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Counterexamples in Analysis (Dover Books on Mathematics)
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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).Read more ›
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.
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.Read more ›
Most Recent Customer Reviews
This book helps unfold so many misters where the mind has difficulties in producing examples that are not intuitive. Read morePublished 6 months ago by Dr. Q
Very useful for students. I like to flip through it, and sometimes my professor mentioned examples/counterexamples that were in this book. I noticed he had one in his office.Published 6 months ago by Fi
Definitely helped "guide" my intuition in real analysis. The format is exactly as I had hoped: very easy to find what you need.Published 9 months ago by QQ
A very advanced mathematical book which constructs lots of count-examples those you can never think of. A good book to study in details.Published 9 months ago by Simon YAU
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 morePublished 24 months ago by Jonathan Puigvert
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 morePublished on June 28, 2013 by Shiyang Li
Graduate student in Mathematics here. Great resource especially for the student looking for easy examples for qualifying examinations. A must have book!!Published on January 4, 2013 by JIMLODGES
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 morePublished on November 24, 2011 by Verrückter