called strict convexity, and smoothness. The first section of the chapter is devoted to characterizations of rotundity, its fundamental properties, and examples, including one due to Klee that shows that rotundity is not always inherited by quotient spaces. Section 5. The second half of Chapter 5 deals with smoothness. Simple smoothness is taken up in Section 5. The partial duality between rotundity and smoothness is examined, and other important properties of smoothness are developed.

Uniform smoothness is the subject of the next section, in which the property is defined using the modulus of smoothness and characterized in terms of the uniform Frechet differentiability of the norm.

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The complete duality between uniform smoothness and uniform rotundity is proved. Frechet smoothness and uniform Gateaux smoothness are examined in the final section of the chapter, and Smulian's results on the duality between these properties and various generalizations of uniform rotundity are obtained.

Appendix A includes an extended description of the prerequisites for reading this book, along with a very detailed list of the changes that must be made to the presentation in Chapter 1 if this book is to be used for an undergraduate topics course in Banach space theory.

Appendices B and C are included to support such a topics course. They are, respectively, a list of the properties of metric spaces that should be familiar to a student in such a course and a development of f. Appendix D is a discussion of ultranets that supplements the material on nets in Section 2.

Dependences No material in any nonoptional section of this book depends on material in any optional section, with the exception of a few exercises in which the dependence is clearly indicated. Where an optional section depends on Preface xix other optional sections, that dependence is stated clearly at the beginning of the section.

The material in the nonoptional sections of Chapters 1 through 3 is meant to be taken up in the order presented, and each such section should be considered to depend on every other nonoptional section that precedes it. One important exception is that, as has already been mentioned, Section 1. However, the reader unfamiliar with the Baire category theorem will not want to skip this material. All of the nonoptional sections of Chapters 1 and 2 should be covered before taking up Chapters 4 and 5.

Chapter 4 also depends on the first two sections of Chapter 3. If the small amount of material in Chapter 4 on the approximation property is not to be skipped, then the development of that property in Section 3. Some results about adjoint operators from Section 3.

Except for this, Chapter 5 does not depend on the material in Chapters 3 and 4. Appendix A does not depend on any other part of this book, except where it refers to changes that must 'be made to the presentation in Chapter 1 for an undergraduate topics course.

Appendices Band C do not use material from any other portion of this book. Appendix D depends on Section 2. Acknowledgments There are many who contributed valuable suggestions and various forms of aid and encouragement to this project.

I would also like to express my appreciation for the support and editorial assistance I received from Springer-Verlag, particularly from Ina Lindemann and Steve Pisano.

This book would hardly have been possible without the day-today support of my wife Kathy, who did a wonderful job of insulating me from the outside world during its writing and who did not hesitate to pass stern judgment on some of my more convoluted prose. Finally, my special thanks go to my friend and mentor Horacio Porta, who read an early version of the manuscript and made some extremely valuable suggestions that substantially improved the content and format of this book.

In particular, the material presented in the first twelve sections of this chapter, with the exception of that of Section 1. Whenever a definition contains two or more different names for the same object, the first is the one usually used here.

The alternative names are included because they are sometimes encountered in other sources. The set of positive integers is denoted by N. The fields of real and complex numbers are denoted by R. The symbollF denotes a field that can be either R. The elements of lF are called scalars. In a topological space, the closure of a set A, denoted by A, is the smallest closed set that includes A, that is, the intersection of all closed sets that include A.

The interior of A, dE'noted by A0 , is the largest open subset of A, that is, the union of all open subsets of A. Scalars are usually represented in this book by lowercase letters near the beginning of the Greek alphabet and vectors by lowercase letters near the end of the Roman alphabet, with one major exception being that nonnegative real numbers are often denoted by the letters r, s, and t. Though the symbol 0 is used for the zeros of both lF and X, the context should always make it clear which is intended.

In this book, a subspace of a vector space X always means a vector subspace, that is, a subset of X that is itself a vector space under the same operations. It is presumed that the reader is familiar with linear independence, bases, and other elementary vector space concepts. It is worth emphasizing that in this book the term "vector space" always means a vector space over lR or C.

The terms real vector space and complex vector space are used when it is necessary to be specific about the scalar field. Another important convention is that except where stated otherwise, all vector spaces discussed within the same context are assumed to be over the same field JF, but lF may be either lR or C. For example, suppose that a theorem begins with the sentence "Let X be a normed space andY a Banach space" and that no field is mentioned anywhere in the theorem.

Normed spaces ahd Banach spaces, defined in the next section, are vector spaces with some additional structure. Thus, it is implied that X and Y are either both real or both complex.

In every instance in this book that such notation could cause a problem, the set turns out to be countable, 1 and the difficulty is avoided by writing the set as a finite list or a sequence. Please try again later. Hardcover Verified download.

This is the second book in the Rudin series suitable for the first year graduate student who has completed Rudin's first book, "Mathematical Analysis" Chapters and 11 or any introductory 1-year course in Real Analysis at the undergraduate senior level. However, in order to fully understand the topics in Complex Analysis presented in this book, one should complete an undergraduate course in Complex Variables in addition to undergraduate Real Analysis.

This book also provides excellent preparation for mathematicians planning to study Rudin's 3rd book "Functional Analysis". This is an excellent book that combines real and complex analysis into one course.

A good thing about using this book is that one can complete a course in both subjects in one year affording them room in their graduate corriculum to study an additional mathematical area. Furthermore, it is good to see the two topics combined into one course showing applicability of Real Analysis in areas of Complex Analysis, such as Fourier Transforms.

Also, topics in Functional Analysis are provided later in the book. Rudin writes in a very user-friendly yet concise manner, and at the same time he masterfully manages to maintain the high level of formality required from a graduate mathematics text. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting via Folland's book was not successful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do.

Rudin's book however, is a lot more accessible for the beginning graduate students who may not have had any more than some basic exposure to measure theory in their upper division analysis classes.

I will inevitably be making a few comparisons between the two texts in the following. One point to keep in mind though, is that Rudin developes the measure in a more formal axiomatic direction, instead of in the more concrete, constructive approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P X of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X a sigma algebra M , obtained via applying the Caratheodory's criterion, results in a "countably additive" set function which is called a measure on X, M The latter is the approach also taken in both H.

The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed some level of mathematical maturity yet. Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory Sobolev spaces, weak derivatives, etc. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study.

For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" aka Henstock-Kurzweil theory which serves to construct a more powerful integral than that of the Lebesgue's.

As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general locally Hausdorff topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets the older general topology books usually prove the Tychonoff theorem using ultrafilters.

All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin.

There are however a few other equally well-written complex analysis books to pick from, for example John B.

Conway's classic from the Springer-Verlag graduate text series, or L. Ahlfors's wonderful monograph, to name just a couple. This is a very nice book. However in my opinion it is not the best of Rudin's three well-known books on analysis Principles, Real and Complex, and Functional.

The third, Functional Analysis, is a better representation of the subject, covering distribution theory alongside Fourier analysis, and containing applications of analysis to other areas of mathematics, such as number theory and PDE.

The first chapter and the chapter on Hilbert spaces are favorites of mine. It makes a good reference, but one should be sure to study other authors as well. The organization of the book is innovative and for that it is stimulating.

It's good, but frustrating at times, when one is drawn in by an elegantly stated theorem only to be inexplicably let down by a proof which does the job, but leaves out the motivating ideas. Rudin has a talent for making difficult things clear, but one should exercise caution, because he also has a talent for making simple things appear difficult. Paperback Verified download. This is not about contents of this great book: The price is good but the paper quality is low.

Walter Rudin is a great expositor. He can one line a proof that would take you three pages, and he could 'one page' a proof that would take your professor three weeks to motivate. Although, you must fill in the steps which he chooses to omit. Don't turn to this book for comfort, if you want someone to baby then choose a different author. If you want pictures, then draw them! Do not use this as a first exposure to analysis or you will certainly hate the subject and take little if anything from it.

If you get stuck, read it again, and again, and again. My only complaint would be the price, charging this much for a book that has been around this long is absurd F you Mcgraw-Hill! Lastly, this book ages like wine so give it a little to warm up to you. Continuing on in the Rudin versus Royden debate, to be fair when Rudin gets compared to "Royden" the comparison really should be to this book and not "The Principles of Mathematical Analysis".

If you want to include complex analysis, for example if you are interested in the characteristic function of the terminal node of a stochastic process, then you should consider this book over "Royden". However, Royden and Fitzpatrick is also very nicely written and is ultimately easier for the reader to understand.

The paper quality is low. One person found this helpful. See all 37 reviews. site Giveaway allows you to run promotional giveaways in order to create buzz, reward your audience, and attract new followers and customers. Learn more about site Giveaway. This item: Set up a giveaway. Customers who viewed this item also viewed. Principles of Mathematical Analysis. Real Analysis: Modern Techniques and Their Applications.

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Learn more about site Prime.All of the nonoptional sections of Chapters 1 and 2 should be covered before taking up Chapters 4 and 5. In a paper that appeared in , Eduard Helly [] proved what is now called Helly's theorem for bounded linear functionals. Customer images.

Continuing on in the Rudin versus Royden debate, to be fair when Rudin gets compared to "Royden" the comparison really should be to this book and not "The Principles of Mathematical Analysis". Stein 4. One important exception is that, as has already been mentioned, Section 1.

The set of positive integers is denoted by N. The rank of a linear operator is the dimension of its range. The last section of Chapter 1, Section 1.

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