Counterexamples in Analysis (Dover Books on Mathematics) |  | Authors: Bernard R. Gelbaum, John M. H. Olmsted
Publisher: Dover Publications
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Rating:  14 reviews
Sales Rank: 49246
Media: Paperback
Pages: 218
Number Of Items: 1
Shipping Weight (lbs): 0.5
Dimensions (in): 8.2 x 5.4 x 0.5
ISBN: 0486428753
Dewey Decimal Number: 515
EAN: 9780486428758
ASIN: 0486428753
Publication Date: June 4, 2003
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Product Description
These counterexamples, arranged according to difficulty or sophistication, deal mostly with the part of analysis known as "real variables," starting at the level of calculus. The first half of the book concerns functions of a real variable; topics include the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, uniform convergence, and sets and measure on the real axis. The second half, encompassing higher dimensions, examines functions of two variables, plane sets, area, metric and topological spaces, and function spaces. This volume contains much that will prove suitable for students who have not yet completed a first course in calculus, and ample material of interest to more advanced students of analysis as well as graduate students. 1962 edition. 12 figures. Bibliography. Index. Errata.
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Showing reviews 1-5 of 14
 You can't understand something without counterexamples. September 24, 2009
Fephisto (Lafayette, IN USA)
1 out of 1 found this review helpful
Like the authors warned, I would've liked it if there were some counterexamples I thought of put in the book, but I realize this is technically infeasible.
Really, this book acts as a handy reference. "Oh, is this true? (look for counterexample) Maybe it is/No it's not." It helps quicken the learning experience quite a bit. The fact that it's only around $10 also makes it handier.
The most important math book an undergrad can buy July 28, 2008
Alice Taniyama (Houston, Texas United States)
5 out of 5 found this review helpful
I wish I had this book when I took my first undergraduate analysis course.
Up until analysis, math is easy and intuitive to just about everyone who pays attention at each step because everything studied is built upon concepts learned in grade school. You know what a triangle is, so trigonometry makes sense. You know what a rectangle is, so low level calculus makes some sense (though as it is normally taught, there appears to be some mathematical voodoo going on when dealing with limits and such).
Then analysis hits, and every student has to deal with concepts that, at the time, appear arcane and bizarre. Open and closed sets? Compactness? Sequences and series? Where did all this stuff come from, and where is the familiar math as used by engineers?
That's where this book shines. Best used as a supplement to standard analysis text, its primary virtue is that it makes all of these strange new concepts easy to grasp. Each chapter gives a brief review of concepts you might vaguely remember from prior reading or a professor's lecture, and after that it launches into useful examples that render the concepts clear and provide motivation for having a good working knowledge of the material.
This results in, as others have pointed out, a good development of intuition for analysis, and that intuition becomes the bedrock for future success. Many students limp away from intro analysis with a shaky grasp of the material that only solidifies when the same concepts show up again in future courses. This book eases that burden and erases some of the feeling of playing catch-up when the really strange stuff comes along later.
Classic book, now IN PRINT from Dover September 14, 2007
Interested Observer (Battle Creek, MI)
All the positive reviews here are true. This is an awesome book that every serious math student should own, especially graduate students preparing for qualification exams. And unlike so many graduate level works this one is a bargain in a well made Dover edition. As one reviewer notes "Just get it".
 Recommended by Analysis Professor March 13, 2007
Stats Student (Raleigh, NC)
2 out of 3 found this review helpful
Counterexamples in Analysis was recommended by our professor as a resource for a course, Introduction to Analysis. It is a support for writing proofs that makes an excellent addition to the home library for mathematics majors. For non-mathematics majors, it makes some important points more clear, but is sometimes not the quickest route to solving problems.
Fascinating and Useful; Maybe a Tad Too Focused June 27, 2006
Kay Linda S. LaVida
5 out of 13 found this review helpful
I have owned this book for years and have quite enjoyed reading in it. I must admit that I have not read it through; it is tough going. Although it is surely a great book, to my taste it is too thoroughly focused on details of pure analysis and not sufficiently attentive to pedagogy and logic. In fact, it is so focused on the internals of pure analysis that it does not even bother to tell the reader what this field is. It does not note that analytic geometry is the application of pure analysis to geometry or that analytic number theory is the application of pure analysis to the theory of numbers. For more on the nature of pure analysis see page 540 of the 1999 CAMBRIDGE DICTIONARY OF PHILOSOPHY, which included an article "Mathematical Analysis" because of the importance of the subject to philosophy of mathematics. John Corcoran might have had this book in mind when he wrote the following in his abstract "Counterexamples and Proexamples", page 460 in the 2005 BULLETIN OF SYMBOLIC LOGIC. I quote the whole abstract because I think that readers will enjoy the book even more if they have Corcoran's ideas in mind--not to imply that I agree with everything Corcoran says there.
"Abstract: Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition "every perfect number is even" is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition that some perfect number is odd. Conversely, every proexample for the proposition "some perfect number is odd" is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition "every true proposition is known to be true" is false - necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample - sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts."
Showing reviews 1-5 of 14
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