Elementary Classical Analysis

Customer Reviews:

Good book - lot's of mistakes.......2007-08-26

Most people taking a course like this use Rudin and this is a reasonable supplement but there is a good bit of errata that should be downloaded from the author's site. Pugh's "Real Mathematical Analysis" is an alternative.

Loved it as a student and as a professor.......2006-10-27

This was my favorite book as an undergraduate student and I've taught from it as a professor. It is an excellent geometric approach to analysis. It can even help students who have difficulty with epsilon delta proofs understand the geometric intuition behind them. The construction of the real line at the beginning is daunting for students who aren't clear about set theory and sequences already but a few supplementary materials can help the students out there (see my webpage notes on real analysis for example). The proofs are hidden which makes it a challenge for students to try prove everything themsleves before peeking at them, but they are available. Just remember to tell your students where they are!

As a student I loved the book because it allowed me to learn everything on the metric space level while allowing students who prefer to stay in Euclidean space to do that. Now I am a metric geometer.

very helpful book.......2006-05-31

I am using this book to teach myself analysis. Because my mathematical background is limited, I cannot assess what the book is missing, or whether alternative methods of presentation would be more insightful. But in terms of clarity and comprehensibility, the book does very well. The authors write very carefully and are not cryptic; the proofs and examples are well-presented, and I rarely feel lost. The book is rigorous but not, let's say, snobbish. I am learning a lot from it.

best suited for the best students.......2006-05-18

Many other reviewers have panned this book. The overall sentiment seems to be that the book is too difficult to follow. Perhaps for them. And, granted, perhaps this is so for many readers. But some students, who are probably majoring in maths or physics and who might be amongst the top in their classes, are likely to appreciate the book.

It is a rigorous explanation of classical analysis. Frankly, for someone who will not major in maths, you are unlikely to need this level of rigour in your understanding and usage of the maths. Even theoretical physicists. But you can regard it as a good part of your maths education. If you have learnt introductory calculus at the level of Apostol or Spivak's books, then that level of rigour is continued here.

The proofs can be quite difficult to follow. It is for good reason that Marsden segregates these into the ends of the chapters. The fact that these proofs are difficult is perhaps misread by some reviewers as a flaw in Marsden's writing. Wrong. Some proofs are inherently difficult, and need a detailed and careful presentation. The Heine-Borel Theorem, for example.

Which is why I find puzzling claims by some reviewers of many errors in the text. Are they referring to simple typos? Or errors in the logic? If the latter, maybe they should cite specific cases. I went through an earlier edition, as a student, and studied carefully most of the proofs. Beyond some typos, I never found any logic errors.

a poor job!.......2005-12-09

Unfortunately this book is used at many good schools, but it should not be! Given that there are many excellent texts it is hard to understand why this book is chosen so often. The author is very brief with his proofs as though the reader is reviewing the material and not learning it for the first time. On the other hand, he explains some simple topics excessively when it's not at all necessary. I think the reason for the style of this book is that Marsden is really an applied mathematican. I would not use this book for self-study, unless I had to. There are also many errors in the book. It is possible to make use of the book by supplementing it with other books (almost any book on analysis would do!). In short, Marsden does a beautiful job of turning analysis into an ugly subject! Be warned!!

History: Fiction or Science? (Chronology, No. 1)

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Calculations are only as good as your numbers
Pants on fire?
Accepted History & Chronology Must Be Changed.
Very Interesting
History as Science Fiction

History: Fiction or Science? (Chronology, No. 1)
Anatoly Fomenko
Manufacturer: Mithec
ProductGroup: Book
Binding: Paperback

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Similar Items:

History: Fiction or Science? Chronology 2 (Chronology)
History: Fiction or Science? Astronomical methods as applied to chronology. Ptolemy's Almagest. Chronology III
Discovering the Mysteries of Ancient America: Lost History And Legends, Unearthed And Explored
Before the Pharaohs: Egypt's Mysterious Prehistory
They Cast No Shadows: A Collection of Essays on the Illuminati, Revisionist History, and Suppressed Technologies

ASIN: 2913621058

Book Description

Recorded history is a finely-woven magic fabric of intricate lies about events predating the sixteenth century. There is not a single piece of evidence that can be reliably and independently traced back earlier than the eleventh century. This book details events that are substantiated by hard facts and logic, and validated by new astronomical research and statistical analysis of ancient sources.

Customer Reviews:

Calculations are only as good as your numbers.......2007-08-03

Yes, we can all agree that mainstream history is nearly 100% BS due to politics, economics, ego, problems with dating techniques, and various conspiracies. Agreed. But, I've been researching the distinct possibility that human history (in terms of civilizations) are much more ancient than we've been told, so coming across this book was very interesting to me. I wondered how Fomenko could be wrong (if at all) because he is very persuasive in his presentations. Then it dawned on me. If at previous times in prehistory, due to the various catastrophies that are well documented (comets, asteroids, planetary disruptions, plasma discharge, pole reversals, etc) the Earth was in a different position in relation to the sun, different tilt on its axis, different orbit, different rotation (in terms of velocity and DIRECTION), and the continents were in different positions, then would this not cause the ancients to see the sky (constellations) differently? In other words, is Fomenko making erronious assumptions about the physics of the Earth in pre-history, which then corrupt his data with regards to dating the relevant astrology? The last event to seriously disrupt our planet occured roughly 3500 years ago, according to other good researchers, so is it possible Fomenko has been confused by this? The vastly different physics of our planet in the not so distant past may explain this confusion, which is not to say the "mainstream" version of history is correct; on the contrary. I am not an expert in these fields, but wanted to see if this idea could spark discussion.

Pants on fire?.......2007-07-19

Will people ever read before spamming? Yes, Jesuits could not rewrite world history alone, they had help. Anyway, Dr Prof Acad A.Fomenko does not point to jesuits as the driving force of world wide history manipulation in published volumes 1,2,3;, actually he barely mentions the poor devils. Check it with 'Search inside' feature, please. China is rarely mentioned either, in fact, Dr Fomenko is completely eurocentric. Right, his theory contradicts all mainstream schools of history, because in their actual state they are all built on blatantly erroneus chronology. You don't need a mysterious cabal (conspiracy) to falsify history, the falsification is its modus operandi. It is inherent to history(ians) to falsify (distort) events, as it is inherent to humans to boast as it is inherent to power (authority) to legimize itself by referrring to glorious past made to its own order. Dr Prof Fomenko and team have identified scores of instances of such manipulation in Russian, European, etc.. history, and delivered valid statistical proof thereof. His own 'reconstruction' is completely another story. Forget c14 as a valid method of dating. W.Libby has initially discovered a brilliant method of INDEPENDENT dating. Too bad, c14 method has become a joke after a forced marrige with dendrochronology with consensual chronological scale inbuilt. Radiocarbon method can't stand blind tests, but is so very productive as a rubberstamp.

Accepted History & Chronology Must Be Changed. .......2007-04-09

There is no doubt that history as most know it is a sham, & institution's version of History both University & Church is fradulent & inaccurate. Everything was established with an agenda, The real "Dark Ages" are now when we have access to incredible amounts of information past authorities & more important 'common folk' didn't have but our institutions & educators are slow to evolve because of what has ignorantly & arrogantly been taught for too long. This is on many subjects not just Chronology.

For anyone to question "Why would a Mathematician have anything credible to say of History?" The answer is from Dr. Fomenko's preface in the book: "It would be worthwhile to remind the reader that in the XVI-XVII century Chronology was considered to be a subdivision of Mathematics." These volumes could possibly be some of the most important works to date & should be read by everyone with an interest in History, especially professors & educators who have a duty to the public. I have read both books & must say that 'Chronology 1' has some very eye opening & revolutionary information. Even if these volumes are part true the implications are profound & opens the doors to further investigations & questions which must be done. I speak several different lanquages & must say the logic Dr. Fomenko uses with "inflection" of words & words being read from left to right in one region & right to left in another then written backwards, the removal of vowels & get down to basics of words, or different cities & locations having the same name etc. is correct. Vowel usage has always been optional & varied, actually complicating linquistics & study. The first thing one has to understand is that words never had a fixed spelling in history like we do now, the spelling of words was mutable & regional, as well as names & titles of people were vast, varied & changed, NOTHING WAS FIXED or understood linear. Matters of Life & Death as well as financial profiteering yesterday & today were & are made with ignorant, illogical & conspiratorial views of history & reality, it's time people get closer to the Truth & society collectively grow up.

Very Interesting.......2007-03-07

It is a good proposal and I believe it will mature into something even better in the future. I think it deserves to be read.

History as Science Fiction.......2007-01-10

Anatoly Fomenko has written a very intriguing book, full of pictures, charts, and computer 'proof' of his thesis: backwards of AD900 we don't really know what happened or when. Between AD900 and AD1600 there is more certainty, but there is still a lot of fuzzy ground, and things don't get reliable until we get past the 1600's where the printing press made it very difficult for the perpetrators of this timeline manipulation to change anything that had been committed to print. The Dark Ages did not happen. Books were burned for a reason. One organization has doubled the actual length of its existence by expanding the real chronology. Read why.

I had always wondered why Christ died about AD33 and yet men waited until the 11th century to form the Knights Templar, the Cathars, etc and go after the Holy Land by force. Why the 1000 year gap? Turns out there wasn't more than a 10-12 year gap and he proves it using astronomy. This also implies that the planet is not as old as we have been told, and current Christian and other creationist scientists are already championing that idea without being aware of Fomenko's book. The two groups, creationist scientists and the Russian mathematical analysts corroborate each other. Fascinating.

Of course, all this flies in the face of what we have been told traditionally is the 'proper' chronology of western civilization, and most readers will experience 'cognitive dissonance' in reading this book. It means that our history going backwards from AD1600 becomes progressively more incorrect and unreliable until it cannot be trusted at all... in the space of 700-800 years.

Naturally, the curious, open-minded reader will want to know WHO did this, WHY, and did any of the events we think of as really ancient ever happen?
Dr. Fomenko is a respected scientist/mathematician at Moscow State University who has already answered these questions to the satisfaction of his initially skeptical colleagues. Most of them are now believers, a few still refuse to believe (the usual diehards), and of course the western press has ignored Fomenko's work -- for obvious reasons when you read the book. The ones who perpetrated this chronology ruse have a lot to answer for. They are still with us. That's why this book is a well-kept secret.

I gave the book a 4-star rating because I was unable to check out some of his claims; those I checked were as he said. But if even 1/3 of his claims are true, this punches a big hole in what we think is our history, the meaning of western civilization, our educational process (for repeating the ruse as gospel), and the trustworthiness of the organization that perpetrated this ruse, well-intentioned or not.

This book relates to current research into a Young Earth paradigm, to John Keel's discoveries about our planet, and Fr Malachi Martin's insights (in his now out-of-print books). We are indeed sheep who are manipulated and kept ignorant -- for a reason. While knowing what these men have to say may be the "booby prize" (as in: 'what can you do with this knowledge?'), it will provide interesting reading. Didn't someone say: "...and the Truth will set you free."?? For you to judge if this book contains the truth.

White Noise Distribution Theory (Probability and Stochastics Series)

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White Noise Distribution Theory (Probability and Stochastics Series)
Hui-Hsiung Kuo
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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An Innovation Approach to Random Fields: Application of White Noise Theory

ASIN: 0849380774

Book Description

Learn the basics of white noise theory with White Noise Distribution Theory. This book covers the mathematical foundation and key applications of white noise theory without requiring advanced knowledge in this area. This instructive text specifically focuses on relevant application topics such as integral kernel operators, Fourier transforms, Laplacian operators, white noise integration, Feynman integrals, and positive generalized functions. Extremely well-written by one of the field's leading researchers, White Noise Distribution Theory is destined to become the definitive introductory resource on this challenging topic.

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masterpiece
A classic text in theoretical probability
A classic
Yakkkkkkkkkkkk............
one of the best

Real Analysis and Probability
R. M. Dudley
Manufacturer: Cambridge University Press
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Binding: Paperback

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Probability (Graduate Texts in Mathematics)
A User's Guide to Measure Theoretic Probability (Cambridge Series in Statistical and Probabilistic Mathematics)
First Look at Rigorous Probability Theory
Measure, Integral and Probability
Probability and Measure, 3rd Edition

ASIN: 0521007542

Book Description

This classic textbook, now reissued, offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The new edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

Download Description

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The new edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

Customer Reviews:

masterpiece.......2006-09-01

This is a great book. The mathematical exposition is excellent and the historical footnotes are extremely interesting.

A classic text in theoretical probability.......2005-01-28

First of all I should say that this book was written for those interested in the foudations of probability theory (the same is also true for Prof. Kallenberg's book). Therefore beginners learning real analysis and probability for the first time and those looking for applications should look elsewhere to find out appropriate books (instead of underrating such an important text like Prof. Dudley's book).
The second point to be emphasized is that this book fills in an important gap in probability literature as it reveals numerous links between this branch of mathematics and other areas of pure mathematics such as topology, functional analysis and, of course, measure and integration theory, while most books on advanced probability develop barely the latter connection, which is plainly insufficient for (future) researches on probability theory.
Finally, despite the complaint of some reviewers, the book is extremely well written and amazingly comprehensive. The sole prerequisite to reading it is a certain amount of "mathematical maturity" which perhaps these reviewers lack.

A classic.......2004-10-06

This is absolutely a classic book on real analysis and probability, although it is a little hard to read. Highly recommend to people working in machine learning and/or pattern recognition, since it provides almost all mathematical foundations needed to do deep research in these two fields, for example, on statistical learning theory.

Yakkkkkkkkkkkk...................2004-09-23

This is a text book for math major students. I believe nothing is more terrible than a book full of theorems without adequat samples. And this happen to be one. The "A Probability Path" is much better than this one.

one of the best.......2004-07-14

This is one of the best textbooks on real analysis and probability (at the graduate level). You will need a solid undergraduate course in analysis before being able to read this one. In any case, the exposition is quite elegant and clear. All the major theorems are proved. Also provides good exercises ranging from routine to quite challanging. The first half of the book presents the elements of advanced real analysis and topology (including the essentials of functional analysis); the second half presents probability theory (including martingales and stochastic processes). Very comprehensive.

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One of the better books on real analysis

A Course in Real Analysis
Neil A. Weiss , and John N. McDonald
Manufacturer: Academic Press
ProductGroup: Book
Binding: Hardcover

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Problems in Real Analysis: A Workbook with Solutions
Real Analysis
Principles of Real Analysis, Third Edition
Lebesgue Integration on Euclidean Space (Revised Ed.) (Jones and Bartlett Books in Mathematics)
Counterexamples in Analysis (Dover Books on Mathematics)

ASIN: 0127428305

Book Description

A Course in Real Analysis provides a firm foundation in real analysis concepts and principles while presenting a broad range of topics in a clear and concise manner. This student-oriented text balances theory and applications, and contains a wealth of examples and exercises. Throughout the text, the authors adhere to the idea that most students learn more efficiently by progressing from the concrete to the abstract. McDonald and Weiss have also created real application chapters on probability theory, harmonic analysis, and dynamical systems theory. The text offers considerable flexibility in the choice of material to cover.

* Motivation of Key Concepts: The importance of and rationale behind key ideas are made transparent
* Illustrative Examples: Roughly 200 examples are presented to illustrate definitions and results
* Abundant and Varied Exercises: Over 1200 exercises are provided to promote understanding
* Biographies: Each chapter begins with a brief biography of a famous mathematician

Customer Reviews:

One of the better books on real analysis.......2004-04-09

I have several books in real analysis including those by rudin, kolmogorov, reisz, etc. Of all the book I have on real analysis, this is by far the most lucid. It is thicker than all of them but only because the authors (McDonald and Weiss) spend time in explaining and motiviating the material. I have learned more in a shorter amount of time from this book than any other advanced math book I own (and I own at least 50 on everything from topology to stochastic processes to algebra). The proofs are clear and the steps are explained in layman's terms (at the level of something like Scientific American's typical readership). The material covers the Lebesgue theory of integration, abstract measure theory, mesure theoretic probability, hilbert spaces, etc. My goal is to learn stochastic calculus beyond the mere manipulation of Ito's lemma and I must say this book, more than any other, is getting me to that level of mathematical maturity quickly. Now I feel pretty confident I can understand that proof in HJM (for you finance buffs out there) involving the stochastic version of Fubini's theorem that always bugged me!

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British Style Math Book

Real Analysis
John M. Howie
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Accessories:

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Metric Spaces (Springer Undergraduate Mathematics Series)
Mathematics and its History

ASIN: 1852333146

Book Description

Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, Real Analysis covers all the key topics with fully worked examples and exercises with solutions.

Featuring: Sequences and series - considering the central notion of a limit.- Continuous functions.- Differentiation.- Integration.- Logarithmic and exponential functions.- Uniform convergence.- Circular functions All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject.

Customer Reviews:

British Style Math Book.......2002-03-10

This is an introductory text of real analysis and it is kind of British Style (in term of the way they proved the theorems). Also, some advanced topics like "Metric" and "Generalized Riemann Integral" are not covered. If you really want to learn real analysis yourself, try Robert Bartle's "Introduction to Real Analysis", Manfred Stoll's "Introduction to Real Analysis", Apostol's "Mathematical Analysis" and Rudin's "Principle of Mathematical Analysis". Stephen Abbott's "Understanding Analysis" is also an excellent real analysis text.

A Radical Approach to Real Analysis (Classroom Resource Materials)

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Getting there naturally

A Radical Approach to Real Analysis (Classroom Resource Materials)
David Bressoud
Manufacturer: The Mathematical Association of America
ProductGroup: Book
Binding: Paperback

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A Course of Pure Mathematics (Cambridge Mathematical Library)
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.)

ASIN: 0883857014

Book Description

This book is an undergraduate introduction to real analysis. Teachers can use it as a textbook for an innovative course, or as a resource for a traditional course. Students who have been through a traditional course, but do not understand what real analysis is about and why it was created, will find answers to many of their questions in this book. Although this is not a history of analysis, the author returns to the roots of the subject to make it more comprehensible. The book begins with Fourier’s introduction of trigonometric series and the problems they created for the mathematicians of the early nineteenth century. Cauchy’s attempts to establish a firm foundation for calculus follow, and the author considers his failures and his successes. The book 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. Mathematica ® commands and programs are included in the exercises. However, the reader may use any mathematical tool that has graphing capabilities, including the graphing calculator.

Customer Reviews:

Getting there naturally.......2005-08-28

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.

Models for Investors in Real World Markets (Wiley Series in Probability and Statistics)

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Models for Investors in Real World Markets

Models for Investors in Real World Markets (Wiley Series in Probability and Statistics)
James R. Thompson , Edward E. Williams , and M. Chapman, III Findlay
Manufacturer: Wiley-Interscience
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Binding: Hardcover

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What Works on Wall Street

ASIN: 047135628X

Book Description

* Considers neoclassical models in light of results that can go wrong with them to bring about better models.
* Questions the assumption that markets clear quickly.
* Offers a timely examination of the LTCM collapse.
* Written by a group of well-respected and highly qualified authors.

Customer Reviews:

Models for Investors in Real World Markets.......2002-11-27

When I looked at the cover of this book, I knew it was going to be contrarian. It shows volatility moving in the opposite direction to growth. Looking inside the book, I found the cover figure in Chapter 9 where it was described as volatility versus growth for 75 years of the Ibbotson Index starting in 1926. I looked at the Ibbotson table, and, indeed with the authors' eleven outlier years eliminated, the correlation is still negative (-.142). (With all 75 years included, I found the correlation to be (-.317).) At any rate, the Markowitzian notion of finding how large you can stand for volatility to be and then finding the portolio which maximizes growth is stood on its head.

The authors come up with an alternative to the Markowitz approach for portfolio selection based on something they call a simugram, which looks to be computer intensive.

Much of the book is spent on fundamental analysis, and indeed the authors do not seem favorably disposed to technical analysis. They dump on Black-Scholes and blame its use for the collapse of LTCM and Enron.

Some finance professionals will find much of this book annoying, since it attacks many standard concepts, such as the Efficient Market Hypothesis. And it seems to attack some of the basic tools in the finance tool kit, such as "risk neutral" evaluation.

Book Description

Volume 1 provides a comprehensive review of differential analysis in multidimensional Euclidean space.

Customer Reviews:

The Mother of All Multivariable Calculus Books.......2006-02-22

(Note: As I write this I'm halfway through the first volume)
Think of this two volume series as the Mother of All Multivariable Calculus books. It's NOT an intro to multivariable calculus for someone who has finished a couple semesters of calculus; you'll need a good stiff course (see my review of Derivatives and Integrals of Multivariable Functions by Guzman) in m.v. calculus, a dose of linear algebra, and mathematical maturity at the junior/senior undergrad level to tackle it. But if you want to go deeper - much deeper - than a first course in multivariable calculus, this is a great book.

You can look inside the book and see the contents for yourself, so I'll limit myself to general comments. The book is refreshingly free of errors (a few trivial typos are about the extent of them, at least as far as I've gotten) and well translated from the Dutch. There are hundreds of problems after the main text; although solutions aren't given in general, the material is well enough explained so that the reader should be able to solve them. (one of the authors has a web site giving corrections to the text plus some solutions - see http://www.math.uu.nl/people/kolk/books.html ).

All theorems are proved in full detail, but be aware of two things: first, the proofs are "mathematicians proofs"; short and slick methods are favored over pedagogically softer ones. (example: one of the main theorems of m.v. calculus is the chain rule. Most undergrad texts would simply prove it head on from the definitions of derivative and composition of functions, but here the authors rely on a slick piece of machinery in the form of something called Hadamard's theorem.) Second, the reader WILL have to take out her pencil and paper and fill in some details, but the good news is that the text gives enough information to make this possible in all cases.

It's a beautiful book, really - so why only 4 stars? It has to do with the old tension between the reader and writer of math texts: how explicit should the writing be? I'm dubious about the educational value of having to fill in the algebraic details, and in any case I think the text should be as explicit as possible, leaving the reader to develop his mathematical muscles in the exercises. There are a number of points where 5% more explanation by the authors would have saved the reader 50% of the effort in understanding a theorem - so I'm witholding one star as a protest. Mathematical authors beware!

Average customer rating:

Excellent textbook for review, but this is not a first course.
Decent Text on Real Analysis and Measure Theory
Ought to be popularized.
Sales Rank is Depressingly Low
Very good introduction to Real Analysis!!!

Real Analysis
N. L. Carothers , and N.L. Carothers
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Analysis | Mathematics | Science | Subjects | Books

Problems in Real Analysis: A Workbook with Solutions
Counterexamples in Analysis (Dover Books on Mathematics)
Basic Topology (Undergraduate Texts in Mathematics)
A Course in Real Analysis
Counterexamples in Topology

ASIN: 0521497493

Book 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.

Customer Reviews:

Excellent textbook for review, but this is not a first course........2007-08-30

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 mention that 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 but I 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.......2007-06-27

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........2007-01-15

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.......2003-08-15

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.

Very good introduction to Real Analysis!!!.......2001-11-06

Very good introduction to Real Analysis!!! I highly recommend it to those starting to study the subject.

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