Understanding Analysis (Undergraduate Texts in Mathematics) 2nd Edition by Stephen Abbott | (PDF) Free Download

Description

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation ­ Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

User’s Reviews

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

⭐This is the most beautifully written book I have ever laid my eyes on. Abbot sent us a gift from above with this book. There never has been, and very well may never be, another textbook so well written as this one. I used this book in my first semester of real analysis as an undergrad and can confidently say that I understand the bulk of what analysis is about after having read this book. The reason I love this book is that Abbot presents an introduction at the beginning of each chapter that motivates what is about to come. Then after completing each section, he caps off the chapter with some sort of mind-blowing conclusion that builds on what you have been just studying for the last 3-4 sections. The definitions are consistent throughout the book as well. The definitions for convergence of a sequence, functional limit, continuity, uniform continuity, convergence of a sequence of functions, etc are all written with intimately close language and symbolic representation making it easy to see the similarities and differences between the definitions. Sorry Rudin, but this is the one true way to learn analysis. I highly recommend this to any professor who is thinking about using this text for their class. Anyone who attempts to use a text other than Abbot as the first exposure to analysis is doing their students a huge disservice.

⭐I got this book for an “Advanced Calculus 2” course which did integration and beginners measure theory. This book covers everything from functional limits, continuity, differentiation, integration, fourier series, weierstrass approximation theory, and so much more. I waited a long time to review this book and there aren’t enough stars in the sky to convey how I would rate this book. It is an absolute gem and worth reading from cover to cover and there is so much to learn from every problem. Shroeder-bernstein theorem is walked through, cantor sets, convergence of functions (uniform and pointwise), completion of R, Baire category theorem, and so so so much more. The amazing thing is all of these really advanced concepts are conveyed through the language of that taught to people who are learning analysis for the first time and so going back and reading this, even as a graduate student, I always find a new deeper understanding in things that I learn now (functional analysis, measure theory, etc.). This book has given me so much through the many years I’ve had it, it makes no sense to hold off on reviewing it any more. If you are serious about being successful in analysis then it is never too late to pick this book up and start reading. It is worth it for the exposition the author gives alone, explaining concepts in words and plain english sentences instead of line after line of definition and proof. Pass this gem along to your friends too, this book will be remembered as a classic when we are all old and can’t remember which way the triangle inequality works anymore

⭐It is an absolute failure of mathematics culture that a book such as this is outside the norm.Mathematics culture seems to enjoy trying to make learning mathematics as non-intuitive and painful as possible. Mathematicians seem to take sadist pleasure in only giving trivial examples (an example of a group is the real numbers. An example of a ring is the real numbers, and example of blah blah blah, is the real numbers…..)Conversely, even many many graduate level physics textbooks focus on clarity, pedagogy, and understanding. Why make things harder than they need to be?Abbott takes the opposite approach and has produced an absolute gem of book. I hope he has ushered in a new era of mathematics pedagogy. One can clearly be both rigorous AND pedagogical. It is a shame it has taken this long for a mathematician to realize this is so.Commendable

⭐We are using this text for my 400-level Real Analysis class. Having survived some terribly written texts, I appreciate the readability of this text, but I am finding lots of little spelling errors, sometimes even worse. More troubling, in terms of editing, is that whoever digitized the book occasionally seems to have gotten symbols wring, or orders wrong. For instance:Definition 1.3.4: A real number a_0 is a maximum of a set A if a_0 is an element of A and a_0 =