Understanding Analysis (Undergraduate Texts in Mathematics)

Please note that this is written from the perspective of an undergraduate student with the only fundamentals in higher level math coming a proof writing class. I picked this up to supplement the book, "The Way of Analysis, Revised Edition" by Robert S. Strichartz for a first course in Real Analysis.The good: The book is compact, easy to read, and somewhat easy to find the results you are looking for. The occasional diagram really help develop geometric interpretation.The bad: The book itself is of lower quality. The pages are thin and feel like printer paper.The bottom line: If you are bad at math, like me, and need a book to hold your hand with a list of definitions and the theorems that emerge as a result with the occasional example sprinkled in? This book does just that.Heres some more info about me. I am bad at math. I don't know how I got to this point in my life taking this class. I don't know what is going on until I go home and re read my notes and watch videos and read through examples and sit on results for a week or two... You get the picture, I need a lot of extra help. This is where this book comes in. For some overarching topic where things are never as clear as I would like them to be, I know that I can turn to this book and find the results I am looking for, my hand held the entire time. One of the most frustrating parts of analysis, for me, is that at times statements are made that seem self evident or that you take for granted. However, the whole point of analysis is that you build a strong foundation in order to justify your thinking at every step. This book offers a great scaffolding for your own thoughts or offers you a template when you feel have nothing else to work off. One definition at a time.

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 am taking real analysis at my university and I found this book to be an amazing supplement. The book that my university provides is lacking in its explanations and exercises. This book is absolutely amazing because it gives good explanations, examples and problems. It even provides some visual examples when you're having a tough time visualizing all of the inequalities. Absolutely love this book, and I recommend it to every Math major who isn't very skilled at pure mathematics yet.

This is a classic and every math major should own this book. My copy arrived with a lot of dents and the spine was cracked. They are sending a replacement.

Once in a while, a book comes along that is so wonderfully written, the reader reflexively searches for other books by its author. Understanding Analysis is a prime example of this rare breed (Unfortunately, this is Abbott's only book as far as I know: write more!).Undergraduates often begin analysis courses with dread and finish in a state of utter confusion,knowing the definitions of key phrases, and sometimes even being able to supply proofs for some elementary results, but having no intution as to why the main theorems are pertinent.But it does not have to be so. 'Understanding Analysis' has the distinction of being so readable, it is sometimes difficult to pry oneself away from its pages and attempt the exercises. On multiple occasions I found myself skimming through the book and reading the various 'special topics' (e.g. Cantor Sets, Integration, Fourier Series) interspersed throughout the book to pique the readers' interest. But most importantly, a reader will come away with an understanding of many theorems in analysis. He or she will begin to develop a vocabulary of results that make sense both mathematically and intuitively, be able to use the results to complete the exercises (which are by no means simple 'plug-and-chug' problems), and be excellently prepared for study at a more advanced level.Bottom line: Abbott's book may not be encyclopedaic in content, but it, without a doubt covers a sufficient amount of material to warrant its use for a one-semester course in analysis. My only concern is that after such a fantasticly lucid treatment, students may have difficulty adapting to the vast selection of more advanced, less pedagogical texts available. I sincerely hope Abbott writes a sequel.