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Understanding Analysis
Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it.
257 pages, ebook
First published January 1, 2000
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Displaying 1 - 30 of 48 reviews
July 25, 2025
Euclid alone...
According to John Derbyshire, mathematics is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. You know what arithmetic and geometry are, and you probably have taken a high-school algebra class. "Analysis", however, is a little obscure. The word has a specialized meaning in mathematics. It is that branch of mathematics that includes calculus. More properly, analysis is the mathematics of the continuum.
The calculus was developed in the late 17th century by Isaac Newton and Gottfried Leibniz. Newton and Leibniz however, didn't quite know what they were doing and inevitably they were a little sloppy about defining things. (This is usual when a new area of mathematics is developed.) At the heart of the problem was this: calculus is all about continuous things -- in calculus space and time are continuous. What that means roughly is that we assume in calculus that every point on the line between two points A and B exists. (There is reason to believe this may not be physically true, but that is not relevant to the mathematics under discussion.) Furthermore, we assume that a number can be assigned to every one of those infinity of points.
That is not a precise definition of continuity. Defining continuity is surprisingly difficult. The ancient Greeks were aware of the problem -- this is what Zeno's paradoxes are all about. Furthermore, the Greeks knew that no number (as they understood numbers) could be assigned to the length of the diagonal of a 1 ? 1 square.
In the 19th century this problem was figured out by European (mostly French and German) mathematicians. Some names to conjure with here are Weierstrass, Dedekind, Cauchy, Riemann, and Cantor. These are names every mathematician knows. Over the course of several decades they figured out how to rigorously define the continuum and to assign a number to every point on the line. These are called the real numbers, symbolized ?. The 19th century analysts did work of astonishing beauty, which, sadly, most people will never perceive. Analysis is now a course that every undergraduate math major is expected to take. It is generally regarded as the most difficult such math class.
In 2015, I was a professor with a 40-year career as a scientist behind me. I decided to retire and go back to school for an advanced degree in mathematics. I had never taken a course in analysis. That was a gap in my education I needed to remedy. I therefore worked my way carefully through Stephen Abbott's Understanding Analysis. This worked. In fall 2015 I took my first actual analysis course -- Functional Analysis, a postgraduate course. I don't remember my exact grade, but it was in the 90s.
So that was good -- it was why I read Abbott -- I got what I hoped from it. But I got much more than that. I was not prepared for the aesthetic experience. Math students don't talk about the beauty of analysis -- generally they are too traumatized by the effort to get through the most difficult course they have ever taken. Abbott does, though. In his preface he writes,
Blog review.
According to John Derbyshire, mathematics is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. You know what arithmetic and geometry are, and you probably have taken a high-school algebra class. "Analysis", however, is a little obscure. The word has a specialized meaning in mathematics. It is that branch of mathematics that includes calculus. More properly, analysis is the mathematics of the continuum.
The calculus was developed in the late 17th century by Isaac Newton and Gottfried Leibniz. Newton and Leibniz however, didn't quite know what they were doing and inevitably they were a little sloppy about defining things. (This is usual when a new area of mathematics is developed.) At the heart of the problem was this: calculus is all about continuous things -- in calculus space and time are continuous. What that means roughly is that we assume in calculus that every point on the line between two points A and B exists. (There is reason to believe this may not be physically true, but that is not relevant to the mathematics under discussion.) Furthermore, we assume that a number can be assigned to every one of those infinity of points.
That is not a precise definition of continuity. Defining continuity is surprisingly difficult. The ancient Greeks were aware of the problem -- this is what Zeno's paradoxes are all about. Furthermore, the Greeks knew that no number (as they understood numbers) could be assigned to the length of the diagonal of a 1 ? 1 square.
In the 19th century this problem was figured out by European (mostly French and German) mathematicians. Some names to conjure with here are Weierstrass, Dedekind, Cauchy, Riemann, and Cantor. These are names every mathematician knows. Over the course of several decades they figured out how to rigorously define the continuum and to assign a number to every point on the line. These are called the real numbers, symbolized ?. The 19th century analysts did work of astonishing beauty, which, sadly, most people will never perceive. Analysis is now a course that every undergraduate math major is expected to take. It is generally regarded as the most difficult such math class.
In 2015, I was a professor with a 40-year career as a scientist behind me. I decided to retire and go back to school for an advanced degree in mathematics. I had never taken a course in analysis. That was a gap in my education I needed to remedy. I therefore worked my way carefully through Stephen Abbott's Understanding Analysis. This worked. In fall 2015 I took my first actual analysis course -- Functional Analysis, a postgraduate course. I don't remember my exact grade, but it was in the 90s.
So that was good -- it was why I read Abbott -- I got what I hoped from it. But I got much more than that. I was not prepared for the aesthetic experience. Math students don't talk about the beauty of analysis -- generally they are too traumatized by the effort to get through the most difficult course they have ever taken. Abbott does, though. In his preface he writes,
Yes these are challenging arguments but they are also beautiful ideas. Returning to the thesis of this text, it is my conviction that encounters with results like these make the task of learning analysis less daunting and more meaningful.So, I will dare to challenge Edna St. Vincent Millay -- it is not Euclid alone who has seen beauty bare. Weierstrass, Dedekind, Cauchy, Riemann, and Cantor have also seen her. And thanks to Abbott, I, too am one of those fortunate ones
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
Blog review.
May 9, 2020
Easily the best math textbook I've ever used. One major frustration I have with university math texts that challenge their readers to think deeper is that the student's work can often feel useless or misguided. The writing can be unfocused or too presumptuous of what the the student can do. Yet, a balance must be struck against the other extreme: so simple and repetitive that a student begins to think of the concepts as a collection of algorithms to complete.
This book achieves a perfect balance of asking the reader to think deeper and persistently, gradually guiding them along. This book should be a gold standard of how to begin each topic (and the entire book) with simple concepts and problems, and grow each with complexity and difficulty until the student reaches such great heights of mathematical thinking and capability. Yet all of this happens without the reader dreading its difficulty. Material is presented so clearly, intuitively, and lightly. I highly recommend this book to anyone who wants the clearest picture of introductory analysis or an understanding of why calculus works so well.
This book achieves a perfect balance of asking the reader to think deeper and persistently, gradually guiding them along. This book should be a gold standard of how to begin each topic (and the entire book) with simple concepts and problems, and grow each with complexity and difficulty until the student reaches such great heights of mathematical thinking and capability. Yet all of this happens without the reader dreading its difficulty. Material is presented so clearly, intuitively, and lightly. I highly recommend this book to anyone who wants the clearest picture of introductory analysis or an understanding of why calculus works so well.
March 13, 2016
This is a fantastic introduction to real analysis. All the concepts are explained and motivated very well. It is great for self study, and requires only as much as the calculus sequence and a previous exposure to proofs. This is a relatively easy book, though some of the exercises are really good. Do them all if you can. My only gripes with this book is that it could at least attempt to be more ambitious in its complexity. For example, it describes the properties of the real numbers in the first few chapters really only implicitly, and doesn't mention fields. I was confused by this, as fields aren't all that difficult to understand as a general structure at this level (especially if one has, say, seen vector spaces in a linear algebra course). Ultimately, I would recommend this book to someone who is struggling with analysis, though you could certainly argue that this book isn't really necessary if you are studying from a more advanced book, with a stronger mathematical background. That is really my ultimate recommendation, as I think that students coming into real analysis should already have a fair amount of mathematical maturity. If that isn't you, then go for this one.
February 13, 2022
I wish we had more maths textbooks like this. An approachable and simple introduction to Real Analysis, written in an engaging style and always focusing on the big picture and keeping track of where a certain theorem or fact fits in the scheme of things. The book is suitable for self-study, and is comprehensive as far as a first-course in analysis is concerned. Some exercises are usually part of the analysis which forces one to work on them. I have done almost half of the exercises but I recommend doing all of them if you can.
To get the best results out of your self-study, you may supplement it with "counterexamples in analysis" by Gelbaum and Olmsted. There is also a great playlist on youtube by Prof Christopher Staecker that follows this book.
To get the best results out of your self-study, you may supplement it with "counterexamples in analysis" by Gelbaum and Olmsted. There is also a great playlist on youtube by Prof Christopher Staecker that follows this book.
January 11, 2021
Quite clear, approachable, nice to follow. Filled in a lot of gaps after Tao's Analysis.
October 17, 2021
ouchie hurtie brain
December 20, 2024
You say you wanna prove calculus from nothing? Okay. God endowed us with the naturals. From there, this book shows you the way.
Perhaps the last math textbook I’ll read a significant portion of. Pretty good if you like to over analyze numbers. I thought I did.
I don’t.
Perhaps the last math textbook I’ll read a significant portion of. Pretty good if you like to over analyze numbers. I thought I did.
I don’t.
December 20, 2024
Honestly, Abbott explains analysis in an accessible way that is also interesting for those not in the field. A great introductory textbook.
July 2, 2023
Great intro at the level of 3rd or 4th year undergrad.
June 30, 2019
This is the best book I found for learning analysis the first time, specially for self-study! The exercises evolve in difficulty, which gives you confidence that you are actually learning the subject (because you are able to do the easy exercises as you read the chapter). Also, the examples and the writing are just great! It felt like a novel! I just wish I could find other textbooks with the same “spirit”.
February 21, 2021
Brilliant. Covers the essence of concepts that are easy to miss in other texts.
Update 1: it's really amazing. I highly recommend reading it even if you have read other analysis texts. I feel like a lot of other books make you work hard but don't reveal the fundamental insights at play. If you have done the hard work from other texts, please have an amazing experience of skimming though this book.
Update 1: it's really amazing. I highly recommend reading it even if you have read other analysis texts. I feel like a lot of other books make you work hard but don't reveal the fundamental insights at play. If you have done the hard work from other texts, please have an amazing experience of skimming though this book.
June 16, 2019
This is the best presentation on Analysis I've read.
He does a really great job of motivating the ideas, and gives really interesting and unusual examples, not just the normal basic examples.
Lot's of exercises for each chapter ranging from simple to pretty involved proofs (no solutions).
Above all that it's just well written and a pleasure to read.
He does a really great job of motivating the ideas, and gives really interesting and unusual examples, not just the normal basic examples.
Lot's of exercises for each chapter ranging from simple to pretty involved proofs (no solutions).
Above all that it's just well written and a pleasure to read.
February 22, 2016
concise, crystal clear, complete core of calculus/real analysis.
July 27, 2025
After having finally read through the whole text and completing almost all the problems, I finally feel myself ready to give my opinion on “Understanding Analysis”. Yet, far before the point of completion, as might be natural for such a long undertaking, I kept thinking of how best to summarize my experience with the book. A few phrases kept coming to mind, yet as I trudged along through the book’s contents, I found each to be inadequate in some way. “Understanding Analysis is a phenomenal book for all students of undergraduate level math”. A decent start but a bit too restrictive for my liking. Hell, I’m not even a math major myself and yet here I am, giving it my endorsement. “Understanding analysis is a fantastic read for anybody looking to learn math”. Better, but surprisingly enough, I came to decide that this too was still an overly narrow description.
Over the 8 months of my life which I spent grappling with this book; on morning commutes, between spare seconds at work, even somewhat ashamedly while scarfing down food during lunches, many a curious friend or acquaintance, with math backgrounds as variable as x, have asked what all the fuss is about. And time and time again, a conversation (at least if I had to guess) borne from some morbid curiousity about my self-masochistic form of leisure morphs into a sincere and genuine interest in whatever problem I have at hand, and often even becomes a genuine attempt to try to solve it themselves.
I’d love to say that these chats happen because I have such a captivating way of talking about math. But in truth, all the intrigue is I generate is a reflection of the book’s quality. Almost every analogy, counter-example, proof, or question I’d provide comes straight from the book because even the often paradoxical and unintuitive results of real analysis are conveyed with such simplicity and lucidity that practically anybody can understand it. After so many such conversations, I find it hard to dismiss all of them as just feigned interest or one-offs, which signaled to me that the book may be for more than just a math student or enthusiast. After much reflecting on these experiences, this lead me to the summary of the book I’ve settled on for this review:
“Understanding Analysis is the perfect book for anybody wanting to *think*”. Despite the book’s obvious use as a tool for learning real analysis, I’m sincerely of the belief that practically anybody can stand to benefit from reading it. This is far from just an endorsement of the benefits of learning math, but specifically the book’s unique approach to pedagogy which I have yet to see done so well elsewhere. Each chapter in the book consists of largely what one might expect from a math textbook – A motivating example, a set of assumptions made from that, and the proofs for theorems which follow, with some questions at the end of each subsection for good (Lebesgue) measure. This is hardly groundbreaking. What sets the textbook apart in my view is the final section of each chapter, which is structured as a long proof that you, the reader, must solve, with questions being integrated directly into the section’s contents, the results of which are further used to pose more questions until eventually the desired proof is finally done. This structure is ingenious, not just because it forces the reader to employ all the strategies from previous chapters since these proofs are often so involved, but also because it leaves you with a feeling that you are “discovering” the same math that the greats of real analysis, Cauchy, Euler, Weierstrass and so on did before you.
I truly feel as though I cannot give this book enough praise, and I honestly feel like I can credit it significantly for my plans to pursue a master’s in mathematics at some point in the future.
Over the 8 months of my life which I spent grappling with this book; on morning commutes, between spare seconds at work, even somewhat ashamedly while scarfing down food during lunches, many a curious friend or acquaintance, with math backgrounds as variable as x, have asked what all the fuss is about. And time and time again, a conversation (at least if I had to guess) borne from some morbid curiousity about my self-masochistic form of leisure morphs into a sincere and genuine interest in whatever problem I have at hand, and often even becomes a genuine attempt to try to solve it themselves.
I’d love to say that these chats happen because I have such a captivating way of talking about math. But in truth, all the intrigue is I generate is a reflection of the book’s quality. Almost every analogy, counter-example, proof, or question I’d provide comes straight from the book because even the often paradoxical and unintuitive results of real analysis are conveyed with such simplicity and lucidity that practically anybody can understand it. After so many such conversations, I find it hard to dismiss all of them as just feigned interest or one-offs, which signaled to me that the book may be for more than just a math student or enthusiast. After much reflecting on these experiences, this lead me to the summary of the book I’ve settled on for this review:
“Understanding Analysis is the perfect book for anybody wanting to *think*”. Despite the book’s obvious use as a tool for learning real analysis, I’m sincerely of the belief that practically anybody can stand to benefit from reading it. This is far from just an endorsement of the benefits of learning math, but specifically the book’s unique approach to pedagogy which I have yet to see done so well elsewhere. Each chapter in the book consists of largely what one might expect from a math textbook – A motivating example, a set of assumptions made from that, and the proofs for theorems which follow, with some questions at the end of each subsection for good (Lebesgue) measure. This is hardly groundbreaking. What sets the textbook apart in my view is the final section of each chapter, which is structured as a long proof that you, the reader, must solve, with questions being integrated directly into the section’s contents, the results of which are further used to pose more questions until eventually the desired proof is finally done. This structure is ingenious, not just because it forces the reader to employ all the strategies from previous chapters since these proofs are often so involved, but also because it leaves you with a feeling that you are “discovering” the same math that the greats of real analysis, Cauchy, Euler, Weierstrass and so on did before you.
I truly feel as though I cannot give this book enough praise, and I honestly feel like I can credit it significantly for my plans to pursue a master’s in mathematics at some point in the future.
December 12, 2024
I first became interested in Middlebury College when I met a girl who chose to attend there over Yale. Her decision intrigued me, and as I learned more about the school, I was impressed by its intellectual rigor and the meaningful academic environment it fosters. Middlebury has a stellar reputation in every discipline, with many influential alumni, especially in finance. It’s a place where ambition and purpose align, making it a haven for those seeking to do meaningful work.
Although I didn’t get into Middlebury, I’ve continued to admire the school for its commitment to cultivating both intellectual depth and practical impact. One example of this academic excellence is Stephen Abbott’s Understanding Analysis, a textbook that perfectly represents the type of rigorous, clear education Middlebury offers. Abbott’s ability to make complex mathematical ideas feel intuitive is truly remarkable. He doesn’t just present formulas; he guides readers through the logic behind each concept, helping them build a deeper understanding of the material.
Reading Abbott’s work made me wish I could’ve had him as a math professor. His approach to teaching analysis is exactly what I would have loved in my own education—rigorous but accessible, with an emphasis on understanding over memorization. His book is an excellent resource for anyone tackling analysis, and it’s a perfect example of the type of instruction that draws me to Middlebury’s esteemed math department. Abbott’s work is a testament to the kind of education that challenges students to think deeply and engage meaningfully with the material, and it only reinforces my admiration for Middlebury's intellectual culture.
Although I didn’t get into Middlebury, I’ve continued to admire the school for its commitment to cultivating both intellectual depth and practical impact. One example of this academic excellence is Stephen Abbott’s Understanding Analysis, a textbook that perfectly represents the type of rigorous, clear education Middlebury offers. Abbott’s ability to make complex mathematical ideas feel intuitive is truly remarkable. He doesn’t just present formulas; he guides readers through the logic behind each concept, helping them build a deeper understanding of the material.
Reading Abbott’s work made me wish I could’ve had him as a math professor. His approach to teaching analysis is exactly what I would have loved in my own education—rigorous but accessible, with an emphasis on understanding over memorization. His book is an excellent resource for anyone tackling analysis, and it’s a perfect example of the type of instruction that draws me to Middlebury’s esteemed math department. Abbott’s work is a testament to the kind of education that challenges students to think deeply and engage meaningfully with the material, and it only reinforces my admiration for Middlebury's intellectual culture.
May 4, 2023
Probably the opposite of most Analysis textbooks, it does a great job of presenting the intellectual, philosophical and historical motivations behind the theory of analysis. It's an interesting book insofar as it's much better suited towards a general audience while covering a topic which is probably if very little interest to a general audience (though it almost certainly should be).
I read this book as part of an introductory analysis course I took. Out of all the courses I took during the 3 years of my undergrad, this was by far one of the best, if most difficult, courses I've taken. We went through this book almost cover to cover, and got quite a lot in, including the topology of R and even a short week on Lebesgue's integration. Though there is certainly more to learn, this course and book really helped me see the beauty of analysis, even if I myself am probably never going to be an analyst.
I read this book as part of an introductory analysis course I took. Out of all the courses I took during the 3 years of my undergrad, this was by far one of the best, if most difficult, courses I've taken. We went through this book almost cover to cover, and got quite a lot in, including the topology of R and even a short week on Lebesgue's integration. Though there is certainly more to learn, this course and book really helped me see the beauty of analysis, even if I myself am probably never going to be an analyst.
December 13, 2019
I like this book very much as an introduction to analysis, because it motivates the concepts much more strongly than most books from the traditional canon. I use this book for a reading course in analysis: the chapters are well structured with a hook to motivate the content, and a nice summary at the end. I agree with another reviewer's comment that the treatment of the real numbers is a little confusing due to the order, and one student noted that with so many parts of proofs left as exercises (a great idea in principle), readers have to suspend disbelief if they are unable to fill in those blanks. But overall a great book., motivating some important concepts in modern analysis
February 25, 2019
Honestly the thing that pisses me off most about math books is when they're disorganized and written for the author, rather than for the students. This one has all the definitions, theorems, proofs, and problems arranged in a really digestible way. I really struggled with the concepts in this class (fuck series, am I right) but the book made it possible to teach myself a lot of the harder concepts, which is a tough feat especially for a proofs class. This is one of the notoriously hard classes at Cal Poly, but this book made it a lot more palatable.
July 24, 2024
I spent around 2.5 months studying this book daily. It's really great. I had difficult with some proofs but there were plenty of YouTube videos that walk through these proofs which was super helpful. When I was still suck, I used Real Analysis by Jay Cummings (https://www.goodreads.com/book/show/4...) to help more understand the topic I'm studying. This is a great book for self-study with access to Youtube and Real Analysis by Jay Cummings. I suspect if you take this formally in the classroom, you won't need any other extra resources!
November 17, 2020
I wish he covered the transition into higher analysis better. If you have a decent point set topology background this book mostly covers just that. Finding a book that transitions, more strongly, into Lp Space and Fourier Series would be great. So, for example, you can't easily read this book and then jump into Rudin's Real and Complex Analysis.
Otherwise, this is a 5 star book and very readable.
Otherwise, this is a 5 star book and very readable.
April 16, 2025
I only worked through the first three chapters before switching out for Rudin (since it didn't have the difficulty/rigour I was looking for), but it's a beautiful textbook. Super visual and intuitive, and obviously designed with the beginner in mind. One note is that this book only covers analysis on R (and topology only in terms of R), so you'd be missing out on analysis in R^n and metric spaces if this is your only analysis reference.
November 30, 2019
Great introductory book in analysis. Presents the material in a straightforward manner accessible to those with introductory calculus, and the many stories, backgrounds, and explorations the author offers help ground understanding. Just an all-around great introductory math text for building conceptual understanding of analysis as a first look.
March 23, 2022
I think this is very clearly written, with good exercises. The only thing I wished it did was treat the topological concepts in general metric spaces. This does not change how most of the proofs are done or make the book any less intuitive, but makes the material a lot more general. See for example how rudin does it in ch 2.
May 14, 2017
Very nice introduction to analysis with good background and motivation of why we are interested in various theorems or problems. Even the chapter on Fourier series was excellent. Naturally, this book doesn't go too far so more difficult analysis must search elsewhere.
January 21, 2023
The theory is very well introduced, but the book is "for dummies" in some regards. If you want to really learn analysis (i.e., if you desire to persue graduate studies in pure math) then pick something like Folland's Real Analysis or Rudin's Real and Complex Analysis.
May 27, 2025
The single best book on undergraduate analysis. There are no superfluous exercises - It covers precisely what you need to know. This can be seen as a disadvantage, as it has far fewer exercises to practice with than Spivak's Calculus. I also highly recommend that book.
July 2, 2017
Simple and great book to get started with real analysis.
November 26, 2019
Beautifully written.
June 28, 2021
Great book. This text, and Calculus ny Michael Spivak, were my main references during my first calculus course.
March 4, 2022
Good introductory undegrad analysis text
Displaying 1 - 30 of 48 reviews