Product Description
Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, series, the derivative, higher derivatives, the integral and more. Each chapter contains a problem set (hints and answers at the end), while a wealth of examples and applications are found throughout the text. Over 340 theorems fully proved. 1973 edition.
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Customer Reviews (9)

The price is right...
Can't beat the price, and the material is well-presented and organized, but it's stripped down to the bare essentials - theorem, proof, lemma, corollary, etc. It's not a book on proof methodology, for sure. I graduated with a degree in computer science, but I haven't done a proof for a while and never took a class beyond linear algebra, and I wanted to teach myself analysis. While I don't find the material too difficult to follow, I really don't find it all that great for self-study. The book yields conclusion after conclusion, but among all the results, I find the text doesn't do a great job of conveying its methodology. In other words, the book spends the vast majority of the time developing new results (the "what" of analysis), but it does little to prepare the reader to understand the "how" of analysis. I feel as though the book is giving me a fish, rather than teaching me to fish.

And there are some idiosyncracies. You need to be wary of an occasionally swapped subscript, for instance. And in chapter 1 problem 5: Which is larger, Sqrt(3) + Sqrt(5) or Sqrt(2) + Sqrt(6)? The answer in the back of the book is plain wrong. And the book proves something as fundamental as the uniqueness of 1; and yet it invokes the binomial theorem out of the blue?

Anyway, the price is right, but beware that it might make a better reference or a collection of examples than a primary self-study guide.It's not that it's "too easy" as one reader put it; rather, it doesn't integrate the material with exercises and explanations well enough for my liking.

It is one very interesting book
To me, the best chapters of this book are that about series and integrals. The text is plenty of interesting notions, like that of direction that is related with the notion of limit. I appreciated very much the study that Shilov does about parameter-dependent proper and improper integrals. The topologicalnotions are placed in one intuitive manner. Without doubt, this is one very good and clear exposition about the subject. However, I think that the problems are not easy. Also sometimes Shilov states the theorems with additional conditions that are not useful. For example, this happens usually in the chapter about derivatives because the definition of derivative given by Shilov imposes that any function with derivative in the interval of the domain has continuous derivative in the interior points of its domain. However, Shilov charges some theorems with the extra condition of continuous derivative.
When the Taylor's formula is presented in page 252 - Theorem 8.22, it is stated that the error of the approximation is computed in some interior point of the interval, what is not completely correct. For example, take the second degree Taylor's approximation around x = 0 of the function x raised to the third power, and you will see that in this case the error is computed on one extreme point of the interval.
Also the proof of the theorem 10.49b (page 415) has logical problems of the kind that may arise during the translation.
However, these remarks are small questions without consequences for the course of the exposition.

An excellent pure maths text.
I purchased this book to study some complex analysis.Being a physicist I would like to brush up on this.The book was completely different to what I expected.

Some applications would have been nice, but this text is pure maths.The book is well written, easy to follow and concise.I ended up reading it and gained and appreciation for the thorough consideration of elementary real and complex numbers.

Shilov is thorough and avoids making leaps and assertions.This would make the book readable to lower undergraduates.However the significance of some things is not explained, or explained in a very dry manner so people might miss this.

I highly recommend this book if you are interested in real and complex analysis from a pure mathematics perspective.

Getting started in math analysis
This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time.
Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book.
Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin.
There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.

Possibly too simple
As Shilov write in the introduction "I have tried to accomodate the interests of larger population of those concerned with mathematics" and at that he seems to do. However, the book does require some mathematical background as he appears to omit defining a few things. I believe the book would be ideal for those who want a handy reference, or an easier book when struggling with an analysis course.

However, for the more mathematically inclined readers, the problems are often too easy, and many things are proved that could be better left as exercises. For a more difficult Analysis book, I would reccomend Rudin. ... Read more

Product Description
Comprehensive, elementary introduction to real and functional analysis. Self-contained, readily accessible to those with background in advanced calculus. Cover basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, much more. 350 problems.
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Customer Reviews (32)

Clarity and examples
This is a clearly written book with lots of examples at an amazingly low price. I once asked a professional economist what books would cover the math used in his work and he pulled two books off his shelf, Kolmogorov/Fomin's Analysis and Luenberger's Optimization. So I tend to think pretty highly of this book in terms of usefulness. That said it's tough to go through as the authors don't spend much time on motivation and seem to jump from topic to topic. I found it easier to understand on a second perusal after I had taken an analysis course.

Good intermediate math book
This is a very good intermediate math book. I used it to write my undergraduate monograph and it actually helped a lot (I'm an economics student). However, it is difficult to understand without the help of other books. In fact, if you want to use this book I recommend to get also: "Topology" by Munkres and "The Way of Analysis" by Robert Strichatz. They all make a very useful math kit and if you are thinking in a Ph.D. in economics they can help you a lot if you read them (not all, buy selected chapters) before you start the math review at the begining of the Ph.D. program.

DON'T buy this book!
This is not the original work. Just like Emily Dickinson's poems, sometimes you get an "edited" version of a beautiful mathematical work. I am SO sorry that I bought this book. Besides the problem with messing with a finished work, there are serious errors in proofs all over the book.

Get Elements of the Theory of Functions and Functional Analysis instead (also by Dover).

Not A Bad Book
I didn't like this book at first because it wasn't what I expected. I think the word "introductory" should be removed from the title. It's actually not an introductory book. I would recommend this book for graduate studies. Most real analysis courses at the graduate level focus on the measure theory and integration. However, I appreciate this book more now.

A good read with too many mistakes
The advantages of this text have been pointed out by other readers, so I will attempt to exhibit the problems of this book.

There are a lot of mistakes. And by 'a lot', I mean that the careful reader should be able to find at least 5 mathematical mistakes in each chapter. I used this text mainly as a supplement to a fairly advanced analysis course, and we'd often have problems from it used in our problem sets. At first, it appeared as if this were a very well-written text, but once we started with our problem sets, there were at least 2 e-mails sent out per week addressing a concern a student had pointed out. After a while, students stopped e-mailing the professor with their concerns, instead just assuming that they were correct whenever they spotted something weird.

Let's take an example:

Problem 1, pg. 137: Let M be the set of all points x = (x1, x2, ..., xn, ...) in l2 satisfying the condition \sum^{\infty}_{n=1} (n^2) (x_n)^2 \le 1. Prove that M is a convex set, but not a convex body.

The problem with this is that M IS easily a convex body, precisely because x = (0,0,...) is in M.

There are many more big mistakes and little mistakes throughout the exercises, oftentimes destroying the entire POINT of the problem. Take, for example, Problem 1 of pg. 76: Let A be a mapping of a metric space R into itself. Prove that the condition p(Ax,Ay) < p(x,y) (x\ne y) is insufficient for the existence of a fixed point of A.

Now, a counterexample here can be easily produced, even by the most elementary reader. But the exercise quickly becomes worthwhile if we make R complete. It's the little things that count in mathematics, and the small errors like these are clearly detrimental to the student.

But the errors in the text aren't limited to exercises. I was reading independently at the front of the book to get some info on Zorn's Lemma and ordinal numbers, and as I read, I found the following definition:

Let M1 and M2 be two ordered sets of type 01 and 02, respectively. Then we can introduce an ordering in the union M1 U M2 of the two sets by assuming that

1) a and b have the same ordering as in M1 if a,b are in M1
2) a and b have the same ordering as in M2 if a,b are in M2
3) a < b if a in M1, b in M2

The set M1 U M2 ordered in this way is called the ordered sum of M1 and M2, denoted by M1 + M2.

There is a clear problem with this definition pointed out here: http://www.physicsforums.com/showthread.php?t=200985. How is the student expected to learn such important material when even the definitions have loopholes?By the end of our experience with this book, our professor was giving out exercises to correct Kolmogorov/Fomin's incorrect definitions.

Note also that Silverman uses very weird words. For example, 'countably compact' is used instead of 'sequentially compact'. ... Read more

Product Description
The well-known and respected authorship team of Geltner and Miller bring you a new edition of what has become the undisputed and authoritative resource on commercial real estate investment. Streamlined and completely updated with expanded coverage of corporate and international real estate investment, this upper-level text presents the essential concepts, principles and tools for the analysis of commercial real estate (income producing) from an investment perspective. This new book continues to integrate relevant aspects of urban and financial economics to provide users with a fundamental analytical understanding and application of real estate investments ? now using ARGUS software. Contributing author Piet Eichholtz from the University of Maasstricht contributes an entire chapter that explores international real estate investments, both opportunistically and structurally, by outlining elements for developing and implementing real estate investments successfully abroad. Jim Clayton from the University of Cincinnati thoroughly revised and updated the finance coverage and real-life applications throughout. Geltner and Miller enhance their pedagogy by adding in a discussion of the real options application to real estate development and streamlining the discussion of data returns. ... Read more

Customer Reviews (19)

Great Book, Great Exercise
This book was all that I expected. It is an excellant source book for commercial real estate analysis.

An added benefit is that it weighs a ton and therefore provides a good weight training workout.

Great book, but you need to compliment it with others...
Overall, my favorite real estate book and the one I consult most often.

At the risk of oversimplification, commercial real estate investing can be broken apart into 3 activities:

1) Figuring out how much cash a property will generate (e.g. pro-forma property financials)
2) Figuring out how to discount that cash for time and risk (e.g. discount/cap rates)
3) Figuring out how to slice this asset value into various claims against it (e.g. mortgages/CMBS, preferred equity structures, etc.)

This book is amazing at #2 and only slightly less amazing at #3. In fact, I find its treatment of some materials (e.g. risk-neutral discounting as it relates to development properties) to be an excellent complement to and in some ways more informative than standard finance textbooks (e.g. Brealey / Myers). The authors do admit that their approach is slightly more academic than rules-of-thumb used in practice, but it never hurts to be more informed than your peers. If you're concerned about being too academic, also pick up the text by Peter Linneman Real Estate Finance & Investments: Risks and Opportunites; (2nd Edition), which covers similar material in a less academic, though still logically sound manner.

I found, and many of you may find, the treatment of #1 (cash flow forecasting) somewhat limited. The book does deal in Part II with urban economics and in Part IV with property-level pro-formas, but these topics fly well above the depth available in other texts.

I recommend that you compliment this text with something on market analysis...I own Real Estate Market Analysis: A Case Study Approach. A text like that gets you up the learning curve on key supply-and-demand metrics for each collateral type. There are also books on real estate operations (e.g. property management and loan servicing) that could compliment the Geltner text if you're into that level of detail.

Side note: make sure to print out the appendices. They are a very useful supplement to the text.

Great book for the commercial real estate professional
This is an excellent text book- I found it very enlightening. Please note that I said TEXT book. I would never suggest this book to the novice investor. With some of the material, it helps to have had a college level finance course to fully benefit from all the information. However, this book covers just about everything related to commercial real estate including the commercial mortgage backed securities market and other concepts. The sections are well written and provide relatively simple and understandable examples that help illustrate the concepts. This book should be within the reach of any commercial real estate pro!

lots of good information
Although I haven't finished reading this text, I think I can accurately comment on it. The information comes from an economics perspective. that's a good thing, if you have an economics background, which I do. I like the way this book reads. It offers a good deal of information about commercial real estate using both financial and urban economics. I don't believe this book is appropriate for an undergrad student. This book assumes you have a basic finance background and some understanding of urban economics. In short, if you have an econ degree or even a finance degree, I think you'll find this text very useful in helping you to understand what drives the commercial real estate market, how to predict where this market is going, how to analyze it, and how to valuate it.

Great Advanced Text on Commercial Real Estate
This is one of the most comprehensive, analytical and thorough texts on topics concerning commercial real estate investment that I have come accross. It would be useful for both advanced students with real estate interests and finance professionals, who seek exposure to fundamental real estate (and real estate securities) analysis. Participants in the real estate industry mishgt find the text useful, but basic. The book is also useful as a reference guide for real estate professionals looking for a particular mathematical formula.

The authors cover a very broad range of topics - from urban economics, to fundamental (supply/demand) real estate analysis, to real estate valuation techniques as well as more specialized topics, such as commercial mortgage backed securities and real estate development. The book really stands out in the breadth of its disourse both on qualitative and quantitative topics.

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Product Description

Real Analysis, Fourth Edition, covers the basic material that every reader should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis.

Product Description

Customer Reviews (2)

Excellent Introductory Text for Real Analysis
The best thing about this book is its clarity and simplicity in defining real analysis concepts. The author conveys his point in a convincing and understandable way without loosing the details involved. I think its a very good text specially for those who are newer to this field.

comprehension look at this book.
This book is an excellent look at real analysis using the philosophical basis around the application of Topology.This book is also one of the easier readers that one will experience when it comes to mathematics. ... Read more

Product Description

Offering the tools needed to evaluate trends and understand key factors affecting the real estate market, this book explains how to get started, where to get information, and how to apply the basic techniques to a variety of development types. This practical primer offers a step-by-step approach to developing property—whether public or private sector—and shows how market-analysis methods have been employed in real projects. The 13 case studies written by top market analysts provide models that can be applied to multifamily, hotel, office, industrial, entertainment, mixed-use, and/or master-planned communities. This is an excellent reference for real estate students and professionals to maximize potential in a troubled market.

Customer Reviews (2)

Not Quite a Doorstop
Critic and curmudgeon Ambrose Bierce once dismissed an author's efforts with the review: "The covers of this book are too far apart."Schmitz and Brett tempt me with the flip side of Bierce's proposition; the covers of this book should be about 300 pages further apart. Its 220+ pages of well written, well organized coverage of how market analysis applies to vital development markets remains - utterly unsatisfying.

The book seeks to cover everything an analyst could hope for:

* basic real estate market analysis goals and concepts (Chapter 1),
* basic approaches to real estate market studies (Chapter 2),
* sector specific considerations for various market segments
(residential, office, industrial, retail, hotels, and mixed use
developments - Chapters 3 through 7), and
* a baker's dozen case studies to illustrate the concepts.

I started with high, soon to be dashed, expectations.

Real Estate Market Analysis - A Case Study Approach is published by The Urban Land Institute (ULI). ULI also published the brilliant but stodgy landmark treatise on real estate development by Miles, Berens, and Weiss (MB&W). Most notably, ULI is recognized as the ideological vessel of James Graaskamp. Graaskamp posited the (then radical, now established) tenet that the public and the body-politic are major stakeholders in development. Either can propel or sink a project. Hence, a case study approach from ULI that devoted only three paragraphs to the importance of public and political awareness in market analysis was monumentally disappointing. Ironically, Schmitz and Brett preface their minimal coverage of the topic with the observation:

"Given the often contentious environment associated with the proposed closure and reuse of a military facility, public participation is extremely important."

They then present, without comment, a two bullet point chronology of this "extremely important" topic. Worse, the topic was the El Toro Marine Base reuse/mixed use plan. Public debate about El Toro persisted for seven years before a tentative plan was finally adopted. That largely unexecuted and heavily litigated proposal remains controversial eight years later.

Lapses of analytical and logical rigor are pervasive.

For a book on market analysis, the Schmitz and Brett work contains startlingly little analysis. Not for them digressions about how and under what circumstances one approach may be preferable over another. Schmitz and Brett content themselves with vacuous observations like:

An economic model is used to evaluate the relative benefits of each scenario over a twenty year period, or to buildout; the model typically presents five year development phases. It is based on assumptions generated by the market analysis and yields the following outputs by land use: total units or square feet, on site employment, total employment generated, land sale revenue, total output, and total income.

Without bothering themselves or their readers with hoary little details like: Which economic model, what assumptions, why are they relevant, how do we know that, and why are we concerned with that particular set of outputs? Unhelpful "directives-without-direction" further undercut Schmitz and Brett. I find the directive that an analyst should analyze some data and reach some conclusions about something singularly purposeless.

Schmitz and Brett are equally cavalier with logic. They can reason from the specific to the general based on a single observation couched in a unique example. For balance, they sometimes reverse the algorithm and reason from the general to the specific with equally little basis. They are particularly prone to this annoying form of ir-ratiocination when drawing conclusions from their case studies. None of their conclusions seem false, but it would be nice to know that their results are based on more than a wing, a prayer and the authors' say-so.

It's a shame that the authors offered no after-action evaluations or any other effort to validate those conclusions. To know that they employed (an unspecified) model on (unspecified) data using (unspecified) analytical techniques based on (unspecified) assumptions is exciting. It sounds so scientific. It would be nice to know whether it worked. Never mind finding out that it didn't work and why not.

That being said, the book does not descend to doorstop status. According to their preface; "Real Estate Market Analysis was conceived as a practical guide for analyzing the market potential of real estate development" and "explains the nuts and bolts of how to collect and organize data and analyze demand and supply." Their aims are laudable, but its best use is as text for guided discussion with a knowledgeable facilitator. The material is there, and for the novice, it outlines a useful conceptual framework and presents excellent information about major data sources, but it is woefully weak on the nuts and bolts. It's an easy if not very satisfying read.

A Thorough Reference
This book is written like a textbook - starting with the most basic concepts - but moves on to discuss the tools and techniques of market analysis in considerable detail. Separate sections deal with residential, office and industrial, retail, hotels and resorts, and mixed-use developments. Case studies are given for each type of property. Technologies and data sources to assist analysis are discussed.

The two primary authors are a real estate consultant and an author/director of the Urban Land Institute. Many other professionals are listed as having contributed to the case studies.

The book includes an appendix of data sources of use to market analysts, a glossary of real estate terms and a thorough index. ... Read more

Product Description
In recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context. Like the first two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral. ... Read more

Customer Reviews (29)

Real Analysis
This book came in great condition. It looks brand new, no marks, and half the price as anywhere else I could find.

If i wanted a book with bent corners, i could have bought used
I had many options at the time that i purchased this book. I could have ordered a used book in excellent condition for far less than a new one..one in good condition with only bent corners for even less.. But on the other hand,if i bought a new one i could sell it when my class was over and recover most of my money. I chose to go new. When i recieved the book the corners were all bent because there was nothing in the box to protect it. I guess just beware..if your buying a very expensive text book and plan to resale it when your done..look for the best deal on used because the high price of new will not get you a much better book!!

A relatively difficult but extremely rewarding book
I used this book all through my freshman and sophomore years. I struggled with it and really tried hard to solve the problems. I mulled over every word and every proof of the first few chapters for days. However at the end of this rather trying exercise, I had a real sense of satisfaction and thought I had a firm grasp of the basics of real analysis. This is not one of those volumes that will instantly get you hooked but it's a volume that rewards patience like few other math books do. If you master the first few chapters of the book including those on series and sequences, you should have as good as grasp of elementary real analysis as anyone else. The book is for the serious student of mathematics and it provides a rigorous and comprehensive introduction to real analysis. To master it you will have to read and understand the proofs as carefully as possible; don't be discouraged and become impatient if you cannot do this easily, since the time spent on doing it is worth it. However, having a good instructor to help out will be enormously useful.

Way better than Pugh
What a breath of fresh air after dealing with Pugh's book!The language is clear.The proofs are concise and easy to follow.The illustrations are good without being overwhelming.I cannot say enough good things about this book.Poor math teachers are obsessed with the most general case and introduce it first.A good teacher starts with a specific case, relates it to what the student already knows, and then begins to generalize it slowly, layer by layer until the most general case is achieved.This is how the mind works, this is how mathematics really developed over time, and this is how math should always be taught!Bartle and Sherbert do a outstanding job of this!

One word of caution.Don't let real analysis be your first proofing class.Take a proofing class first and if your university doesn't have one, demand one!.Real analysis is not the place to learn propositional logic, quantifiers, truth tables and the like!.Learn that stuff first and do your first proofs in elementary number theory or geometry, then when you have a repertoire of proofing tools and some skill in proofing, then take real analysis.You cannot learn proofing and real analysis at the same time.First learn to proof, then take real analysis! If not you will be miserable and you will take it out on your teacher and text!

Really hard, but Really good
Like some others, I really disliked Real Analysis at first b/c the proofs were so much more complex than anything else I had seen. I struggled, ordered other analysis books to help me, only to find that this one really is good! You do need a great instructor to go with this book or you may be lost. That said, the appendices are fantastic and the authors give "hints" (and some answers) to selected problems. The proofs themselves are terse, so without an instructor who understands the gaps, you may not connect the steps solo. Good text which is now part of my math library. ... Read more

Product Description
Focusing on one of the main pillars of mathematics, Elements of Real Analysis provides a solid foundation in analysis, stressing the importance of two elements. The first building block comprises analytical skills and structures needed for handling the basic notions of limits and continuity in a simple concrete setting while the second component involves conducting analysis in higher dimensions and more abstract spaces.

Largely self-contained, the book begins with the fundamental axioms of the real number system and gradually develops the core of real analysis. The first few chapters present the essentials needed for analysis, including the concepts of sets, relations, and functions. The following chapters cover the theory of calculus on the real line, exploring limits, convergence tests, several functions such as monotonic and continuous, power series, and theorems like mean value, Taylor's, and Darboux's. The final chapters focus on more advanced theory, in particular, the Lebesgue theory of measure and integration.

Requiring only basic knowledge of elementary calculus, this textbook presents the necessary material for a first course in real analysis. Developed by experts who teach such courses, it is ideal for undergraduate students in mathematics and related disciplines, such as engineering, statistics, computer science, and physics, to understand the foundations of real analysis. ... Read more

Product Description
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. ... Read more

Customer Reviews (23)

Great Learning Experience
This is truly a well-crafted book. The organization is tight and the book is largely self-sufficient, really only calling upon material covered in his previous book, Principles of Mathematical Analysis. Rudin only proves such lemmas as he needs to get to the major results; peripheral facts and theorems are left to the exercises, which are far from routine and require a lot of thought and patience to get through. The book is rather terse, but there's a powerful elegance behind it. The proofs are rigorous and avoid hand-waving, albeit Rudin intentionally leaves gaps for the reader to fill. Working carefully through this book and doing as many exercises as possible, one can get a good grasp of the material.

However, this is not a book I would recommend for people who have not had prior experience with measure theory or complex analysis. Rudin provides little motivation. His arguments are elegant, but his methods are designed to give the quickest route to the results, which is seldom the most intuitive one. For example, where other authors explicitly construct Lebesgue measure, Rudin opts for a difficult proof of the Riesz Representation Theorem from which he pulls Lebesgue measure as a magician might pull a rabbit out of a hat. A lot of the material has a lot of geometric intuition behind it, but Rudin seldom goes out of his way to point it out. The beginner would be much better served getting an introduction to the material elsewhere before tackling Rudin. It will enable one to get much more out of the book than slogging through it blindly. It's one thing to know the definitions and be able to follow the proofs, and another thing to understand what it all means; Rudin doesn't hold the reader's hand where the latter is concerned.

For those with adequate preparation and sufficient daring, this book makes for a great learning experience and I daresay is even fun in places.

One of a Kind
I normally don't review books that already have this many reviews, especially when I agree so much with the reviews that already exist.But I'm teaching out of green Rudin for the first time this semester, 20 years after getting to know the book well as a student, and I find myself so enthusiastic about it again, that I just had to chime in with an "Amen" to the other positive reviews.When it comes to mathematical writing, it doesn't get any more exquisitely elegant than this.

Probably all our reviews are irrelevant, however, because there are probably very few discretionary purchases of this book:There will be nearly a one-to-one correspondence between buyers of the book and students in classes for which it is required.For them, I can only recommend skipping the outrageously expensive hardback (which even at its high price is pretty cheaply constructed nowadays) and opting for the more reasonable international paperback edition.

Rewarding, but has limitations
This is a very nice book, but in my opinion it is the worst of Rudin's three well-known books on analysis.The third, Functional Analysis, is a better representation of the subject, and (as opposed to this book) contains applications of analysis to other areas of mathematics, such as number theory and PDE.

The chapters here are very good for reference, but they are not well-written (at all) for self-study.So my two cents is: it's good, but frustrating at times. Rudin has a talent for making difficult things clear, but he has a more sinister talent for making simple things appear difficult.

I love this book!
I love this book, even though I have not absorbed more than a small portion of it yet.I find this to be a much better book than the "baby Rudin", which struck me as dry, overly concise, and without motivation.This book provides ample motivation, and although it proceeds in great generality, proceeds at a reasonable pace.

The best thing about this book, however, is the spirit of it--the integrated approach to analysis that Rudin takes is unique and greatly appreciated--Rudin is, like Lang, a testimony to the fact that the best mathematicians do not draw artificial lines between different areas within mathematics.Rudin presents the material in ways that connect to other areas of mathematics and will help the reader become a better mathematician, even if she never directly uses any of the material contained in this volume.

I would not recommend this book as a first exposure to measure theory or complex analysis--it is advanced and requires a great deal of background to fully understand and appreciate.But I think this is a book that any serious mathematician should add to their collection and eventually work through.People wanting to learn measure theory might look to the book by Inder K. Rana, or to the classic book by Royden.For more elementary treatments of complex analysis I would recommend the classic by Ahlfors, Theodore Gamelin's book, or the book by Greene and Krantz.

My 2 cents
There are some excellent reviews here for this outstanding book, so I will try to avoid repetition. In preparation for my qualifying exams in graduate school, two of my colleagues and I did all of the exercises in Rudin (give or take a couple, no more). What I found striking at the time was how Rudin took three subjects -- measure theory, functional analysis, and complex analysis -- and weaved them together seamlessly. It is not that I believed them to be separate subjects, but until then I hadn't realized just how they all fit together. Really, this book is superb.

A word of warning, though. Rudin's prose is concise, and his proofs leave you wondering if you'd ever be able to reproduce them on your own. It is what we in the business are used to call 'elegant'. It pays to work in groups, persevere, and go over everything twice or more. Good luck. ... Read more

Product Description

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.

After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.

As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.

Also available, the first two volumes in the Princeton Lectures in Analysis: ... Read more

Customer Reviews (6)

Great book
This book is very nice, concise and still clear to read. I still did not do a lot into it (only chapter one so far).
I taken a course in Analysis before and decided to read this book just to review and to study the subject through a different perspective.

Not as good as the classics
I just completed a first-semester graduate course in which we used this textbook, and I was very disappointed by this choice. The authors too often gloss over details and omit definitions. Plus there are a few minor mistakes or non-standard definitions (check out the definition of "limit point" on page 3!). It reads much more like a lecture than a textbook, and I found it frustrating not to have a thorough resource to fall back on when my own professor's lecture was unclear. I have always prided myself in my ability to learn from a textbook, as I had no difficulty following Munkres in his "Topology" or Dummit and Foote in their "Abstract Algebra." However, I found this real analysis text to be quite challenging to follow time and time again--even our professor commented on how some proofs were unnecessarily complicated and how certain "trivial" details that had been omitted were not quite so trivial and indeed deserved mention. The only reason I did not give this book one star is that I found the problems to be good.

I am getting ready to purchase a copy of Royden's "Real Analysis" to help me study for my qualifying exam. I wish we had used it all along!

great book
i found the first three chapters of this book very clear and well written. i'd strongly recommend it for someone looking to learn about analysis on the real line.

Good book for reading and as a graduate student
Easy to read. My university is using this book to get the graduate students ready for the real analysis qualifying exam. So go ahead and buy this book if you're planning to work on a PhD in mathematics. If you're not planning to work on a PhD in math, this is still a good book to read if you enjoy studying about the real line.

The book begins with measure theory, integration and differentiation. These are included in Chapters 1 to 3. Then in Chapters 4 and 5, we look into Hilbert spaces. This is similar to studying finite-dimensional inner-product spaces, but here, Hilbert space is infinite-dimensional. However, the analysis is very similar. If you know some linear algebra, it should feel like as if you have already read these two chapters.

Finally in Chapters 6 and 7, we see abstract measure theory, including Hausdorff measure, and we study fractals and self-similar sets. And this concludes the book.

Also recommend Walter Rudin's Real Analysis.

Suffers from all the flaws of a 1st edition
This book has a lot of problems. Several sections are poorly written/edited. Several important named theorems are not clearly labeled. Also some of the proofs contain typos or errors. The chapter on differentiation is particularly lacking. The chapter is poorly organized and presented. There is also a glaring TeX error in the chapter.

At Princeton this book is used as part of an undergraduate course, and it shows. This is not the ideal book for a graduate level course in real analysis(though I think it would be very well suited for advanced undergrads). Too much time is spent on Lebesgue measure and integration in the first 2 chapters, and abstract measure theory is not intoduced until chapter 6. Also the Monotone Class theorem is lacking from the chapter on abstract measure theory. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).

On the other hand, the presentations of Lebesgue measure/integration and Hilbert spaces in the book are pretty good. The exercises and problems in teh book (when stated properly) are very good and instructive. Overall this book has a lot of potential to be very good, but seems to be suffering from a lack of revision. Hopefully these issues will be fixed in later editions. ... Read more

Product Description
A collection of problems and solutions in real analysis based on the major textbook, Principles of Real Analysis (also by Aliprantis and Burkinshaw), Problems in Real Analysis is the ideal companion for senior science and engineering undergraduates and first-year graduate courses in real analysis. It is intended for use as an independent source, and is an invaluable tool for students who wish to develop a deep understanding and proficiency in the use of integration methods.
Problems in Real Analysis teaches the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in Principles of Real Analysis, Third Edition. The problems are distributed in forty sections, and cover the entire spectrum of difficulty.


... Read more

Customer Reviews (4)

Very good for struggling in Real Analysis
There is no problem solutions for Real Analysis texts available in U.S. since most of the teachers believe that Math students in grad level should be more creative. However, not all students in Real Analysis are potential mathmatician. If they lost in class and must study by theirselves, they may feel frustrated missing all the stuff contained in problems. The material in "Principles of Real Analysis" may not superior much than other famous text(like Royden, I think Royden is clear enough but too much mysterious things lie in problems.). But use it with this workbook, you will find much comfortable in self study. It helps a lot not only in my homework assignment, but also in my understanding.

A must have!!!
This book rocks! It covers practically all the major topics of an introductory course in graduate Real Analysis. Excellent solutions that aid in the understanding of the material. This book's worth is immeasurable, or should I say, non-measurable.

Must have one
Great guide, must have for anyone taking Real Analysis.

Must have for anyone taking Real Analysis
Before buying this book, I was failing Real Analysis.Now I have a prayer of passing.Thank goodness I found it in time.For it to be of use, you need to buy the companion book "Principles of Real Analysis",same authors.On the down side, it doesn't have an index, but overall,well worth the money.Besides, it is the only source I've found ofworked-out Real Analyis problems outside of borrowing from other studentswho have already taken the course. ... Read more

Product Description

This is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration.

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Customer Reviews (24)

Good candidate for the worst math book ever
This book is absolutely horrific .Royden proves (mostly handwaves) a few theorems , and then expects the student to be able to solve the exercises easily given this background.Can we even call this a Math Book?? As it is unfortunately the case with most graduate level textbooks in the U.S , the exercises are minimally related to what the author develops (or tries to develop).Books like this do not take much effort to write , while the students trying to learn this hard subject are the real victims.I agree with another reviewer that this book was written by Royden just to impress his peers.The important theorems are just given with the most minimal explanations possible ,and no effort whatsoever is made to get to the intuitive roots of the subject.Important examples are also close to nonexistent.Can anything be worse??

Classic text, but a poor reference.
Halsey Royden's Real Analysis has become the de facto standard for teaching a graduate course on real analysis and integration.It has, however, become a bit dated.First off, the method of developing the Lebesgue integral before general measure theory is out of style.It is now generally accepted that learning the (relatively easy concept of) general measure theory first, and then the Lebesgue measure as an example, is a superior pedagogical approach.

That said, Royden is very good at explaining things in more detail; it is both a complement and a criticism that the book manages to cover a good deal less than Folland's text in almost twice the length.Complementary in the sense that the book motivates the material and gives explanations without leaving the reader to any important developments, critical in that the book is more or less useless as a reference.What's more is the fact that in the book's 444 pages it only manages to cover about half of what Gerald Folland goes through in his shorter book; this makes the ridiculous price of the book even less justifiable.

All of this said, I would still recommend this book for study.It explains well and would be a good read for self study.As for the criticisms that label the book as either "too difficult" or "too dense," disregard them.Those who make these claims are probably just not very good with Analysis.For a book that is truly awful, see M.A. Armstrong's Basic Topology.

Really Great for Certain Topics
Great for the bookshelf but really pretty hard - you need a good proof course and a year suffering through baby Rudin - and believe me, you will suffer - but will be a better person (and mathematician) for it.

Maybe good as a supplement, or a first time looking at the material
There are three books that are usually used for a first graduate course in analysis, including measure theory, namely Rudin's Real and Complex Analysis, Folland's Real Analysis, and Royden's book. Of the three, I would say Royden's book is the easiest, both in terms of the exposition, material, and exercises. Of the three, Royden is the only one to fully develop the Lebesgue measure and the associated integral before developing a more general theory of measure and integration. Furthermore, he does not develop Hilbert and Banach space theory, the very basics of functional analysis, to anywhere near the extent that Folland and Rudin do.

There is some debate as to whether it is better to start with the Lebesgue integral, and then talk about abstract integration, or the other way around. Personally, I found the development of the Lebesgue integral a bit tedious; the whole thing works a bit better when you first talk about abstract integration, which really isn't a terribly difficult concept, prove the basic integration theorems, then show how to construct an outer measure, and suddenly, the Lebesgue measure and integral just falls into place. I'm not sure anything is lost in the process.

The biggest shortcoming in this book would have to be the exercises: for the most part, they are not very difficult, particularly when you compare them to say, Rudin's text. For the most part, the exercises are fairly trivial, and if they are difficult, or require a bit of creativity, Roydenoften gives you lots and lots of hand-holding, sometimes even in the form of sketching out the proof for you. In spite of the relatively low difficulty level, most of the exercises are fairly instructive, in so far as they highlight, elucidate, and expand upon the material.

For the most part, this book is not bad. It makes a good supplement to a book like Rudin or Folland, as it is less abstract, and does a better job motivating the material. The exercises here can work well if you want some extra practice that won't take up too much time. If you're a student of econ, or physics, or you just feel like learning graduate-level real analysis, then this book is probably adequate (although I should qualify that statement by saying that I know nothing of econ and little of physics). But if you are a serious student of mathematics, particularly the pure variety, this is really not the book you should be using. It is just too easy.

Classic text on measure & integration theory
Many people criticize this book as unclear and unnecessarily abstract, but I think these comments are more appropriately directed at the subject than at this book and its particular presentation.I find this classic to be one of the best books on measure theory and Lebesgue integration, a difficult and very abstract topic.Royden provides strong motivatation for the material, and he helps the reader to develop good intuition.I find the proofs and equations exceptionally easy to follow; they are concise but they do not omit as many details as some authors (i.e. Rudin).Royden makes excellent use of notation, choosing to use it when it clarifies and no more--leaving explanations in words when they are clearer.The index and table of notation are excellent and contribute to this book's usefulness as a reference.

The construction of Lebesgue measure and development of Lebesgue integration is very clear.Exercises are integrated into the text and are rather straightforward and not particularly difficult.It is necessary to work the problems, however, to get a full understanding of the material.There are not many exercises but they often contain crucial concepts and results.

This book contains a lot of background material that most readers will either know already or find in other books, but often the material is presented with an eye towards measure and integration theory.The first two chapters are concise review of set theory and the structure of the real line, but they emphasize different sorts of points from what one would encounter in a basic advanced calculus book.Similarly, the material on abstract spaces leads naturally into the abstract development of measure and integration theory.

This book would be an excellent textbook for a course, and I think it would be suitable for self-study as well.Reading and understanding this book, and working most of the problems is not an unreachable goal as it is with many books at this level.This book does require a strong background, however.Due to the difficult nature of the material I think it would be unwise to try to learn this stuff without a strong background in analysis or advanced calculus.A student finding all this book too difficult, or wanting a slower approach, might want to examine the book "An Introduction to Measure and Integration" by Inder K. Rana, but be warned: read my review of that book before getting it. ... Read more

Product Description
This course in real analysis is directed at advanced undergraduates and beginning graduate students in mathematics and related fields.Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and nonspecialists alike. The text covers three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal, down-to-earth style, the author gives motivation and overview of new ideas, while still supplying full details and complete proofs.He provides a great many exercises and suggestions for further study. ... Read more

Customer Reviews (6)

Very Good Book
I highly recommend this text to learn from.This book is like taking a course from your favorite teacher, who loves the subject and knows it very well.And what is more, she wants to communicate her knowledge to you.She has anticipated your difficulties and is prepared to guide you. The material is well motivated; the historical notes help you understand and appreciate the material even more.There are a lot good exercises. I wish I had this book when I was studying this subject.

Excellent textbook for review, but this is not a first course.
In the author's preface, he states that the prerequisites are "one semester of advanced calculus or real analysis at the undergraduate level". So, this book cannot be judged as an 'intro to real analysis'.

I just want to comment on how I have experienced this book. Let me mentionthat I am using this for self-study after completing a course using Rudin's Principles of Mathematical Analysis (we covered every chapter except Ch. 10 on integration in R^n). I picked this up to review analysis with the goal of covering function spaces and measure theory with more emphasis that Rudin. This book does just that! But, I also wanted a book that stays in R for the Lebesgue measure. Having read the first 3 chapters of Folland, I didn't really think I 'understood' the material even though I could do the exercises (but not without a lot of sweat and coffee). (At one point I felt I became a function: [input] facts, assumptions then [output] proofs, ie hw exercises.) Folland does everything for abstract measures and treats the Lebesgue measure as a corollary.

Having said that, this books hits the spot.

A previous reviewer said this book was informal, unprofessional, and chatty. I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask.

I think this book shines for the purpose of an intermediate course between Baby Rudin and graduate real analysis ala Folland. As such, the exercises are at the perfect level and include standard, important, and interesting results and extensions. I don't think this book is rigorous enough for a real course at the graduate level, however.

A final note, the editorial (why?)'s placed throughout do get annoying butI feel they make sure you do not take results for granted, an all too common habit when reading advanced math.

Decent Text on Real Analysis and Measure Theory
I used this text for a two term (on the quarter system) course on Real Analysis (we covered Complex Analysis using another book in the third term).We used Walter Rudin's classic text, Principles of Real Analaysis, as a supplementary book and to fill in some gaps not covered in this text (which I'll address later in this review).The course covered every chapter in the text except for Chapters 12 and 20, which cover the Stone-Weierstrass Theorem and Differentiation respectively (the chapter on differentiation is the last, and I think it specifically covers how differentiation relates to the theory of Lebesgue Integrals).

This was my first actual course as an undergraduate studying Real Analysis and I think the book did a fair job of exposing me to the subject.It begins with a review of calculus (covering the definition of limits of sequences and functions, explaining the real numbers using the Archimedean property, the density of rational numbers, least upper bounds and greatest lower bounds, etc.), and simple notions regarding continuous functions between real numbers.The second chapter covers countability, which doesn't really come into play too much until a few chapters later, but is a nice diversion, which many students are often interested in.The book never really formally defines many things that are expected to be known by the reader from before (such as the definition of a derivative), whereas other introductory Real Analysis texts such as Rudin's Principle of Real Analysis often revisit these topics from elementary calculus in greater detail.

Soon after the introductory chapters, which cover some basic point set topology, the book exposes the reader to generalized metric spaces and continuous functions between such spaces, but then the text devotes attention specifically to continuous functions in C[a,b] or C(R) (continuous functions with domains on an interval or the entire set of real numbers, being mapped into the real numbers).One problem (if it may be called that) with the book is that it rarely if ever address multi-dimensional spaces (R^2, R^3, or the generalized R^n) in the text, and only occasionally assigns problems dealing with these important metric spaces.As a consequence, I'm quite sure that the Implicit Function Theorem and Inverse Function Theorem are not covered (which doesn't matter per se, since they're not needed for the Lebesgue chapters, which is what the book really builds up to), so our instructor taught these subjects out of Rudin.Looking only at the real line may make sense from a Real Analysis point of view, but I believe covering R^n would have helped broaden the scope of the book without any loss of focus.

When I first began using this book, I felt uncomfortable, since the tone of the author was so casual and might I say unprofessional.He often uses exclamation marks (!) or sometimes even two (!!) at the end of a theorem or result that he himself finds particularly interesting.Sure, this is an attempt to try to motivate people learning the subject, but it isn't all that necessary in my humble opinion.Also, the author heavily resorts to using paranthetics such as (Why?) in many parts of a proof, which at first was frustrating, as it appeared to be a nasty little tactic used in order to avoid more complete proofs, but I grew to like this approach since it got me thinking about some of the more basic ideas that we sometimes tend to overlook, and it's a nice check to make sure you're where you need to be with the material at any given time.A few chapters in, I grew to not mind the style, and eventually come to even like it, though I think I still would have appreciated something more traditional.Being talky is one thing and many same some texts such as Dummit and Foote's Abstract Algebra reads like a novel, but this text was especially awkward since it read like a conversation with an overenthusiastic friend.

More on the style: the author also continually "foreshadows" things to come and tries to motivate everything before putting all the rigorous bits and pieces in place and formalizing the ideas at hand.This was sometimes distracting and caused too much filler text (many times he gave "examples" before you really knew what the examples were going to be about), but sometimes rewarding so you could see what "the point of learning it all" really was, if there was ever any doubt.

Like most books the exercises range from trivial to difficult, but the author kindly puts little triangle symbols next to the most important exercises which allow one to check ones understanding of the basics.Many exercises have helpful hints where necessary.No answers are provided, but I assume that's fairly standard.

The verdict?I think this is a 3 ½ star book, maybe a bit better, so I'll go ahead and give it 4, especially since it's so affordable.

Ought to be popularized.
I used this book for a semester course in Analysis II. We didn't read the book in a linear fashion from start to finish, but we managed to thoroughly cover the material first on Banach spaces, then functions of bounded variation, then Stieltjes integration, then Lebesgue measure.

The book's biggest asset is that the majority of its many problems are worth attempting. He scatters them throughout each chapter instead of lumping them all at the end which presumably is more pedagogically sound. I was able to do most of the problems I attempted but not some. I really cannot overstate how good the exercises here are.

Also, Carothers will not hold you by the hand - he inserts a parenthetical "why?" everytime he skips over a detail. I agree with this approach but I think the "why's" ought to be omitted since that one should actively read math is implicit, so such parenthetical remarks are superfluous (cf. Rudin).

When I was taking the course, I said the book was too chatty, but I recant this now. Carothers includes extensive historical commentary when appropriate, which is a refreshing departure from monotony, and enlightening in its own right.

The one drawback to this book is that everything is done on the line R^1. Nonetheless it's done well and thoroughly.

Carothers' book is definitely different from most introductory analysis texts, so I wouldn't expect all students or professors to like it as it's admittedly somewhat idiosyncratic, but ultimately it's first-rate. Moreover, it's only a third of the price of certain canonical introductory analysis books that it may even better.

Sales Rank is Depressingly Low
It is sad to see the sales rank of this book so low, currently
700,000+.The author does a good job explaining real analysis.
This book goes well with R.P. Burn's book. ... Read more

Customer Reviews (2)

Great seller, highly recommended
Book was in great shape, shipped quickly which was greatly appreciated as my class had already begun.Thank you for the great service!

Very Hard to understand
This book is terrible.It is very hard to understand.Not recommended. ... Read more

Product Description
Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises. ... Read more

Customer Reviews (12)

One of the Best Books in Analysis, or even Maths
I dont know how to say how good this book is: it not only teaches us the technical aspects of mathematics, it also teaches us intuitions, ideas behind the proofs, styles, and philosophy.

I would like to also start with a comparison of the classic baby Rudin:

While Rudin's little book is also a real gem, I would say Pugh's belongs to a slightly higher level (due to its problems mainly and topic selection and coverage). Rudin's book could either be used in 1st year one semester as a strong first course in basic analysis (1st 7 chapters) for extremely motivated and hardworking students (such as those at MIT, Harvard, Princeton, and many other good institutions), or ought to be supplemented in an honour's undergrad real analysis (in 2nd year or 3rd year). Its presentation of Lebesgue theory is rather incomplete and no one virtually uses it for lebesgue theory. On the other hand, pugh has a full chapter on it, covering almost all the standard undergrad lebesgue materials.

Pugh's book on the other hand can be the last reading before attempting Folland and Big Rudin. Knowing pugh well and having solved its problems would make Folland and big Rudin not hard, whereas little rudin may not surve this purpose that well. Many Rudin's problems are hard but standard (Prove, Show this. Very few is it true? what about?), whereas Pugh's is more thought provoking (Is it true? What about? What do you think? which mimics a key part in maths research).

Moreover. mathematics is not just about formalism and logic, especially in analysis ang geometry. The ideas and our feeling about how the objects behave are at least equally important. (Anyone can write proofs well with sufficient training; yet not everyone feels that a measurable function is no more complicated than continuous ones in a sense; why lebesgue's definition of length and integral are powerful; weirstrass approximation is as simple as "taking expectations of functionals", etc.) Amazingly Pugh's book trains people to this direction very well.

I prefer Bartle and Sherbert
I used this book in my first Real Analysis course and thoroughly disliked it.I seems that everyone else who reviews this book mentions Rudin.I haven't had a chance to read Rudin yet but I prefer Intro to Real Analysis by Robert G. Bartle and Donald R. Sherbert over this book.Many people like Pugh for it's conversational tone but I found it annoying.This might be a good secondary book but I wouldn't recommend this as your first book in real analysis.Pugh makes the cardinal mistake of mathematicians in introducing the most general case first.The most important thing in mathematics is not the most general case but the process of generalization itself.This is like saying the journey is as important as the destination.To generalize one must start with a specific case and then work, layer by layer, to the most general case.That's one reason I prefer Bartle and Sherbert.It starts with functionsfrom R to R and generalizes from there.It takes up where undergraduate calculus leaves off.I also prefer the exercises in Bartle and Sherbert better. They are challenging without being infuriating. They are still general proofing exercises but are specific enough to deal with specific functions, series, sequences, and so forth.I was also annoyed by the way Pugh qualifies his proofs like Chapter 1 Theorem 2: "Proof Easy" ,or theorem 9: "Proof Tricky!", or Chapter 2 Theorem 10:"Proof, Totally natural!".I feel his language is imprecise and sloppy. I feel the section on cuts is superfluous. It seems that cuts are a lot of work and headaches just to prove that everything I learned in elementary school is correct. I was worried that x+0 didn't really equal x but now with cuts, I can rest assured that it does! Whew, what a relief!The only plus to Pugh is the thorough chapter on metric spaces helps put things into a broader context.All and all I dislike Pugh's book and highly recommend Bartle and Sherbert. as the best introduction to Real Analysis.

Excellent problems and diagrams -- great book
This is an excellent introductory text on real analysis. It is very approachable, and he does a very good job at supplementing the traditional "definition-theorem-proof" style with intuitive explanations and wonderfully descriptive diagrams (the diagrams are one of the strongest points of this book -- and are something that are sadly left out of many otherwise good books on analysis).

My only (minor) complaint is with the layout/formatting of the book -- it is very jumbled together, the typesetting is poor, and it looks like it was printed on a low-resolution $10 printer.

Other than that, it is an excellent companion to a more in-depth/advanced treatment. As far as more "advanced" books go, I would recommend -- Apostol's "Mathematical Analysis" and/or Shilov's "Real and Complex Analysis" -- both of which are incredibly well written and informative.

Pugh is wonderful. Rudin is good too, but both texts working together is the best.
I wish that I had discovered Pugh in my first semester of undergraduate analysis. The assigned text was Rudin and it was a great choice. The exposition there is excellent. The exercises are incredibly well done. Pugh covers just about the same material as Rudin, and in the same rigor, but is more likely to give you paragraphs before and after important theorems/definitions that help to clarify things. I must admit I am not too familiar with the first half of Pugh's text as I didn't discover it until I was well in chapter 10 of Rudin ~~ chapter 5 of Pugh. But, if the first chapters are as good as the fourth and fifth, you can get just as much from Pugh as from Rudin, if not more.

Sometimes, you get a picture (this would have been really helpful back when I was learning what an open cover was). Other times, Pugh actually gives a better presentation. For instance, when discussion the rank theorem, Rudin's statement of it is hard to follow. The proof is about as difficult. Pugh, however, introduces C' equivalence and then gives an alternate statement of the theorem which is much more intuitive. AND some pictures after the proof. Some think having pictures in analysis books is bad--Pugh gives evidence otherwise.

It is difficult to say which text has better exercises as I have not attempted them all. But Pugh definitely has more of them. I think the best thing for any undergraduate to do is to just own both books. Rudin is the standard for a good reason. Pugh's or someone else's exposition may become the standard in the future, but Rudin will always be an excellent reference. Doing Rudin's exercises will help prepare you for your qualifying exams if you ever take them. Pugh has some UC Berkeley good prelim exam questions in his book which prepare you for future math endeavors as well. So I say just buy both. But if you can only buy one.... probably get Pugh because he's cheaper. Or you can get International Edition Rudin for cheaper still.

Brilliant
The style is friendly and fun, and the presentation is really intuitive! My personal favorite! ... Read more

Product Description

Customer Reviews (5)

Gift for an economist -- he loves it and is reading and re-reading it
Bought this as a gift -- the economist/recipient loves it and is reading and re-reading it.

One of a kind
This book has a wealth of material on analysis. A quick perusal of the contents list should be enough to convince anyone, even working mathematicians, that this is a book worth having (see author's website for the content list.) Don't let "with economic applications" fool you, this is a highly rigorous, compendious, well-exposited tome on real analysis.

The economic focus means that some of the topics that are covered exhaustively in this book are rarely seen in books of this level (e.g., emphasis on fixed point theorems and correspondences). There is also a very good chapter on differential calculus on normed spaces -- a topic that is (inexplicably) left out of many other functional analysis books. Another excellent book which covers differential calculus is Zeidler's book on applied functional analysis, Zeidler's exposition is actually better than Ok's for those who are interested in this topic specifically.

Any economist who is already fairly comfortable with analysis will enjoy this book. Mathematicians will also find things they've probably not seen before in standard analysis courses or texts. As an added bonus, the economic applications provide some direct motivation for the material. Normally, it is rare to find a textbook on analysis which includes accessible and non-mathematical applications, but this book is filled with them. You rarely have to learn any economics to be able to appreciate the applications in this book.

However, that said, this book is not suitable as an introduction to analysis. Although it is self-contained, a level of mathematical maturity is necessary. Ideally, you should have a couple of undergraduate analysis courses behind you before you attempt this book. That is, you should be fairly comfortable with limits, continuity, convergence, metric spaces, etc.

The only serious complaint I have about the book is that occasionally, rigor gets in the way of clarity. Sometimes, proofs can get cumbersome. Certain parts of the book seemed to be unnecessarily complicated by excessive formalism.

Excellent Book
This is an excellent real analysis book with a lot of material that fits perfectly any one's interests in economic theory. Other real analysis books out there do not cover things that are very important in economics, e.g, fix point theorems, correspondences, and convexity. This books covers all that and much more in a rigorous way so it also fits perfectly the needs of any math grad student, particularly if he/she has some interest in economics. I strongly recommend this book to any econ grad student who wants to learn the tools needed in economic theory.

A fantastic book which fills a gaping hole.
A fantastic book which fills a gaping hole. I have yet to find a comparable book. Incredibly well-written with an embarrassingly large wealth of material. The ideal book for graduate students in mathematics, economics or mathematical economics. Any mathematician with a strong interest in Analysis and curiosity about economics (or any economist with a strong interest in mathematics) would do well to read and re-read this book!

Great book for mathematical economics
This is a very interesting book that explains real analysis focusing on economics issues and, I must say, it does its job beautifully and with no lack of rigour. When it comes to the mathematical aspects of microeconomics, the book turns out to be even better. A great book that will help very much Mas-Colell's Microeconomic Theory readers. ... Read more

Customer Reviews (4)

Great Text with Great Proofs
In some presentations rigor is sacrificed for an intuitive overview of the material. This is a mistake in my view, particularly for an undergraduate course in analysis, which is the perfect environment for rigorous proof-writing skills to be developed.

Goldberg makes no such sacrifice. His proofs are great. His exposition is very clear. If I have a complaint, it's that it's not colorful enough, there aren't enough different kinds of pictures and graphs. The text DOES contain pictures and diagrams, and they are good, but they are not great. On the other hand, this might be seen as a feature, for it forces the student to work on coming up with her own, which is a very important skill to develop.

Great introduction to Real Analysis
This book reads like an instructor would teach in class. It derives all the important theorems quite rigorously and throws in a few lines of intuition which is very helpful when you are trying to self-study something as intense as real analysis. He also has two or three examples following every major result and shows clearly how to "use" the result just derived to solve an actual math problem.
I went through lots of great analysis books (Rudin, Shilov, Kolmogorov, Aliprantis, Johnsonbaugh, Rosenlicht and Protter among others) until finally learning from this. After getting my foundations and intuitions right, I now feel like I am better equipped to read and understand the results from the above books which, in general, treat proofs more tersely and cover a lot more material in the process.
This book does not cover the origin of real numbers and it's axioms. It covers standard results from elementary set theory, sequences, limits, metric spaces, open and closed sets, completeness, compactness and connectedness, the derivative, Taylor Series, exps and logs, the Lebesgue integral and Fourier Series.

Well worth it.
I come to real analysis from a non-mathematics background.I began with Steven Lay's "Analysis with an Introduction to Proof" which was great for an absolute beginner. Goldberg's text is the next step up.Andwell worth the money!

Clear and to the point
This is an excellent textbook for an introductory course in Real Analysis.The text is rigorous, but contains enough examples to be readily understood.If you honestly want to learn the subject matter, this book isworth the money.However, if you would rather struggle with it, there areplenty of those books floating around . . . ... Read more

Product Description
In the second edition of this MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, or as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier s introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy s attempts to establish a firm foundation for calculus, and considers his failures as well ashis successes. It culminates with Dirichlet s proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet s proof. ... Read more

Customer Reviews (3)

Excellent sequel
If you had calculus in high school or college then you learned about Newton, Leibnitz, and Riemann but probably did not encounter Lebesgue (pronounced le-bek). At the University of Alabama Huntsville learning about Lebesgue integration is key to advancing into graduate studies in mathematics. The natural follow-on course after Calculus I and II, etc. is Real Analysis. This book, using Lebesgue integration methods, is a good sequel to Lebesgue calculus.
I purchased this book, after reading about it in the Mathematical Association of America (MAA). For autodidacts like myself, it is a good first introduction to the topic.

A radically false account of history
This is not a bad book, but it does everyone a huge disservice by pretending to be historically informed when in fact it is propagating harmful and stupid myths that have no basis in historical fact whatever. An example should make this clear.

"Daniel Bernoulli suggested in 1753 that the vibrating string might be capable of infinitely many harmonics. The most general initial position should be an infinite sum of the form
(2.71) y(x) = a_1 sin(pi x / l) + a_2 sin(2 pi x / l) + a_3 sin(3 pi x / l) + ...
Euler rejected this possibility. The reason for his rejection is illuminating. The function in equation (2.71) is necessarily periodic with period 2l. Bernoulli's solution cannot handle an initial position that is not a periodic function of x. Euler seems particularly obtuse to the modern mathematician. We only need to describe the initial position between x=0 and x=l. We do not care whether or not the function repeats itself outside this interval. But this misses the point of a basic misunderstanding that was widely shared in the eighteenth century. For Euler and his contemporaries, a function was an expression: a polynomial, a trigonometric function, perhaps one of the more esoteric series arising as a solution of a differential equation. As a function of x, it existed as an organic whole for all values of x. ... To Euler, the shape of a function between 0 and l determined that function everywhere." (p. 53-54)

There is not a single line anywhere in any pre-19th century mathematical work that comes anywhere near making this sort of claim. Self-righteous "mathematicians" have invented these myths to justify their dogmatic and authoritative mode of "teaching" and their passionate hatred of intuition. In falsely lending these propaganda fabrications a veneer of historical truth, Bressoud is perhaps the worst lier of them all. It is not Euler who is "obtuse," but Bressoud. There was no "basic misunderstanding widely shared in the eighteenth century"; rather, the "basic misunderstanding" lies with Bressoud and his fellow poseur historians of today.

All of this is easily established by simply reading Euler. The relevant paper is E213, which is readily available online. Let me summarise what you will find if you read that paper.

First of all, Bernoulli never claimed that (2.71) can express any initial position of the string. He merely argued for a general series solution of the vibrating string equation which *implies* that the initial position is of the form (2.71). Hence Euler's main objection, which is this: I can bring the string into any position whatever, let go, and it will move according to the vibrating string differential equation. Thus Bernoulli's solution is not completely general insofar as (2.71) does not express any possible initial position of the string. And since Bernoulli has provided no argument that (2.71) can in fact express any initial position, nor in fact any method for calculating the coefficients a_i, we have no reason to believe that his solution is completely general. At this point Euler preempts a hypothetical counterargument: perhaps, says Euler, some might argue that "owing to the infinite number of undetermined coefficients," equation (2.71) "is so general as to include all possible curves." This, however, is plainly false, Euler points out by noting the periodicity properties of (2.71). Now, at this point it would be possible for a Bernoullian to retreat still further and say that (2.71) can express any function, not on the real line, but on the interval [0,l]. This is a perfectly valid argument, but it is an argument which Bernoulli never raised and which Euler never claimed to have refuted. So much for the periodicity argument, which Bressoud has obviously distorted most unfairly. But worse still is Bressoud's generalisation from this case to the alleged "basic misunderstanding." This is sheer stupidity and fabrication, as is plain to anyone capable of reading at a fourth grade level. In fact, every last word of it is plainly and unambiguously rejected by Euler in the very article in question when he points out that the initial position of the string can be any curve, which "often cannot be expressed by any equation, be it algebraic or transcendental, and is not even included in any law of continuity."

Getting there naturally
I am a topologist by training who was Shanghaied into being an analyst when I was hired as a teacher.As a consequence of this, the Advanced Calculus course I taught was rather heavy on topology.

Over the course of time--having been transformed into more of an analyst that I would have ever dreamed--I've come to the conclusion that analysis is best learned before topology.

This is a text that accomplishes that by using the historical approach.

One learns how Newton approached problems, how Euler did, how Cauchy did.Not only is it interesting, it is enlightening.I've taught this course for 15 years now, and of all of the approaches I've taken, this has been the most fruitful.

My students have come from calculational courses, and the historical approach of this book provides a bridge over which they may come into the land of proof.They also see the issues that caused the need for modern rigor face to face

I do supplement the course with material that is more modern (Hardy's book A Course of Pure Mathematics) and material on the Riemann integral, but I've been spoiled for any other approach.

... Read more

Product Description
This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters. ... Read more

Customer Reviews (3)

One of a kind !
Up to my knowledge, this is the only book that constructs the Lebesgue integral for functions to a general Banach-space instead of the real numbers (thus saving us from the unnecessary and esthetically dissapointing construction through positive and negative functions).
I don't know how Lang does it, but eerytime you'll pick up one of his books, you'll marvel at the beauty of mathematics !

Much better than Royden!
It drove me up the wall, in my first course on measure and integration, that integration was first done for positive functions, then for real functions by writing them as a difference of positive functions, then complex functions in terms of real and imaginary parts.Why couldn't you just integrate real-valued functions
intrinsically, without the silly decomposition into positive and negative parts?

After that course, I found Lang's book.What a blessing to see that you can just integrate in infinite-dimensional spaces right from the start.I can't understand why virtually all books on integration theory still succumb to the "positive functions first" approach.

Overall, A Good Book
I've read several analysis books and this is one of the better ones that I have read.It covers a variety of interesting and useful topics and the exposition is clear. It's presentation is a bit more abstract than some others starting with some functional-analytic concepts before doing integration in that framework. However, if you want to study stochastic analysis, getting in this frame of mind will definitely help your understanding of stochastic integration. For a truly thorough understaning of the subject, I recommend purchasing this book as well as the somewhat easier "Lebesgue Integration on Euclidean Space" by Frank Jones - the two together cost about the same as Royden, Rudin, or the terrible book by Aliprantis. ... Read more

Product Description
"Mun demystifies real options analysis and delivers a powerful, pragmatic guide for decision-makers and practitioners alike. Finally, there is a book that equips professionals to easily recognize, value, and seize real options in the world around them."
--Jim Schreckengast, Senior VP, R&D Strategy, Gemplus International SA, France

Completely revised and updated to meet the challenges of today's dynamic business environment, Real Options Analysis, Second Edition offers you a fresh look at evaluating capital investment strategies by taking the strategic decision-making process into consideration. This comprehensive guide provides both a qualitative and quantitative description of real options; the methods used in solving real options; why and when they are used; and the applicability of these methods in decision making. ... Read more

Customer Reviews (15)

Real Options are not derivative contracts
What is a "real option"? This kind of option is not a derivative instrument, but an actual option (in the sense of "choice") that a business may gain by undertaking certain endeavors. For example, by investing in a particular project, a company may have the real option of expanding, downsizing, or abandoning other projects in the future. Other examples of real options may be opportunities for R&D, M&A, and licensing.In corporate finance, real options analysis or applies put option and call option valuation techniques to capital budgeting decisions.

This book is written for both the beginners and as those skilled in real options applications. The book is organized in such a way that each chapter has a brief summary and a list of questions, so you can construct a small exam based on these questions.

The book provides a refreshingly new view of evaluating capital investment strategies by taking into consideration the strategic decision-making process. The book provides a qualitative and quantitative description of real options, the methods used in solving real options, why and when they are used, and the applicability of these methods in decision-making. In addition, multiple business cases and real-life scenarios are discussed. This includes presenting and framing the problems, as well as introducing a stepwise quantitative process developed by the author for solving these problems using the different methodologies inherent in real options. Included are technical presentations of models and approaches used as well as their theoretical and mathematical justifications.

The book consists of two parts. The first part looks at the qualitative nature of real options, providing actual business cases and applications of real options in the industry, as well as high-level explanations of how real options provide much-needed insights in decision-making. The second part of the book looks at the quantitative analysis, complete with worked-out examples and mathematical formulae.

Pros:
- Has a good introduction to real options, suitable for the beginners
- Lots of remarkable, real-world examples
- Gradually progresses from basic to advanced techniques
- Provides some basic financial statement analysis concepts used in applying real options
- Compares financial options with real options; provides major similarities and differences

Cons:
- The accompanying CD disk includes the demo version of "Real Option Analysis Toolkit" (demo) and "Crystal Ball Software" - both expire briefly after the installation.

This book can help me to investment after MBA finance class.
It is a real-hand-on book! Although the book looks very "huge", the analysis helps me to prepare MBA class.

Real Options
Mun's book shows how real options problems, like those faced in the real world, can be solved.Other books may provide a better introduction to real options concepts, but the methods employed are suitable only for very simple problems.Where other approaches require that you develop your own lattices (or other solutions),Mun shows you how to use his Supper Lattice Solver and Monte Carlo simulation software to solve these problems.I am convinced that his approach will not only facilitate the solution of these problems, but will also be more readily accepted by management.I look forward to acquiring Mun's software and applying it in practice.

The Second Edition - A Great Practical Guide through the Real Option Debate
As practitioners and academics continue to grapple with quantifiable uncertainties in real asset decision making, the debate about real option models will no doubt continue.

Johnathan Mun's second book and more specifically his case study approach allows practitioners from diverse industries to enter the debate with simple excel asset pricing skills. To my mind there is no better pragmatic work on the topic than the second edition of Real Options Analysis. With the book in one hand and the robust SLS software up on the screen - framing, pricing and understanding real options is pretty straightforward.

Two points to note: After 30 days, just as you begin to get hooked on the superb software it is likely to gently expire. That's when you are saved by the second point; the author is hugely supportive - His `one line insights' in response to specific queries made this a great purchase.

Edinburgh. Scotland.

On average: a good book
This was my first book on Real Options. After this, I complemented my knowledge with more accurate researches on the theoritic foundations on the subject (eg. Trigeorgis and Copeland).
What I liked of this text is that it was a soft landing into the Real Option world, with a simple and easily understandable description. Its major pro is to present transparently the basics of a concept that is often approached at a too high and formal level.
What I did not like is the fact that few chapters at the end were not really useful but full of stuff and formulas with no explanations that cannot practically be used. I had the sensation they were out of place, since I could grasp their meaning only after passing to more comprehensive books.
One more criticism is that you don't understand the effect of the difference between private and public risk in real options evaluation as you do with other texts. However, I still consider this the book where I formed my basics before being able to master some other more detailed book (but also more difficult to master). The Crystall-Ball package was also a nice surprise. At the end, if you consider the price and the content it was surely good value for money even though it's not a masterpiece. ... Read more