Product Description

Thorough introduction to an important area of mathematics

Contains recent results

Includes many exercises

Product Description
This book is an English translation of the last French edition of Bourbaki's Functions of a Real Variable. The first chapter is devoted to derivatives, Taylor expansions, the finite increment's theorem, convex functions. In the second chapter, primitives and integrals (on arbitrary intervals) are studied, as well as their dependence with respect to parameters. Classical functions (exponential, logarithmic, circular and inverse circular) are investigated in the third chapter. The fourth chapter gives a thorough treatment of differential equations (existence and unicity properties of solutions, approximate solutions, dependence on parameters) and of systems of linear differential equations. The local study of functions (comparison relations, asymptotic expansions) is treated in chapter V, with an appendix on Hardy fields. The theory of generalized Taylor expansions and the Euler-MacLaurin formula are presented in the sixth chapter, and applied in the last one to the study of the Gamma function on the real line as well as on the complex plane. Although the topics of the book are mainly of an advanced undergraduate level, they are presented in the generality needed for more advanced purposes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependance on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc. ... Read more

Customer Reviews (2)

A Very Useful text
This text is an interesting and rigorous presentation of advanced calculus for undergraduate mathematics and applied mathematics students. It is definitely worthy of thorough study; it will prepare the reader for further study of many areas of mathematics and its applications that require a basic knowledge of functions of a real variable or functions of a vector variable. There are many illustrative examples throughout the text.

W.W. Wittgentstein
This book has the wost format of any mathematical text I have ever seen. The author's favorite Greek letter is small Xi. A page littered with these presents a real mess for the eye to decipher. All his other notation is hard on the eyes. How the author expected this text to be widely used I do not know. Still if you can stand the eye strain this book has some good things in it, but not enough to be worth the effort of thorough study. ... Read more

Product Description
Volume 2 provides a comprehensive review of integral analysis in multidimensional Euclidean space. ... Read more

Product Description
This book builds upon the earlier volume Problems in Analysis, more than doubling it with a new section of problems on complex analysis. The problems on real analysis from the earlier book have all been checked, and stylistic, typographical, and mathematical errors have been corrected. The problems in complex analysis cover most of the principal topics in the theory of functions of a complex variable. The problems in the book cover, in real analysis: set algebra, measure and topology, real- and complex-valued functions, and topological vector spaces; in complex analysis: polynomials and power series, functions holomorphic in a region, entire functions, analytic continuation, singularities, harmonic functions, families of functions, and convexity theorems. ... Read more

Product Description
Students preparing for courses in real analysis often encounter either very exacting theoretical treatments or books without enough rigor to stimulate an in-depth understanding of the subject. Further complicating this, the field has not changed much over the past 150 years, prompting few authors to address the lackluster or overly complex dichotomy existing among the available texts.

The enormously popular first edition of Real Analysis and Foundations gave students the appropriate combination of authority, rigor, and readability that made the topic accessible while retaining the strict discourse necessary to advance their understanding. The second edition maintains this feature while further integrating new concepts built on Fourier analysis and ideas about wavelets to indicate their application to the theory of signal processing. The author also introduces relevance to the material and surpasses a purely theoretical treatment by emphasizing the applications of real analysis to concrete engineering problems in higher dimensions.

Expanded and updated, this text continues to build upon the foundations of real analysis to present novel applications to ordinary and partial differential equations, elliptic boundary value problems on the disc, and multivariable analysis. These qualities, along with more figures, streamlined proofs, and revamped exercises make this an even more lively and vital text than the popular first edition. ... Read more

Customer Reviews (3)

Not a bad choice.....
I have studied Real Analysis from both this book and Bartle and Sherbert. Personally, I like both for different reasons and could recommend both.

Krantz, in my opinion is very good at providing a means to developing a geometric intuition, as he introduces basic point set topology very early and uses it throughout the text (this is a strength of Baby Rudin as well). I feel that Krantz's proofs are not as complicated as those in Bartle and Sherbert, and thus are much easier to follow. I feel that this book is easy to understand and includes a helping or two of complex analysis as a bonus.

Now, I would not be an honest person if I did not mention the mistakes in this book. Yeah, Krantz makes a few significant errors. They are sprinkled around so, just watch for them. I don't recall Bartle and Sherbert having as many.

Now, to contrast with Bartle and Sherbert. I feel that Krantz is lacking in its teaching of analysis techniques- Bartle and Sherbert achieve this well. Their proofs are much more complicated, but in that complication you learn a lot about how to DO analysis. Point set topology is mainly introduced in the last chapter so you lack that power through out the book (a poor choice of writing style in my opinion) but the machinery you develop in this book beats Krantz into the ground. Bartle and Sherbert is lacking a little on conceptual development in my opinion (but does have some strengths over Krantz even on this front).

Neither book is perfect, both are incomplete. As I said before, I could easily recommend both without reservation. I do recommend getting a copy of Rudin's Principles of Real Analysis as a supplementary text to either book. Rudin's book is very difficult for most undergraduates (It was very hard for me the first time I looked through it) but EXCELLENT. Given the choice, personally, I would have chosen Krantz because I felt that I understood analysis better after working through it. But having read Bartle and Sherbert first well prepared me for Krantz! Moral of the story: read both (if time permits).

To summarize:
1) Go with Krantz for concepts and to UNDERSTAND analysis
2) Go with Bartle and Sherbert for techniques and to DO analysis
3) Go with Walter Rudin to MASTER (undergraduate) real analysis after on of the other! (Unless you go to Berkeley or Harvard and are really smart- then just jump strait to Rudin)

Not Satisfied
The Krantz text- while it covers many of the topics of real analysis- its notation is not concise. I would look elsewhere.

Real Analysis and Foundations
This book is an excellent text for understanding the foundations of Analysis. I have used this book as a suplement for several math classes. It is a good refrence tool for math majors. ... Read more

Product Description
The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash--Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve. "The Implicit Function Theorem" is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas. ... Read more

Customer Reviews (1)

could have been a great book
The book is full of typos and mistakes which make every proof a headache to read and understand. This is my third book by the same publisher and they are all of similar quality, full of typos and logical mistakes. Both are deadly sins but they are even deadlier in math books so just stay away from Birkhäuser books when it comes to math. ... Read more

Product Description
In this volume we study the generalized Bessel functions of the first kind by using a number of classical and new findings in complex and classical analysis. Our aim is to present interesting geometric properties and functional inequalities for these generalized Bessel functions. Moreover, we extend many known inequalities involving circular and hyperbolic functions to Bessel and modified Bessel functions. ... Read more

Product Description
An in-depth look at real analysis and its applications-now expanded and revised.

This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.

This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension. ... Read more

Customer Reviews (13)

The Best! book on the subject, hands down.
The book by Folland is, by far the best book on the subject. All of the proofs are succinct, and the problems are extremely appropriate. Comparing with other books: it has much more material than Rudin's book, regarding Real Variable Theory; it' much more general and substantial than the Royden and WHeeden/Zygmund books; of course, as a modern book, it blows the Kolmogorov/Fomin book out of the water. You won't find another book which presents Real Variable Theory as comprehensively or as general as this book; also, you won't find these techniques in any other book. It functions as an introduction to Measure Theory/Integration, Topology, Functional Analysis, and Fourier Analysis. I will always cherish this book. Many thanks to the author, Dr. Folland.

Outstanding text, poor typesetting quality
This is a great analysis book as others have discussed.

However, the publisher has done a really shoddy job
on the new edition.The print looks like a cheap photocopy
of an original manuscript.Or even a copy of a copy.
It is outrageous that they can charge so much for such
poor quality work.If I were Folland, I'd be really mad.
Also, the new cover is weak- bring back the old
brown cover!

5+ stars for the text, 1 star for the book itself.

I get it now
This is the second time I've re-reviewed this book.

First off, I am not a mathematician. I was trained as an engineer, and have recently started studying more advanced mathematics to apply it to my research. The only undergrad math course I'd taken before using this book was the standard analysis course. I initially used this book for a first graduate course in real analysis. Even with a professor, going through the book was incredibly difficult, and I had to resort to another book (Wheeden and Zygmund) as well as extensive notes provided by the professor. This experience made me loathe the book.

A few months after the course, having gained more exposure in this area, I returned to the book, and was surprised to find that I had finally started to understand why the author had organized it the way he had. Now, 6 months and another grad course in analysis later (operator theory), I think the book is worth its weight in gold.

First off, let's outline the cons. At first sight, the book takes brevity to the brink of lunacy. A (very) respectable first graduate course in analysis is covered in the first 100 pages. Dense doesn't even begin to cover it. Major results are relegated to the exercises, whole topics are compressed into a section (sometimes two or three are crammed into one), and even the proofs are presented with the barest minimum of explanation. The whole book is about 370 pages, and has enough material for about 4-5 courses. The exercises range from doable to extremely difficult. You also have chapters on everything from point set topology to harmonic analysis (abstract and otherwise) to probability to functional analysis. Heck, even fractals and manifolds pop up by the end.

The truth, however, is that all of these cons are actually pros in disguise. I know most engineers secretly think that the word 'elegant' used to describe mathematics textbooks is basically code for 'stupendously bad exposition', but the simple truth is this: analysis is hard. Sometimes brutally so, and the more you beat your brains against it, the better. If you're looking for a quick and easy explanation of the Lebesgue integral, this really isn't your book. Especially if you're not used to thinking as abstractly as is required here (I certainly wasn't).

I freely admit that I wouldn't recommend this as a first textbook to my worst enemies (my worst enemies are other engineers). But for mathematicians, the brevity might actually be useful. And once you know the basic material (the first three chapters), the book becomes an invaluable resource.

Splendid
This is such an excellent textbook on the so called graduate real analysis materials. All major concepts are well motivated. In short a pleasure to read. Read it along with Big Rudin and Stein's Real Analysis; they are all excellent texts. Dr. Folland is a master on the subject and makes the materials very understandable and intuitive.

This book deserves at least 4 in my opinion. This is an advanced undergraduate or graduate level textbook. Lacking examples? Create yourself. This is one of the ways to know how maths objects behave. Definitions are not motivated? Good, motivate yourself. Too terse? It should not be that terse, after one has seen concrete treatment of Lebesgue integration, also having accquired the mathematical maturity from Weirstrass, Arzella Ascolli, Baire Category Theorems, and Fourier Analysis (Pugh's Real Mathematical Analysis, Carother's Real Analysis, Stein's Fourier Analysis are excellent) Being terse can be a good thing: intuitionise and geometrise the materials. Try to think of a way to present them in a manner so that people with (honours) 1st year calculus (Spivak, Apostol) can listen as stories.

Regarding someone who reviewed the book, yes a picture is worth a thousand words. But eventually, seeing others' pictures is not sufficient; we need to learn to be mature enough to draw pictures ourselves. Folland provides the platform.

I believe the negative reviews come from the fact that the readers have not yet had the sufficient preparation before coming to Folland (Poor organisation of courses exists everywhere, not students' fault). The book itself is actually a very good book to learn from.

too dense, no example
I am an engineering graduate student. however, I am not afraid of rigorous math at all and I had passed quite a few courses in graduate math before taking real analysis (with this as a text book). This book is too dense for a self study, even with a help of a professor you cannot grasp the idea until a few examples are given. The problems are difficult which is a plus, but you cannot solve them by reading 2 or 3 pages full of theorems and proofs with no intution or good explanation (even some proofs were left as an exercise!). this book is excellent if you know the subject and you want a book as a reference in your library which has everything in the shortest possible way with the proofs as short as possible, but never ever try to rely on this for a text book to learn the subject ... Read more

Product Description
The theory of the Lebesgue integral is a main pillar in the foundation of modern analysis and its applications, including probability theory. This volume shows how and why the Lebesgue integral is such a universal and powerful concept.

The lines of development of the theory are made clear by the order in which the main theorems are presented. Frequent references to earlier theorems made in the proofs emphasize the interdependence of the theorems and help to show how the various definitions and theorems fit together. Counterexamples are included to show why a hypothesis in a theorem cannot be dropped.

The book is based upon a course on real analysis which the author has taught. It is particularly suitable for a one-year course at the graduate level. Precise statements and complete proofs are given for every theorem, with no obscurity left. For this reason the book is also suitable for self-study. ... Read more

Customer Reviews (2)

couldn't be better than this
This book is great for graduate students studying real analysis.
There are tons of problems and proofs helping you understand the details.

Excellent book!
It's a good book for 1st year grad students who try to learn the technique of proof! ... Read more

Customer Reviews (1)

Good for Review
Like the other books of the Schaum's Outline series, this book goes through the theories and shows you how to work the problems step by step.Although this is very helpful, I like to be able to test my knowledge.I say this is good for review because you will remember as you read this.You will also have a good idea of whether you are doing the supplementary problems correct or not.Here, most the supplementary problems do not have answers.Those that do, just have the final answer rather than an illustration of how the writers got the answer.This is difficult if you are learning about real variables for the first time.

As well as giving you the basic concepts, this book also goes into Lebesgue integrals, Riemann integrals, and Fourier series.The explanations are fairly well written, and the examples are easy to follow. ... Read more

Product Description

This text is a single variable real analysis text, designed for the one-year course at the junior, senior, or beginning graduate level. It provides a rigorous and comprehensive treatment of the theoretical concepts of analysis. The book contains most of the topics covered in a text of this nature, but it also includes many topics not normally encountered in comparable texts. These include the Riemann-Stieltjes integral, the Lebesgue integral, Fourier series, the Weiestrass approximation theorem, and an introduction to normal linear spaces.

The Real Number System; Sequence Of Real Numbers; Structure Of Point Sets; Limits And Continuity; Differentiation; The Riemann And Riemann-Stieltjes Integral; Series of Real Numbers; Sequences And Series Of Functions; Orthogonal Functions And Fourier Series; Lebesgue Measure And Integration; Logic and Proofs; Propositions and Connectives

For all readers interested in real analysis.

Customer Reviews (3)

Real Analysis purchase
Super fast shipping!Was exactly what was described.Would luv to buy from this seller again

Professors Should Choose Another Book
This was my undergraduate textbook for Advanced Calculus I and II (as they were called at my school).I am returning to school to start my master's degree this next term and am going through the book to refresh my memory.

Wow, it is just the way I remember it.Frankly, I can't believe the other reviewers ratings.So, I thought I'd balance the average rating a bit with a review of my own.

When I was in college, this was my most dreaded reading material.It is a difficult subject to master, sure, but the author does not help matters by using failing to use a clear structure with emphasis on key points.Instead, there is barely any structure at all.Headings consist of unenlightening phrases such as "Theorem."Pragraphs are downplayed by the typesetting style as well, making each section almost an undifferentiated block of information.(The author has not even used an end-of-proof symbol!)

And not only is this book unfriendly, it is dry.The author tends to use strictly symbolic language when explaining in words would be so much clearer.In fact, he frequently skips the explanatory material altogether and moves straight to the examples.What is the context or object for these examples?The reader is mystified.

If you are a professor, please do not choose this book for your analysis class.I have a feeling it is only comprehensible to those who already thouroughly understand the material.

If you are a student who has come here to buy this book, you have my sympathy.

Another excellent Real Analysis Text
The style of this book is a bit like Robert Bartle's Introduction to Real Analysis. It is detailed and rigorous. It is an excellent book for those who want to learn Real Analysis. ... Read more

Product Description
This volume provides a comprehensive introduction to the theoryof spline functions. Emphasis is given to new developments, such asthe general Birkhoff-type interpolation, the extremal properties ofsplines, their prominent role in the optimal recovery of functions,and multivariate interpolation by polynomials and splines.
The book has thirteen chapters dealing, respectively, withinterpolation by algebraic polynomials, the space of splines,B-splines, interpolation by spline functions, natural splinefunctions, perfect splines, monosplines, periodic splines,multivariate B-splines and truncated powers, multivariate splinefunctions and divided differences, box splines, multivariate meanvalue interpolation, multivariate polynomial interpolations arising byhyperplanes, and multivariate pointwise interpolation.
Some of the results described are presented as exercises and hints aregiven for their solution.
For researchers and graduate students whose work involvesapproximation theory. ... Read more

Product Description

This revised and enlarged second edition is devoted to asymptotical questions of the theory of entire and plurisubharmonic functions. A separate chapter deals with applications in biophysics. The book is of interest to research specialists in theoretical and applied mathematics, postgraduates and students who are interested in complex and real analysis and its applications.

Product Description
This volume provides a comprehensive, up-to-date survey of inequalities that involve a relationship between a function and its derivatives or integrals. The book is divided into 18 chapters, some of which are devoted to specific inequalities such as those of Kolmogorov-Landau, Wirtinger, Hardy, Carlson, Hilbert, Caplygin, Lyapunov, Gronwell and others. Over 800 references to the literature are cited; proofs are given when these provide insight into the general methods involved; and applications, especially to the theory of differential equations, are mentioned when appropriate. This volume will interest all those whose work involves differential and integral equations. It can also be recommended as a supplementary text. ... Read more

Product Description

Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Many developments of the basic theory since its inception arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.

The theory will probably enjoy substantial further growth, but even now a connected account of the mature parts of it makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.

This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.

Product Description
The main goal of this book is to demonstrate the usefulness ofset-theoretical methods in various questions of real analysis andclassical measure theory. In this context, many statements and factsfrom analysis are treated as consequences of purely set-theoreticalassertions which can successfully be applied to measures and Bairecategory. Topics covered include similarities and differences betweenmeasure and category; constructions of nonmeasurable sets and of setswithout the Baire property; three aspects of the measure extensionproblem; the principle of condensation of singularities from the pointof view of the Kuratowski-Ulam theorem; transformation groupsand invariant (quasi-invariant) measures; the uniqueness property ofan invariant measure; and ordinary differential equations withnonmeasurable right-hand sides.
Audience: The material presented in the book is essentiallyself-contained and is accessible to a wide audience of mathematicians.It will appeal to specialists in set theory, mathematical analysis,measure theory and general topology. It can also be recommended as atextbook for postgraduate students who are interested in theapplications of set-theoretical methods to the above-mentioned domainsof mathematics. ... Read more

Product Description
This monograph is devoted to the study of Köthe-Bochner function spaces, an area of research at the intersection of Banach space theory, harmonic analysis, probability, and operator theory. A number of significant results---many scattered throughout the literature---are distilled and presented here, giving readers a comprehensive view of Köthe-Bochner function spaces from the subject's origins in functional analysis to its connections to other disciplines. Key features and topics:* Considerable background material provided, including a compilation of important theorems and concepts in classical functional analysis, as well as a discussion of the Dunford-Pettis Property, tensor products of Banach spaces, relevant geometry, and the basic theory of conditional expectations and martingales * Rigorous treatment of Köthe-Bochner spaces, encompassing convexity, measurability, stability properties, Dunford-Pettis operators, and Talagrand spaces, with a particular emphasis on open problems * Detailed examination of Talagrand's Theorem, Bourgain's Theorem, and the Diaz-Kalton Theorem, the latter extended to arbitrary measure spaces. ... Read more

Product Description
The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results. ... Read more