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.

I had studied math as a young man and had long felt I could do a better job. I started with Understanding Analysis and I'm nearly done with chapter one. The pedagogy is at a nice level for self study and I really like how the exercises are woven into the topic. First rate text book.

I currently own Baby Rudin, Kolmogorov and Fomin, Marsden, John Royden, Probability and Analysis and this book I think if this book provides a solutions manual, it will be the best self-study book. Sure Rudin is elegant and very general in a sense that it covers huge amount of materials through pithy and concise yet rigorous proofs but this is exactly why i don't think it is a good book to start. When i started with Rudin because of its fame, I was very frustrated because of the density of the book. It is very heavy reading that you need to explicate in your head to understand. This is all good and well if you already grasp the idea but for first time learners rudin should be used as a reference. ( i am sure lots of people disagree, this is just my view). This book on the other hand is super friendly and when you read it, it does the explicating of the ideas for you. it is as if you are listening to a professor who does the thinking for you. OF COURSE this is not GOOD for learning how to proof stuff. That, one must learn by himself. However, When one wants to learn the ideas and grasp the general overview and the beauty of analysis, it serves one well. I recommend this book to anyone who wants to leisurely yet rigorously learn analysis. usually, those two words are oxymorons but this book combines them - leisure and rigor come together.

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.

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 =<a for all a in A. Without having the benefit of subscripts and a proper less than/equal to sign--"a_0" is "a naught" and "=<" is "less than or equal to"--that is the definition verbatim. Because of sloppy editing, I have a textbook that is telling me that a set may have elements greater than the maximum element. Otherwise, I find the book to be the most readable math book I've ever had. The homework exercises are appropriately challenging. One thing that I really like with this Kindle edition is that when past theorems and definitions are invoked, there is a hyperlink. There is also a handy search function. All in all, I'm really very happy with the book and have no problems using it for my class. My only issue is with the editing.

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.

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

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.