Reviews (10)

Excellent introduction to analysis
Of the many analysis books I have seen, I think this is one of the best for the student approaching the subject for the first time.

It is mathematically rigourous, yet develops the major concepts of analysis in a leisurely (in the good sense of the word) way with interesting and sometimes unusual examples.

Beginners will especially appreciate the quality exercises and the solution guide in the back.

The style of this book is a bit similar to Spivak's *Calculus* in that the author is a bit wordy.I find Ross' presentation more direct and less pretentious than Spivak--and far less intimidating.

This is definitely the best introductory analysis book I know of for self-study.A student who masters the material in this book will be well prepared to tackle Rudin and other classic works in real analysis.

Entirely mediocre
A solid, if limited, textbook. The author provides an entry level approach to analysis without much use of more abstract topological concepts like metric spaces. A decent book for the reader who desires this type limited approach. Even in that light it is hardly spectacular as the author's logic often does not flow as clearly as in alternate texts

This book might be better viewed as an auxiliary text beside a classic like Rudin or the very good Real Mathematical Analysis by Charles Pugh.

Isbn: 1899332022
Sales Rank: 1812205

Reviews (11)

more substantive than most books of this type
This book has what I want in terms of content, a treatment of the basic language of mathematicasl proof, i.e. logic, sets and functions, and then three chapters actually using that language to do significant proofs.

Those chapters treat cardinality of sets both finite and infinite, basic algebra of groups including LaGrange's theorem, and then elementary analysis such as convergence theorems using completeness of real numbers.

Unfortunately this seems to make it too hard for weak students, and there are a lot of those out there, as evidenced by some of these reviews.I.e. this book is written seriously, and in a bit of a dry style, as books were in the 60's.

Enjoyable selection of problems
Although this text tends to be a bit thin, it introduces many important concepts in a way that is easily learned. The selection of problems is interesting, and provide ample food for mathematical thought for the curious student. Enjoyable to firmly introduce varioius forms of induction, well ordered principle, and some introductory group theory. WELL WORTH THE MONEY

Editorial Review

Physics and computer science genius Stephen Wolfram, whose Mathematica computer language launched a multimillion-dollar company, now sets his sights on a more daunting goal: understanding the universe. Wolfram lets the world see his work in A New Kind of Science, a gorgeous, 1,280-page tome more than a decade in the making. With patience, insight, and self-confidence to spare, Wolfram outlines a fundamental new way of modeling complex systems.

On the frontier of complexity science since he was a boy, Wolfram is achampion of cellular automata--256 "programs" governed by simplenonmathematical rules. He points out that even the most complexequations fail to accurately model biological systems, but the simplestcellular automata can produce results straight out of nature--treebranches, stream eddies, and leopard spots, for instance. The graphicsin A New Kind of Science show striking resemblance to thepatterns we see in nature every day.

Reviews (1)

Reviews (2)

This ONE is for REAL
No matter what you think about differential equations, you just got to have that book. The software that comes with it, is dynamite, and fully adds to Mathematica's Dsolve. Fact is, after you read the book you may understand what Ordinary Differential Equations are all about. "By the way there is nothing ordinary about Differential equations". These guys really know their stuff. Even better they can really explain it well. The book reads like someone is constantly talking to you or guiding you. The presentation is great. My only problem is that I finally understood the stuff. [ A pleasant oneat that] The other fascinating thing about the book is the historical backgound of the mathematicians who contibuted to the discovery of DEs. This makes the book so enjoyable.Read this book and your girlfriend/boygriend willthink you are a true inspired genius. However make it fast. They are bound to find out for themselves. This book will be true classic. BTW, you c! ould also look up another great book on DE.Authors are Blanchard, Devaney and Hall. Published by Brooks/Cole. These two books are all you should ever need.

Reviews (4)

A incomplete work
It's rather frustrated to see the book with this title only values plain call options. It didn't give a general approach for valuing general payoff securities. The transform formula only works for plain call option, and even this need to use put call parity once to convert the payoff function so that the fundamental solution can apply.

More than just stochastic volatility
Other reviewers discussed the virtues of this book as a first book devoted to option pricing under stochastic volatility. And, indeed, the book provides a detailed exposition of stochastic volatility models. What I wantto add to the other reviews is that this book is more than just aboutstochastic volatility. The book gives a careful exposition of theapplication of the two important mathematical methods to contingent claimvaluation: the method of integral transforms (Fourier and Laplace inparticular) and the method of eigenfunction expansions. Long the core toolsin mathematical physics, these important methods now find more and moreapplications in financial economics. They can be applied to option pricing,interest rate modeling, and, more generally, any problems in economics thatinvolve calculations with diffusion processes. The author clearlydemonstrates how to use these powerful tools for calculations in finance.Researchers working in the area of derivatives pricing, both in academiaand on the Street, will not want to miss this point.

the first book on stochastic volatility models
The Black-Scholes model for the pricing of derivatives whilst very effective is also known to be imperfect. A number of ways of generalizing the model to cope with these imperfections have been developed. One popularmethod is to allow the instantaneous volatility parameter to follow astochastic process. This allows the possibility of observed volatilities inthe market to evolve from day to day and also to produce market type"smiles" that is graphs of volatility against strike which aresmile shaped rather than the horizontal line implied by the Black-Scholesmodel.

In this book, Lewis develops pricing formula for options understochastic volatility models. This is mainly via the use of transformmethods, that is a closed form solution is developed for the Fouriertransform of the price as a function of log of the spot.The actual priceis then obtained via a numerical inverse Fourier transform.

Thestrengths of this book are that it covers an important area that heretoforehas been restricted to research papers and that it provides a large numberof carefulderivations and formulas.

The principal weakness is that theapproach is too formula-based. The reader does not gain many conceptualinsights from the author. Indeed one gains the impressionthat the authoris technically strong but does not have a good conceptual understanding ofthe subject. The author does not really make a case for stochasticvolatility models as opposed to other generalizations of the Black-Scholesmodel.

The book is restricted to vanilla options with no discussion ofhow using astochastic volatility model impacts on the price of exoticoptions.

Reviews (6)

Terrific Book
Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.

Very Good.
I read the FIRST EDITION. This is definitely the best introductory mathematical logic book I've seen. It's the most rigorous, most advanced (a reasonably strong form of Godel's theorem is given), and is well-organized and very clearly written. It would be suitable as 1)an introduction for students with some mathematical experience- say a little abstract algebra and perhaps some previous exposure to logic. 2)a refresher for advanced students 3)a nice reference for basic topics. The exposition is great- Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading. Chapter 1 covers propositional logic, with a general-purpose discussion of inductively defined sets, unique readability, and recursion. Many books these days do a sloppy job justifying recursive definition, or dont bother at all- Enderton does it right and is fairly detailed. Chapter 2 begins first order logic and has the most detailed proof of the completeness theorem I've seen. Sect 2.7 concerns translating between theories in different languages, something i hadnt seen developed explicitly before. 2.8 is a great exposure to nonstandard analysis- long enough to give you an idea how it works and why its useful. Chapter 3 begins with an analysis of some reducts of number theory- (N,0,S) (N,0,S,<) and (N,0,S,<,+) and shows how to eliminate quantifiers in them. Next, toward Godel's theorem, a finite set of axioms for a subtheory of number theory is given, and a host of relations and functions are shown to be representable in this theory. In 3.5 we get the fixed-point theorem, Tarskis thm, a weak Godels thm, a stronger Godels thm, and Church's Undecidability thm, and an introduction to the arithmetic hierarchy. 3.6 lifts Godels thm to show set theory is incomplete, and discusses Godels 2cd thm. Chap 4 is 2cd order logic, skolem normal form, many-sorted logic (a first order logic with different sets of variables ranging over different universes), and general 2cd order logic (restrictions are placed on the subsets "X" ranges over in the 2cd order formula \all X \phi). Basic recursion theory is developed throughout the book- Enderton begins with informal notions of computation, then defines a relation R as recursive iff it is representable in some consistent finitely axiomatizable theory, and discusses Church's thesis. 3.8 quickly covers universal computers, partial functions, Kleene normal form, unsolvability of the halting problem, the smn thm, Rice's them, and a register machine model. All this seemed a bit disorganized, so familiarity with computation and automata theory would be a plus. Heres the contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS. . . .1
Chapter One - SENTENTIAL LOGIC
1.0 Informal Remarks on Formal Languages 14
1.1 The Language of Sentential Logic . . . . . 17
1.2 Induction and Recursion . . . . . . . . .22
1.3 Truth Assignments . . . . . . . . . . . .30
1.4 Unique Readability. . . . . . . . . . .39
1.5 Sentential Connectives. . . . . . . . . .44
1.6 Switching Circuits. . . . . . . . . . . .53
1.7 Compactness and Effectiveness. . . . . 58
Chapter Two - FIRST-ORDER LOGIC
2.0Preliminary Remarks. . . . . . . . . .65

2.1First-Order Languages. . . . . . . . . .67
2.2Truth and Models. . . . . . . . . . . 79
2.3Unique Readability. . . . . . . . . . . 97
2.4A Deductive Calculus . . . . . . . . . .101
2.5Soundness and Completeness Theorems. .124
2.6Models of Theories. . . . . . . . . . . 140
2.7Interpretations between Theories. . . ... 154
2.8Nonstandard Analysis . . . . . . . . . . .164
Chapter Three - UNDECIDABILITY
3.0Number Theory . . . . . . . . . . . . 174
3.1Natural Numbers with Successor. . . . 178
3.2Other Reducts of Number Theory. . . . 184
3.3A Subtheory of Number Theory. . . . . . 193
3.4Arithmetization of Syntax. . . . . . . . .217
3.5Incompleteness and Undecidability . . . 227
3.6Applications to Set Theory . . . . . . . .239
3.7Representing Exponentiation. . . . . . .245
3.8Recursive Functions. . . . . . . . . . .251
Chapter Four - SECOND-ORDER LOGIC
4.1 Second-Order Languages . . . . . . . . . 268
4.2 Skolem Functions . . . . . . . . . . . . 274
4.3 Many-Sorted Logic. . . . . . . . . . . . 277
4.4 General Structures . . . . . . . . . . . . 281
Index . . . . . . . . . . . .291

Reviews (2)

a comprehensive book on stochastic calculus, yet accessible
I only read about 70% of the text, without essentially touching
the excercise problems. I have to confess I'm pretty much overwhelmed by the myriad topics treated in this book.

From the perspective of a student, I think Revuz/Yor has the following merits:

1. It covers an enormous amount of materials, systematically and
carefully. It thus provides the necessary preparation for a graduate student who's eager to get ready for research.

2. Despite of its scope, this book is accessible to graduate students. By "accessible", I mean any dilligent student with certain mathematical maturity should be able to understand most of the materials in the text.
Two things about this book make possible the accessibility. First, proofs are very carefully written, and a quite few of them may even be called detailed. Second, the authors deliberately chose the "slickest" approaches to many classical results,
while preserving, even elucidating, the fundamental ideas. Examples include the construction of BM from the perspectife of Gaussian processes, the presentation of Markov processes in Chapter 3, the "global" definition of a stochastic integral, etc.
This paves the way for further study of more general cases.

3. The computations displayed in this book can serve as good exercise for "basic" trainings. As the book goes on, the reader is more expected to carry out the details. And some of them, although said to be "easy" by the authors, could take some time to figure out.

4. The exercise problems are wonderful. You lose half of the benefits if you don't work out a substantial amount of them.
Many of them are useful results from current research papers, or classical results from these or those "bibles". I myself
haven't done that, and that's why I feel I'm not in the position to give five stars at this moment.

Here's some of my thoughts for an "easier" reading. First, because of the scope of this book, it might be a good idea to read it with real motivations, and maybe during a prolonged period of time. Otherwise you may easily get tired, esp. when you get stuck with some details the authors claim as "easy".
Second, the reading could be frustrating if you care about every detail and do them all alone. A good way would be skipping over some of the details in the first reading, and then coming back at a later time for a second reading, or even a third reading. Find freinds to form a study group would be surely helpful. But I've never had this luck.

Finally, my review is just intended for fellow students. For the opinions of experts, the wonderful review of Frank Knight should be consulted. It can be accessed at MathScinet.

Reviews (24)

Nice idea, but MANY errors
This book is the "standard" for providing straightforward code demonstrating various options pricing techniques, and perhaps deservedly so -- after all, it really doesn't have too many competitors in that niche. However, even in the course of the fairly straightforward applications I've had for this book (mostly simple equity Cox-Ross and Black-Scholes modeling), I've been shocked by the number of blatant mathematical errors (in the formulas for rho with distinct carry and risk-free rates, for instance). Clearly, the author and editors didn't bother to spend much time verifying that the formulas they cite are actually correct. (I'm actually shocked to see a fellow reviewer praise the "proofreading" -- I'm guessing he or she never actually had to use any of the formulas in this book!)

Bottom line -- if you're looking for a handy, compact reference of option pricing formulas, this is probably what you'll end up with. But be careful. It is SO frustrating to spend hours trying to figure out where you made a mistake in implementing one of these models, only to learn that you DIDN'T make a mistake -- the mistake was in your source. Consider yourself warned...

Numerous technical mistakes
I have reviewed many of the formulas in several sections of this book and have found a number of mistakes.As a result, I can trust no formula from the book without reviewing the literature or some other source.

The author does not use consistent terminology throughout the book.Rather, the terminology of the original journal article is used for each pricing model.This makes referring to the articles convenient, but then you don't need the book if you're going to the source...

I have used few of the computer programs offered, but the ones that I have used have had terrible inefficiencies.For example, a bisectional iterative search was used, which is very simple to write but is also very inefficient.There are many other simple and more efficient alternatives.

THE Reference Manual
Fantastic reference manual for the most basic to unusual options.Well organized making it easy to find what you're looking for.If you're looking for derivations of models, you've got the wrong book.If you're looking for formulaes and perhaps even spreadsheets, this is the book for you.

Reviews (33)

Real bookfor traders
really great to figure out the intricacies of options. if you need something more conventional just get hull; this book is bysomeone who does it for a living and has different intuitions than what you get in the ivory tower.
some of the points are great but may need more updating

Not Worth the Read
This book is surely not worth reading.The formulas and analyses are more often incorrect than they are correct.As much as he likes fluffing people with PhDs (which I have - in mathematics), I can't help but feel he himself lacks the requisite skills for proper option analysis.This is not a book any serious trader, quant, risk manager, or academic should bother adding to their library.

Derivatives Theory meets Practice
This book provides a healthy dose of practical wisdom for options traders so that they don't blindly follow their mathematical models into oblivion. The author (Taleb) has a PhD in finance, but also has traded in the pits, he knows both theory and practice and where they diverge.

Taleb focuses on hedging, which is a trader's main task when running a large portfolio of options.Instead of using a flood of equations, Taleb relies on charts, graphs, and tables to make his points. Most of the equations & heavy mathematics are relegated to the appendix, presumably because quants (or software) will price the instruments.He covers the behavior of the Greeks (delta, gamma, vega, theta, etc.) for vanilla options as well as behavior of exotic options, and delves into the practicalities of volatility, hedging at discontinuities, and various other topics.

The book is very popular on trading desks, and although I found it pretty good, I didn't find it to be outstanding.Also, notably, the book does NOT cover credit & interest rate derivatives at all; hopefully this will be corrected in the next edition.

Reviews (6)

Read this before Shreve
Read this small book before reading Shreve's volume II book.
The sections on conditional expectataions, martingales, and Brownian motion are well written and simple enough to understand. While not packed with finance examples until the last chapter, the author attempts to provide what is needed of the subject matter to successfully complete a first semester course in Stochastic Calculus.Once read, it's a great second reference.

Stochastic Calculus for the Rest of Us
I have recently started to work with stochastic calculus.While I do have a mathematical background and I do understand advanced concepts like measure theory, I wanted something that I could bring down to a level that I could see the wood from the trees.This is pedagogically better.
Unfortunately, there are so few books out there that strive to be anything but a way to demonstrate the author's cleverness that when I do find a book that is understandable by someone who has had no training in that field, I can't sing its praises enough (For physics people Feyman's lectures are easily the best).
Mikosch explains things in a clear easily understandable way.He goes over probability theory, stochastic processes, the Ito Integral and all the basic ideas.While the book is not necessarily rigourous, someone starting is probably more interested in understanding the concepts and saving the rigour for later.
Understanding the core concepts now allows me to read other textbooks in more detail and it even lets me play with the concepts.
For starting quants, this book rocks.

Reviews (7)

Junk !
I bought this book based on the other reviewers' comments. This book is a piece of junk !Don't buy it and waste your money.These professors should not be allowed to write books and force it on their students.

There are better books on probability and Measure Theory.Try Marek Capinski and Ekkehard Kopp.Yes, I bought this too through Amazon.

some more details
I've been reading this book along with Durrett's PROBABILITY: THEORY AND METHODS and Williams' PROBABILITY WITH MARTINGALES. I also have Billingsley's PROBABILITY AND MEASURE. All of these are good books, pitched at roughly the same level. Here are a few more specific reactions:

1) Measure theory background: Resnick & Billingsley assume no background in measure theory and interleave the relevant measure theory with probability. Durrett & Williams have appendices on measure theoretic results which cover more or less the same ground.

2) Mathematical level: Resnick is a easier than the others. He spells out lots of details in the proofs that are either left as exercises or omitted altogether in the other books. I found myself reading a statement in Resnick, asking myself why the statement was true, working out the answer easily--only to find that Resnick provided the details shortly thereafter. Sometimes this is a good thing, sometimes a little tedious.

3) Style: I'd rate Resnick below Williams and Billingsley. Williams has very elegant proofs and covers as much material as Resnick in half as much space. Billingsley is wonderfully eclectic and makes connections to lots of other areas of mathematics. Resnick is easy enough to understand, but is much more workman-like.

I think Resnick fills an important niche in this literature. I think it's a good book for teaching. I also refer to it frequently when I'm confused by something in the other books. It's thorough, relatively easy to understand, and seems to be accurate.

Reviews (14)

I Hate It When Books Lie About Mathematical Requriements
The book says that its only prerequisites are calculus and probability.This is not true.To be able to understand everything that's going on, you'll need to have a very good grasp of subjects like measure-theoretic probability, Hilbert spaces, and functional analysis.I quit reading the book in the early chapters, when Steele starts talking about things like "spans" and "denseness" for function spaces.I don't know where you went to school, but at my school, I didn't learn these subjects in my intro calculus and probability classes.To summarize, don't buy this book if you don't know measure theory.

If you want to learn quant finance at an elementary level, Baxter and Rennie is much, much better.Moreover, if you're comfortable with measure theory,and you want to learn the math that's necessary for option pricing, you'd be better off buying Oksendal's excellent book, which is at least as rigorous as Steele's book but much more clear.

Riskfreeprofit !!
The book is at the interface of three areas, math, statistics, and finance. While connections between the first two have a long history, it was the connection to finance that caught my attention. Coming from math myself, I needed first to take a closer look at the book to orient myself. The mathematical subjects, smooth sailing, include stochastic differential equations (SDE) as they relate to PDEs; and the ideas from probability and statistics include Brownian motion, martingales, stochastic processes, and the Feynman-Kac connection. Browsing the chapters I found them to be a lovely presentation of ideas with which I am familiar. For me, it was chapter 10 that turned out to have stuff that I wasn't familiar with. That is the finance part, and it is based on a model for Option Pricing developed in 1973 by Fischer Black and Myron Scholes. An arbitrage opportunity [simplified] amounts to the simultaneous purchase and sale of related securities which is guaranteed to produce a *riskless* profit. It was after reading more in this chapter I understood why the book is used in a course at the Wharton School at the University of Pennsylvania. I am impressed with the level of math in this course. Part of the motivation in the applications to finance is that arbitrage enforces the price of most derivative securities. And I learned from ch 10 that the SDE of the Black-Scholes model governs the processes which represent the two variables S, the price of a stock, and B the price of a bond, both S and B representing stochastic variables depending of time t, i.e., both stochastic processes. In the model, S is a geometric Brownian motion, and B is a deterministic process with exponential growth. The two are determined as solutions to the SDE of Black-Scholes.

Reviews (3)

On balance, this is a great book
Grimmett & Stirzaker's book, *One Thousand Exercises in Probability, 2nd ed.,* contains exercises with answers/solutions to each and every exercise covering a wide range of topics from probability. Although this book is a companion to G & S's 3rd edition of *Probability and Random Processes,* the authors strove to make the *Exercises* book stand on its own.

The 2 previous reviewers took a widely different view of this book. I wanted to have a balanced approach to reviewing this book. I hope this helps.

-- This book has exercises that cover a wide range of topics in probability. It starts from basic issues in probability and eventually covers topics like queueing, Monte Carlo (& Markov Chain Monte Carlo), Ito's lemma & financial option valuation, etc. Any one vaguely interested in probability realizes that the topics covered by Grimmett & Stirzaker are *hot topics* and very useful to those who want to try to get a sense of how important probability theory is in real life. I can't think of a comparable book that is so ambitious and covers so much useful ground in one place. [At least not one with answers to every question.]

-- I am not personally familiar with Grimmett's work (I believe he is at Cambridge University). I am, however, familiar with Strizaker's work (he is at Oxford). I consider him to be one of the finest expositers of probability theory. Stirzaker's views on probability theory (which I read in a different work) is one of the most lucid and sensible I have ever come across. Anyone seriously interested in probability should try to get exposure to Stirzaker's thoughts on the matter.

-- I believe this book is GREAT for self-study. One of the major problems I have with many math, science, engineering, and other technical books is that -- even very good books -- do not provide answers/solutions to the questions they pose. As someone who is very interested in self-study, I find a book like this one -- which has the answers/solutions to ALL of the questons -- to be extremely refreshing and welcomed.

To address the prior criticism that this book does not contain a sufficient amount of detail in the solutions .... I would suggest that such a criticism is unfair. As I pointed out above (and most people know) it is very rare to have technical books like this where there are a great deal of interesting and useful exercises given plus answers/solutions to all of the questions posed. For 2 emminent Oxbridge dons to write such a book is even more exciting.

In all fairness, Grimmett & Stirzaker wrote this book with the intent that it be used along with an appropriate probability textbook(s). Any one willing to take the time to look at the solutions given along with a companion text should be able to work out what went wrong (or right) with any question that the reader attempts to work out.

In closing, I highly recommend this book to anyone who is interested in going from a novice level at probability to a point where you can approach and solve useful problems in probability.

Well worth the effort
I appreciate the other reviewer's oppinion as it is indeed true
that some of the problems' solutions are succinct.

However it is clearly not the point of the set of books to provide
the reader with fully worked solutions to every problem. It is
often like this in maths, you just got to go for it, no matter.

If you kick off from the first chapter and attempt the problems
in increasing order of diffculty keeping the course book by your
side, you should be fine, really.

This course is very rewarding.

Reviews (10)

The only way to learn probability
Grimmett & Stirzaker develops probabilty theory rigorously and simultaneously develops a good understanding of how to use the theory in practice.Explains important theorems, uses good examples, etc.I was (inexplicably) assigned Casella & Berger to learn probability and, since that was completely useless, I used Grimmett & Stirzaker very effectively to learn it on my own.

Not good at all
Authors have the knack of throwing in suppossedly profound but obscure comments, probably to show the depth of their knowledge. Doesn't make a pleasant reading at all.

Beauty magazine makes you look ugly, if you get my drift.
Beauty magazine makes you look ugly, if you get my drift.

Reviews (4)

Outstanding book!
An extremely well written, readable and thorough book.Worth every penny

A good textbook for beginers.
This book is a very good textbook for beginers, especially mathematicians or physicists with PDE backgroud. Unfortunately, PDE is not a cure-all. It may be cumbersome in many cases. Like most research papers by the sameauthors, it lacks diversity.

Reviews (3)

Sloppy, with gross errata.
The two stars are for the one good feature this book has: an extensive table of symbols, listing the first page of text on which the symbol appears in the book.This rare feature could well be adopted by all math textbooks.But sadly, while a meaning is linked to each symbol, the grammar of the concatenated symbols is not addressed, to the continued detriment of the reader.There are some really sloppy errors, even in the exposition of axioms, where precision matters most.Very poor print quality. Outdated and massively ovepriced.

An Excellent Introduction
Perhaps because it is a Foundations book -- in my mathematics training it always seemed that the people who did the best job of motivating and explaining (or at least making you feel you understood) the material were Foundations people -- but this book has a presentation polished to the point where the closest genre of mathematics text in level of polish would be intro calculus books, where the problems theorems and proofs have been worked over for many many many years.Here, however, the material is in great part relatively recent - probably the closest to contemporary stuff you can see as an undergraduate -- in Real Analysis, by contrast, you may well just be coming out of the 19th century by graduate school.This polish, I have discovered in later years, facilitates use of this book for self-study and it is a wonderful text for providing rapid refreshment of important concepts.I have over the years referred back to it on a number of occassions and have always been pleasantly reminded what a wonderful book it is.
This is a very nice book and the best introduction to the material I have seen (although, given the number of intro books I have seen on the topic, this may not be a strong statement).

Reviews (50)

Extraordinarily clear and intuitive
I suggest all to hedge your risk of ignorance by taking a long position in this book immediately!

A good first draft not a useful text
I used this text for a graduate course on financial derivatives in an applied mathematics program. The text generally made a good selection of the topics covered, but often key insights and proofs were missing. There were few useful examples and the end-of-chapter exercises were both too few in number and were not well thought out.The material could have been better organized and less meandering. A more compact and rigorous theoretical presentation surrounded and amplified by lots of good examples--including some involving numerical techniques--would have been both deeper and more accessible to students. As an instructor, I found the text was usable, but required me to add a great deal of my own material for the students. Finally, the index is just about useless, making it difficult to use this work as a reference.

Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)