» Counterexamples in Analysis (Dover Books on Mathematics)

Counterexamples in Analysis (Dover Books on Mathematics) Reviews

Customer Rating:
Summary: The most important math book an undergrad can buy
Comment: 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.

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Customer Rating:
Summary: Classic book, now IN PRINT from Dover
Comment: 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".

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Customer Rating:
Summary: Recommended by Analysis Professor
Comment: 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.

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Customer Rating:
Summary: Fascinating and Useful; Maybe a Tad Too Focused
Comment: 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."

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Customer Rating:
Summary: Amazon, the most trusted
Comment: I always trust Amazon.com. I bought the book with low price. Brand new! A very good book which will assist you in understanding of mathematical analysis! Sometimes, if you do not understanding the accurate meaning of a concept, a counterexample would make it easy.

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