Understanding Analysis (Undergraduate Texts in Mathematics)

out of all the intro analysis books this is the most readable. it sticks to R and the usual metric, so no mention of metric spaces. So a more advance is needed after. What is does it does well. I would read Buck advance calculus right after,browder, apostol right after. one more thing, the book lacks a dedicated section for the lim sup and lim inf, something that is needed for Lebesgue Measure/Integral. The best explanation I have seen for lim sup/lim inf is from Gaskill: Elements of Real Analysis. Browder has a section, but it can be hard to read, I have an MS Math from a low ranked skill, and used the above books to get an understanding for Analysis. The analysis at my school is trash, but Abbot allowed me to gain the fundamentals to do so. A must buy!

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 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.

I really like the tone of this book and absolutely love the historical motivations provided by the author for all of the topics presented. In my humble opinion though, the book lacks sufficient rigor and detail to really serve as a complete introduction to analysis. The proofs lack sufficient detail and many of the theorems still rely too heavily on an intuitive understanding rather than a rigorous axiomatic treatment. Although completely lacking the excellent motivation for the theorems provided by Abbot I would highly recommend Apostol's Mathematical Analysis (considered a classic by many) as the ideal introduction to the "heavy" analytic treatment of calculus. The absolute perfect combination would be both books together (which is what I ultimately used to understand the basics of analysis). I don't disagree with the other reviews on how excellent the book is but I don't know that it can really stand alone. For instance, it would be a considerable leap for someone to read this book and then jump in "feet first" into Rubin's Real and Complex Analysis, which someone should be able to do after completing an intro to analysis textbook.

It is great for self-study.

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.

It's a great book!

This book is such a wonderful introduction to analysis that I am wary I cannot do it justice. It stands in its own right and for those who do not appreciate Rudin's brevity this book allows you to appreciate Rudin's brevity.

A starting course on functional analysis and some proof knowlegde would be fine before starting the reading, but it has very good examples and excercises to work through

Writes a dense subject in a very engaging manner

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Well written. Good examples and exercises.