Counterexamples in Analysis (Dover Books on Mathematics) by Bernard R. Gelbaum (PDF)

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. 12 figures. Bibliography. Index. Errata.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐What a fantastic idea for a book. Every math undergrad needs this. Should be required.

⭐I bought this to get help with limits of step functions. I’ve primarily looked at Example 24 of Chapter 2.Given the need to consider many examples, one can’t quibble with a terse presentation. However, in the given example, it’s difficult to see how the sequence converging to g_n(x) can be chosen to be “decreasing”, as opposed to “non-increasing.”Overall it’s a great book and came to me in great condition!

⭐This book helps unfold so many misters where the mind has difficulties in producing examples that are not intuitive.What this book taught me later in my professional life: put a lot of effort to break things that are important: methods, software (yes! software), solution paths etc. For example, in software it is important to spend resources to break a code – we can see very well the merits, but it is hard to look under the hood.

⭐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. Depending on your preference, the notation can sometimes be a little awkward, but it is neither incorrect nor ambiguous; that is, it is still correct and clear, depending on your understanding. Any mathematician ought to have this book in his collection for reference.

⭐This book is an incredible help to those trying to learn analysis. It does a great job showing which “obvious” statements are actually false in a clear and concise way. When theorems seem too abstract the counterexamples in this incredibly cheap book come to the rescue.Even if you have no trouble with the theory you will need to have examples in your head to do later proofs, especially in other subjects such as functional analysis.This should be a companion course text it’s so useful.

⭐One might question the taste of such a book but helpful in conjunction with baby Rudin to prepare for real analysis test in college on the first exposure.

⭐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, above all, this book is dirty cheap! It’s difficult to find a better book for even a heftier sum, this book is a must have for anyone interested in real analysis!

⭐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.

⭐Un riferimento utilissimo per chi vuole addentrarsi in una materia tanto complessa quanto affascinante come l’Analisi (Reale e Complessa). Spazia dagli esempi più comuni a quelli più elaborati e relativi agli argomenti più avanzati.Necessario per una più profonda comprensione dell’analisi. Molto esempi sono veramente significativi e costruiti in maniera originale. Consigliato a tutti i livelli di conoscenza.good book for graduate students and teachers of Analysis

⭐Perfect for everytime that one risks to fall into ingenuos intuitions

⭐Need to give some more examples in a proper way.

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