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Counterexamples in Analysis (Dover Books on Mathematics) [Paperback]

Bernard R. Gelbaum , John M. H. Olmsted
4.7 out of 5 stars  See all reviews (18 customer reviews)

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Book Description

June 4, 2003 0486428753 978-0486428758
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. 12 figures. Bibliography. Index. Errata.

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Counterexamples in Analysis (Dover Books on Mathematics) + Counterexamples in Topology (Dover Books on Mathematics) + Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics)
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 218 pages
  • Publisher: Dover Publications (June 4, 2003)
  • Language: English
  • ISBN-10: 0486428753
  • ISBN-13: 978-0486428758
  • Product Dimensions: 8.5 x 6.4 x 0.5 inches
  • Shipping Weight: 8.8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (18 customer reviews)
  • Amazon Best Sellers Rank: #181,097 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
230 of 233 people found the following review helpful
5.0 out of 5 stars Great for the coffee table July 22, 2003
It can happen to anybody. There you are, minding your own business, when the though hits you: Does every continuous function have a derivative somewhere? You try to prove that it must. It sure seems like it must. How could it not? Hours slip by, and you've made no progress. What do you do? You pick up Gelbaum and Olmsted's classic "Counterexamples in Analysis". There on page 38 is an example of a continuous function that has no derivative; none; anywhere. No wonder you couldn't prove it.
It turns out that questions of the form "Does A always imply B?" entail proofs with two very different flavors, depending on whether the answer is affirmative or negative. The affirmative variety can be very difficult, as it usually deals with an infinity of things. But a negative answer requires only one solitary example of an A that is not a B; this is affectionately known as a "counter-example". These are the slickest little proofs around--often a one liner--and they can provide a lot of insight. Here's a trickier one: Are all linear functions continuous? Surprisingly, the answer is "no", which means there is a counter-example. Gelbaum and Olmsted show how to construct a discontinuous linear function. Case closed. They also provide examples of
A perfect nowhere dense set
A linear function space that is a lattice but not an algebra
A connected compact set that is not an arc
A divergent series whose general term approaches zero
A nonuniform limit of bounded functions that is not bounded
I won't give away any more (although there are hundreds).
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57 of 60 people found the following review helpful
5.0 out of 5 stars Test the limits ! October 4, 2004
In my teaching of the basic tools of mathematical analysis; and even going back to my student days, I noticed the hurdle that separates the beautiful definitions from the `messy' examples. Often students tell me that the theory looks so easy, `but how do we construct an example to illustrate the limits of the theory?' -----A counter example?

Part of the difficulty is that the definitions involve quantifiers; and how do you check the quantifier `for all' ? And on top of that, there are the axioms of set theory: the axiom of choice, or one of its equivalent variants.

The lovely little book by Gelbaum-Olmsted was a savior to many of us when we started out in math, and it appeared first in 1961. But I had almost forgotten about it until by accident (while browsing in the bookstore) I stumbled over a new edition of it about a year ago, a lovely Dover reprinted edition. And so affordable !

In all the other books you learn about the wonderful things that are true about convergence, sets on the line or in the plane, modern variants of the so called Fundamental Theorem of Calculus, and in Gelbaum-Olmstead you learn the things that aren't true. And then there are all the lovely Cantor constructions, The Devil's Staircase, space filling curves, and much more; beautiful, but little known constructions going back to Lebesgue, and some to Riemann.

But more importantly the book gives students an edge when they have to do the assigned exercises in your analysis course. Many told me that the book is a 'secret weapon'.

Palle Jorgensen, October 2004.
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71 of 77 people found the following review helpful
5.0 out of 5 stars A Bestiary of Analysis Monsters December 8, 2003
By A Customer
Format:Paperback|Verified Purchase
For 200 years after it was invented by Isaac Newton, calculus lacked a rigorous foundation. In the 1800's the missing rigor was finally provided by the ingenious theory of limits, developed by Bolzano, Cauchy, Weierstrass, and others. This development, in turn, revealed the need to formulate and understand the structure of the real numbers, which was achieved by Cantor, Dedekind, and Peano, who showed how the real numbers can be constructed from rational numbers, which are in turn constructed from integers, which are defined in terms of set theory.

But it was a Faustian bargain, because immediately a host of bizarre and counterintuitive examples were discovered - continuous functions that were nowhere differentiable, nonmeasurable sets, one-to-one pairing of points between the line and the plane. These peculiar entities were deeply disturbing to many.

Poincare said "Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose... In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that."

These counterexamples displayed features that were nowhere to be found in the physical universe. When Richard Feynman was a physics graduate student at Princeton, he enjoyed teasing his mathematician friends that mathematics was so easy that he could instantly decide the truth or falsehood of any mathematical statement they could give him.
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Most Recent Customer Reviews
5.0 out of 5 stars Awesome book!
Ok this is among the best books in real analysis you can find around. The counterexamples are a big help in understanding the real thing, the explanations are crystal clear and,... Read more
Published 4 months ago by Jonathan Puigvert
5.0 out of 5 stars Really interesting topic
Either historically or logically, it is usually those counterexamples that build our reasoning and lead to the strict and beautiful mathematics world as we see. Read more
Published 12 months ago by Shiyang Li
5.0 out of 5 stars Great!
Graduate student in Mathematics here. Great resource especially for the student looking for easy examples for qualifying examinations. A must have book!!
Published 18 months ago by JIMLODGES
4.0 out of 5 stars A Really Cheap yet Great Resource for Analysis
I really like the compactness of this book. Seriously, It gives almost all the major definitions which you would ever need, and it provides all kinds of nontrivial examples. Read more
Published on November 24, 2011 by Verrückter
5.0 out of 5 stars You can't understand something without counterexamples.
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. Read more
Published on September 24, 2009 by Fephisto
5.0 out of 5 stars The most important math book an undergrad can buy
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... Read more
Published on July 28, 2008 by Hoa Hong
5.0 out of 5 stars Classic book, now IN PRINT from Dover
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. Read more
Published on September 13, 2007 by Interested Observer
4.0 out of 5 stars Recommended by Analysis Professor
Counterexamples in Analysis was recommended by our professor as a resource for a course, Introduction to Analysis. Read more
Published on March 13, 2007 by Stats Student
5.0 out of 5 stars Fascinating and Useful; Maybe a Tad Too Focused
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. Read more
Published on June 27, 2006 by Kay Linda S. LaVida
5.0 out of 5 stars Amazon, the most trusted
I always trust I bought the book with low price. Brand new! A very good book which will assist you in understanding of mathematical analysis! Read more
Published on October 25, 2005 by Xianglu Han
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