Analysis: With an Introduction to Proof (4th Edition)

Book Description

By introducing logic and by emphasizing the structure and nature of the arguments used, this book helps readers transition from computationally oriented mathematics to abstract mathematics with its emphasis on proofs. Uses clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers. Offers a new boxed review of key terms after each section. Rewrites many exercises. Features more than 250 true/false questions. Includes more than 100 practice problems. Provides exceptionally high-quality drawings to illustrate key ideas. Provides numerous examples and more than 1,000 exercises. A thorough reference for readers who need to increase or brush up on their advanced mathematics skills.

Customer Reviews:

Great book.......2007-04-29

Analysis at this level is probably the most challenging class for an undergraduate degree. However, this book made it very manageable. I found the introduction to proof very helpful. I encourage anyone who is using this book to study this chapter ahead of time. It will make the subsequent chapters a lot easier to handle. If it was not for this book and the outsdanting professor I had, I would never have passed this class. Go for it!

good to go.......2005-09-29

The book arrived in good condition and I have not had any problems with it.

Definitely a good first text.......2002-09-05

I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book. For the self-studier, this book is excellent! I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc... For the beginner, this book is superior to all of them. A plethora of examples. Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought. If you already have had some Analysis, then this book is probably not for you. But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat. After this book, one should be able to tackle "Papa Rudin". For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis". This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration. On the whole, this is a great start! If proof-based math is new to you, then you will appreciate the first chapter on proofs. Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration. That's really only nit picking, though.

This book was surprisingly good.......2002-07-03

I didn't think this book was going to be very good, but the author has "proved" me wrong ;-) This book starts out so basic that in my class (which was the first analysis course in our math department) we actually skipped the first 1/3 or so of the book. The first 9 or 10 sections consist of stuff like basic set theory, logic, definition of a function, etc. I would think that even the most elementary Analysis books would completely leave this out and expect that the reader is already familiar with this. So if you need it, this book will be a good resource for you.

Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject.

All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis).

The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems. With enough time though the average student could probably do every problem at the end of every section.

I'd recommend this book for self study as well as a supplement to any introductory analysis course. If you have already have exposure to rigorous proof of calculus theorems, then this book will probably be too basic for you.

Customer Reviews:

Professors Should Choose Another Book.......2006-06-01

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

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

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

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

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

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

Another excellent Real Analysis Text.......2002-03-10

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

Introduction to Calculus and Analysis, Volume 1 (Classics in Mathematics)

Average customer rating:

a superb book
More than an introduction
Absolutely beautiful!
simply the Best Calculus Book
You must have this.

Introduction to Calculus and Analysis, Volume 1 (Classics in Mathematics)
Richard Courant , and Fritz John
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 354065058X

Book Description

From the reviews: "Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)" The Mathematical Gazette (75.1991.471)

Customer Reviews:

a superb book.......2006-04-05

This is a rewrite of the great book by Courant, and it does justice to its origin. I prefer the somewhat more charming original book of Courant myself, but I have taught from this one too and learned something more.

Since the original Courant costs $120 for the 2 volume set, this volume at $33 is a bargain, so snap it up. This is 10 times as valuable as most current $130 calculus books.

More than an introduction.......2005-12-15

Those books (volumes 1-2) can be seen as a new edition of Courant's classical Differential and Integral Calculus, volumes 1-2 (that can still be used for general calculus courses). The first volume was written while Courant was still alive, and the second was postumous. I believe that they are the best work to start understanding analysis. Indeed, for the general scientist (as a physicist) it contains all the theory needed for any application. The book is not easy reading though. Much of the text can be understood on first reading, but there are pretty profound sections, mostly on the appendixes, that turn the book genuinely onto a book of analysis. The second volume requires some mathematical maturity, and I doubt whether it is suitable for beginners, but it is simply the best book of multivariate calculus that I know - and it is really difficult to think of a better presentation. Courant was a giant, and his concept of mathematics shines in every page of those books (although he did not see the publication of the second volume, his hand can be seen in every page). For the serious mathematician, a must-have. For the beginner, the best way to get in love. Courant and John don't lie, they give every proof and guide you most gently in this complicated garden called mathematics. I'd give it aleph stars if it was possible.

Absolutely beautiful!.......2005-01-24

I give 5 stars to this book because in contrast with the majority of the calculus textbooks it gives the reader the perfect combination between rigor and intuiton. Another thing that I also like a lot is the fact that volume 2 has solutions to almost all the excercises, which is great because some of the problems are very difficult. I really think this book is a "must have".

simply the Best Calculus Book.......2002-08-12

An intuitive, rigorous and a beautifully conceptual approach to calculus is what distinguishes this book from the thousands of run-of-the-mill "Calculus I" textbooks published every year.

This is not surprising because 1) Courant and John were both important German-born mathematicians, both schooled in that great mathematical mecca, Gottingen, both making fundamental contributions to many classical branches of pure and applied mathematics. Courant is an especially important mathematician since he not only studied under the greats Minkowski and Hilbert - even serving as the latter's assistant - but founded the Courant Institute of Mathematical Sciences in New York, modelled on the Gottingen Mathematical Institute. 2) That typical German thoroughness and emphasis on the mastery of the "fundamental concepts", so dear to German textbooks, is evident in all sections of the book, particularly in the introductory material on the number continuum, functions, continuity etc.

The exercises at the end of chapters are substantial and excellent, and help to develop proof skills in students as well as a subtle mathematical intuition.

Mathematics is best learnt by studying books written by important mathematicians. Classic books like these should always serve to prove the truth of Abel's dictum that to master mathematics one should 'study the masters and not the pupils'.

You must have this........2002-05-28

My review of the first volume pretty much applies here as well. How many *calculus* texts have an introduction to complex variables, and the theory of analytic functions? This is the only one I've ever seen, and I don't think anyone else could make it more enriching than Courant. Useful material on vector calculus, the theory of matrices, and even introductory material on the *calculus of variations* (something we usually don't see at *all* in the undergrad curriculum) is included. It is refreshing to have an instructor like Courant, who doesn't assume we can't follow higher mathematical roads, but also doesn't sit at the other end of the spectrum, just waving a wand and "poof, here is the result".

Courant also published a standard reference work (also two volumes, I believe) on Mathematical Physics. While the level of mathematics required is post-grad, I was still able to read sizeable sections of it without getting lost.

Book Description

An in-depth look at real analysis and its applications, including an introduction to wavelet
analysis, a popular topic in "applied real analysis". This text makes a very natural connection between the classic pure analysis and the applied topics, including measure theory, Lebesgue Integral,
harmonic analysis and wavelet theory with many associated applications.

*The text is relatively elementary at the start, but the level of difficulty steadily increases
*The book contains many clear, detailed examples, case studies and exercises
*Many real world applications relating to measure theory and pure analysis
*Introduction to wavelet analysis

Introduction to Real Analysis, 3rd Edition

Average customer rating:

Great Book
Its a Solid Introduction
No!!!!!!!!!!!!!!!!!!!!
A good instructor and a will to work the examples and proofs needed
Very terse

Introduction to Real Analysis, 3rd Edition
Robert G. Bartle , and Donald R. Sherbert
Manufacturer: Wiley
ProductGroup: Book
Binding: Hardcover

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ASIN: 0471321486

Book 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.

Customer Reviews:

Great Book.......2007-10-05

I'm using this book for my real analysis course at university of illinois and I love it. Most readers seem to be upset that some of the material isn't presented as easily as it could be, but this book is an introduction to real analysis, not to math. This is not a good book for people who have never written or read formal proofs or who are not familiar with concepts like the triangle inequality. This is a good book if you are familiar with formal mathematics and have interest in real analysis.

Its a Solid Introduction.......2007-08-01

Honestly this is a 4 star book, but like many of the advanced math textbooks the average score is too low, because of reviewers who clearly did not understand what they were getting into.

Probably the best piece of advice with regards to advanced math books like this is given in the "Preface to the Student" in Sheldon Axler's Linear Algebra Done Right, he states: "You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast."

If you study from this book from that standpoint, you will get a lot out of it. But its a serious commitment.

No!!!!!!!!!!!!!!!!!!!!.......2007-06-30

Basically, the book sucks. Expensive, dry, terse, useless, garbage...for tyros, they won't get past the first chapter. For everybody else, they won't get past the first chapter. Either you have a solutions manual, or you can burn the book up.

A good instructor and a will to work the examples and proofs needed.......2007-04-30

At first, I hated this textbook. New to Analysis, I couldn't make any sense of it. But, I was trying do accomplish too much on my own without the help of the instructor. This textbook is terrible for self-study. A poor course instructor (you know the kind - the ones who read the book to you) can make learning Analysis a singularly miserable experience. However, if you're lucky enough to have an instructor that's willing to supplement and clarify the material in class (i.e. get you over the hump), you'll find this textbook quite adequate and worthy of keeping for reference. To get the most out of it, however, you'll need to be prepared to work much harder than you have in your previous math courses.

I know that there are two camps of reviewers of analysis books. There are those who complain that a text is too terse and those who say it's too wordy. This book finds a nice balance. It can be very frustrating at times...but, such is analysis. (Note that the authors warn you of this in the first paragraph of the Preface.) Mastery of the first two chapters is essential for one to succeed in the rest of the book.

I have given this textbook three stars because it let's the reader down on some topics. A couple of the proofs (L'Hospital's Rule is an example) are a bit too sparse in explanation for the beginner who must fill in the missing steps as part of his study. Also, a few topics are more notationally cumbersome than necessary, requiring the reader to be very adept with indexed summations. The chapter on Riemann integration is understandable, but falls short of most other texts in this area. (In my class, we used a different text for the Riemann Integral.)

If this is the required textbook for your upcoming Analysis course, I recommend that you read every section VERY carefully many, many times. Physically work through the examples and given proofs with pencil and paper. Many times the example proofs provide a model for the student that he/she can apply to the problems. If you still find it tough, make a separate notebook in which to write the definitions, thereoms, and even examples. Try diagramming the definitions and theroems, separating the conditions from the conclusions that they imply (don't forget what "if and only if" means).

The authors have included some very helpful appendices that should be treated with the same careful study as the rest of the book. Analysis is not a subject where one can pick up on a proceedure and calculate an outcome as one might do in Algebra or regular Calculus. One must learn to know and apply the definitions and thereoms logically. Well-known problem solving strategies still work here, but the method in which a student may be accustomed is entirely different (see How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) or, for proofs How to Prove It: A Structured Approach). Above all, rise to the challenge and don't get discouraged. All but the most gifted will find this subject difficult. Find help if you need it.

I recommend reading the linked books above before entering into analysis. Another really good book in the subject that will immensely help you is Yet Another Introduction to Analysis.

Very terse.......2007-02-11

Possible one of the best introductions to Real Analysis, compact, serious and with detailed demostrations and explanations (but, some omited or not completed). Not for the faint-hearted. Perhaps second edition was easier because third edition has more integral leaning.

J.R. Abuin R. (Senior Engineer)

Measure and Integral: An Introduction to Real Analysis (Pure and Applied Mathematics)

Average customer rating:

best book to learn measure theory from
One of the best real analysis books
A (quasi) masterpiece
A good textbook for new learners
An excellent choice

Measure and Integral: An Introduction to Real Analysis (Pure and Applied Mathematics)
Richard L. Wheeden
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 0824764994

Book Description

This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given. Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function. Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas.

Customer Reviews:

best book to learn measure theory from.......2006-05-16

this is a great, clearly written book that excels as a book to learn analysis from. the book takes a ground up approach, starting with only the positive real line and generalizing from there. being presented in the most simple context where all the abstraction is stripped away, the essence of the arguments is laid bare and thus the proofs are as understandable as possible. then, once the arguments are made and the intuition is in place, the book proceeds to generalize the results to more abstract circumstances. thus making the motivation for using more powerful tools clear.

this style can be compared to that of rudin's classic book, which is largely a disaster to learn out of. for instance, rudin's book oscillates between C and R for whatever gives the most slick, but often least insightful proof. this is fine, and even enjoyable once you understand the subject, but is terrible to learn from for most people. there isn't a better book to learn measure theory and integration out of than this book

One of the best real analysis books.......2005-07-09

For the past two semesters, our professor used this book as textbook. I didn't buy this book during that time. But when I was preparing for the final exam, I borrowed the book from library, and found it was really well-written. Basic materials are well organized and explained. Some profound and important topics are also covered, like Hardy-Littlewood max function; these materials are also explained in a clear and easy-to-follow way. Thus I decided to buy this book to prepare for my Real Analysis prelim and as future reference.

A (quasi) masterpiece .......2005-01-05

4 stars, which actually means 4.5. I don't rate it the maximum, because I think it lacks a couple of things to be perfect.

The pros:

1. the theory is built from the very ground up to the "ante-room", so to speak, of further and more advanced developments in abstract measure theory and functional analysis, in a deeply logical and clear way with the highest economy of words and of thought. From this viewpoint, for example, I don't see the fact of setting the theory in the R^n environment as a weakness: on the contrary, since it results from a deliberate choice of the authors, it actually ends up in an element of strength, because the reader/learner can take all the time he/she needs to become familiar with the "exact integration" approach of Lebesgue (which is *completely* different from Riemann's), and to visualize how things are going by using the familiar multivariable environment of R^n.

In other words: the reader can take all the time he/she needs to learn to swim, before he/she actually has to swim on the much longer and more difficult track of abstract measure theory (as a branch of functional analysis). I believe such a gradual approach to be better than a direct one, where from the very first page you are thrown into abstract measure theory, with the risk of being almost completely unable to understand what all that stuff is about.

2. The almost perfect way in which the authors build the theory and logically argument actually makes the book a fantastic school to learn the deep essence of the axiomatic method. This is its greatest strength, in my opinion: that is, the fact that in carefully going through the definitions, lemmas, theorems and corollaries (and in fact *working out* them) you can actually learn what the essence of correct mathematical thinking is. As long as I can remember, there are only a couple of other books, at the same level of this one, which are as good: i.e., Rudin 1 & 2 (the "Principles" and "R&C Analysis") and Einar Hille's "Lectures on Ordinary Differential Equations" (too bad it's definitely out of print. It would be such a great thing to have it reprinted in some economic edition).

The cons:

1. The Theorem of Integration by Substitution isn't demonstrated at all, with the possible exception of a particular case in the problems. Since it is a fundamental result and since its demonstration can be very enlightening from a geometric point of view, I think this is a weakness.

2. The part about Indefinite Integral and Differentiation (Vitali's Covering Lemma, and all the results deriving from it) isn't on the same level of the preceding chapters, and isn't as clear and well built as it is on Royden's "Real Analysis" (another great book): maybe because in the latter it fits naturally into the rest of the book (which is, in the first chapters where the theory is built from the foundations, intrinsically one-dimensional) as a necessary development of what comes before, while in Wheeden-Zygmund it seems to be forced in a book which, until that point, had been developing in an intrinsically multi-dimensional way: and this cannot happen at no cost.

Everything considered, it's worth its price (which, btw, is a little too high for a book of less than 300 pages ;) )

A good textbook for new learners.......2001-11-24

This book uses both classical and abstract approaches to introduce Lebesgue measure and integral. It starts with the classical approach and bases its presentation on Euclidean space. This makes it easier for new learners like me since it is more intuitive. In later chapters an abstract approach is also used. I find this repetition natural and helpful. This book has almost no typo. Its exercises are reasonably challenging.

It could be improved in page layout if the end of each proof is clearly indicated.

An excellent choice.......2000-10-17

This is the book we used when I was a grad student. This is indeed quite a nicely written book: logical progression of concepts, a large number of exercises of varying difficulty (hard ones have hints) and no typos (always a big plus with me). All the classical results are included. My only suggestions to make this book better would be to have some longer discussions of the concepts introduced to break the litany of definition-theorem-proofs and to include historical notes. This would make this book a little bit less dry and an even more enjoyable read. Nevertheless, this is one of the best books on the subjects, better than the book by Royden which is also used by some professors.

An Introduction to Measure and Integration (Graduate Studies in Mathematics)

Average customer rating:

Full of small errors. Excellent, and brilliant book. Very thorough.

An Introduction to Measure and Integration (Graduate Studies in Mathematics)
Inder K. Rana
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover

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ASIN: 0821829742

Book Description

Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, $L_p$ spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on.

The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study.

For this edition, more exercises and four appendices have been added.

Customer Reviews:

Full of small errors. Excellent, and brilliant book. Very thorough........2007-01-12

This is an amazing book; its clarity is outstanding throughout. However, I want to voice serious reservations about it due to an abundance of errors; I am reviewing the second edition published by the AMS.

This book has more errors than any other math book I have read. These errors include minor typographical errors like sloppy spacing, to equations with the terms included in the wrong order or on wrong lines, misnumbered references to earlier results, and occasional abuse of notation that hinders mathematical rigour. There are substantive errors as well, including the citing of a source for a proof of a theorem that is not actually proved in the cited source.

Errors aside, this is one of the clearest and best motivated expositions of measure theory I have been able to find. The book moves slowly, but never too slowly; it explores essential questions that a student should consider, like counterexamples, converses, and the subtle distinctions between different strengths of conditions. I find this thoroughness very welcome; most texts in measure theory present the most logically direct path to a bare-bones collection of useful results, an approach that doesn't necessarily help students.

The first chapter, on Riemann integration, is unique. The topic is explored in much more depth than in most analysis texts. Most students feel they understand Riemann integration; this book will likely convince them that they do not--and then it will fill the gaps in their understanding. The counterexamples in this book are outstanding--simple, worked through with clarity, and deep.

I think this book would make an outstanding textbook on measure theory, and it is one of the few texts that is good for self-study. I just wish the errors could be corrected; I would then rate it 5 stars without a doubt.

Real Analysis: An Introduction to the Theory of Real Functions and Integration

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Real Analysis: An Introduction to the Theory of Real Functions and Integration
Jewgeni H. Dshalalow
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584880732

Book Description

Designed for use in a two-semester course on abstract analysis, REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis. Self-contained, with coverage of topology, measure theory, and integration, it offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering. Structured logically and flexibly through the author's many years of teaching experience, the material is presented in three main sections: Part 1, chapters 1through 3, covers the preliminaries of set theory and the fundamentals of metric spaces and topology. This section can also serves as a text for first courses in topology. Part II, chapter 4 through 7, details the basics of measure and integration and stands independently for use in a separate measure theory course. Part III addresses more advanced topics, including elaborated and abstract versions of measure and integration along with their applications to functional analysis, probability theory, and conventional analysis on the real line. Analysis lies at the core of all mathematical disciplines, and as such, students need and deserve a careful, rigorous presentation of the material. REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration offers the perfect vehicle for building the foundation students need for more advanced studies.

Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics)

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Review of 'Lectures on the Hyperreals' by Goldblatt
Well-written, interesting subject
more than an introduction

Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics)
Robert Goldblatt
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 038798464X

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This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation etc.). The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line, Ramsey's Theorem, nonstandard constructions of p-adic numbers and power series, and nonstandard proofs of the Stone representation theorem for Boolean algebras and the Hahn-Banach theorem. Features of the text include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set- theoretic approach to enlargements than the usual one based on superstructures.

Customer Reviews:

Review of 'Lectures on the Hyperreals' by Goldblatt.......2006-03-13

Goldblatt presents a very nice view of Nonstandard analysis and its applications beginning with a historical overview and some of the ways it can simplify the calculus in the standard sense. This book is a great reference for anyone studying this interesting branch of mathematics.

Well-written, interesting subject.......2003-07-22

This is an outstanding book. First of all, the subject matter is interesting. It is often said that the worst thing about nonstandard analysis is its name. "Nonstandard" suggests that the subject is divorced from the rest of mathematics, perhaps relying on some alternative logic. The ideas, however, are completely "standard". One way to construct the reals is via equivalence classes of Cauchy sequences. The hyperreals are also equivalence classes of sequences, modulo an ultrafilter. Hence, the sequence {1/n} is representative of a hyperreal which is positive, yet smaller than any positive real number; i.e., an infinitesimal. Inverses of infinitesimals are "unlimited". These ideas permit a very simple and intuitive development of the calculus, where the derivative becomes a linear map with infinitesimal approximation error, and the Riemann integral becomes a sum over a partition with infinitesimal mesh. All the cumbersome epsilon-delta statements are banished, making for a very clean development.

Later in the book, the author introduces the concept of a "universe". Universes are essentially structures which can encompass most of "standard" mathematics: topological spaces, measure spaces, etc. It is then shown that such universes can be embedded in larger ones: nonstandard universes. This formalizes the idea that nonstandard analysis is an extension of standard mathematics, with new and interesting objects. The notion of transfer allows one to prove sophisticated statements via simple ones; e.g., see the proof of the intermediate value theorem by partitioning the domain into subintervals of infinitesimal width.

Another good aspect of the book is the quality of the writing. Most graduate-level analysis textbooks are deliberately dense, forcing beginners to spend hours per page. In contrast, this book is very easy to read, and the pages fly. This is because the author is careful to motivate the main ideas, and to include most of the logical steps in the proofs. The exercises are also excellent, being strangely both easy and instructive, making the book valuable for self-study, which was my case.

In any first edition, there are bound to be typos. This book contains remarkably few. However, the discussion of hyperfinite summation seems flawed. The author wants to sum functions over their domains, and proposes to do this by summing over the image. The problem with this, of course, is when the function is not a bijection. For example, the sum of a function which is constant and one should count the domain. However, the sum of its image is just 1. This problem is easily fixed. Define the set of all finite sequences and the function which sums them. Transfer this. Then sum functions over finite sets by making a bijection between the domain and a finite sequence. By transfer, one obtains summation of functions over hyperfinite domains.

Another small complaint: for pedagogical reasons, the author has chosen to merely state Los's theorem (on transfer), and then illustrate its use repeatedly. Although I agree with this, after becoming familiar with transfer, I reached the point where I wanted to see the proof, which should have been included somewhere at the end of the book.

more than an introduction.......2001-03-28

Most of the book on hyperreal numbers I've seen use a heavy logic formalism to treat and introduce this subject. This book introduces this concept in a very intuitive way ( which becomes more and more rigourous as the author points out different arising difficulties, and the necessity to use more sophisticated tools to avoid them) , and the first 50 pages wich give the main ideas behind the construction of these numbers can be read very easily. The historic introduction of the book by itself is a jewel. I ( a condensed matter physicist ) highly recommand this book.

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From a student of the Author
comprehension look at this book.

Introduction to Real Analysis
Michael J. Schramm
Manufacturer: Prentice Hall College Div
ProductGroup: Book
Binding: Hardcover

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ASIN: 0132298244

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From a student of the Author.......2006-03-22

I have had Dr. Schramm as a professor for 3 different classes and his teaching style has made things relatively easy and clear to follow as he has done in his textbook. I used this book for my Real Analysis Class and just reading it has allowed me to make the class a success since I tend to miss classes. The book contains a large number of exercises as well as step by step examples making life easier in this new type of math class(meaning consisting mainly of theory). Like he states at the begining of the book the topic of the book is presented through comparing the real and the rational numbers. If you had no former formal introduction to proofs the opening chapter provides solid groundwork for all the proofs that will be done as exercises later on in the book as well as the material presented. The topics of the book you can check from the index provided by Amazon.com.
Overall it is an exelent book for the first meating with Real Analysis.

comprehension look at this book........2001-03-23

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.

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