Understanding Analysis (Undergraduate Texts in Mathemat… (2024)

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 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.

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 ℝ. 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.

Who, though once only and then but far away,
Have heard her massive sandal set on stone.

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.

Quite clear, approachable, nice to follow. Filled in a lot of gaps after Tao's Analysis.

ouchie hurtie brain

Great intro at the level of 3rd or 4th year undergrad.

Cauchy! Weierstrass! I did read this book cover to cover! Maybe you don't want to learn real analysis, I didn't, but I did, and maybe you should too. I would do it again though.

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”.

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.

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.

concise, crystal clear, complete core of calculus/real analysis.

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 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

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 (f*ck 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.

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.

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.

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.

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.

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.

Simple and great book to get started with real analysis.

Beautifully written.

Great book. This text, and Calculus ny Michael Spivak, were my main references during my first calculus course.

Good introductory undegrad analysis text

Easily digestible and engagingly written. I read this for a class in fundamental analysis, but I think it would be fine for self-study, too.

If you struggle with Rudin's, this is the book to fill that gap.

I think I… understand analysis

Superb

book was a lifesaver, easily the most fun I’ve ever had in a math class

One of the favorite mathematics textbooks I've used. Abbott is excellent at explaining the content, and he succeeded at his goal of making sure none of the example problems were trivial. I was fascinated working through each one, feeling like I'd gained a more nuanced view of the real numbers when I was done. This is also a very good book for self-study. I was in a hybrid real analysis class with limited instruction time, and this book was better than the lectures.