# Terence Tao–type books in other fields?

I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.

Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.

That was the question; the following is just why I think Tao's style is so great.

• When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"
• Similarly, there is no bad feeling of "maybe I wasn't supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can't". It makes everything feel "fair game"
• It makes it difficult to be a passive reader
• It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time

I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that "this proof may be too hard, it would not be expected of the reader to come up with this proof". In Tao's book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.

• I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field? Jul 25, 2019 at 17:08
• @Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject. Jul 25, 2019 at 17:15
• @Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $\mathbb{R}$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology. Jul 25, 2019 at 18:14
• Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier. Jul 25, 2019 at 18:17
• That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts. Jul 25, 2019 at 18:56

Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449

• Ah, the youth of today. Hartshorne was written in a (what was considered 40 years ago to be a ) breezy informal style in contrast to the (what was considered terse and traditional nature) of EGA ! I remember how excited I was to receive from Berkeley the typo version of his book in 1977.
– meh
Jul 26, 2019 at 18:57

I know the OP probably knows this but for undergraduates, there are Analysis I and Analysis II by Terence Tao which follow the exact same style as mentioned in the question. The book is self-sufficient and Tao provides all the necessary background needed to solve the exercises. Some sample chapters can be found freely here.

• Yup; and these books contain some material which I haven't seen anywhere else, for example, the construction of the real numbers starts from the natural numbers. It also talks about "the axiom of equality" or something like that.
– Ovi
Jul 26, 2019 at 12:59
• @Ovi He mentions 4 axioms if equality in the appendix on mathematical logic. In no other analysis books, I've seen such a fantastic exposition of construction of real numbers using Cauchy sequences starting from construction of natural numbers using Peano axioms. What's surprising is that the book is accessible by anyone who has completed high-school. Also, this is the only book in my knowledge (except Zorich) which discusses axiomatic set theory. The book is a gem in my opinion. I would love Tao writing other books as well, say converting his notes on Linear Algebra. Jul 26, 2019 at 13:37

One such example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.

Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).

An additional textbook that has lots of discussion and illustration, while leaving a fair amount of results to the reader, is John Lee's "Introduction to Smooth Manifolds," which is one of the standard texts on the subject for graduate students. Although Lee is more proactive in proving results than Tao in most of his books, I'd say this still fits the description, at though to a lesser extent.

I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.

Excerpt from the book's introduction:

The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets.

I believe it may be suitable for you, because:

When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"

As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.

It makes it difficult to be a passive reader

Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.

It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time

As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.

You can check out the book here

I’ve had a very good experience with Marcus’s Number Fields. I haven’t been through the entire book, but the first few chapters definitely follow the style you’ve outlined. Particularly, it is quite difficult to read it passively. Many important results are relegated to exercises, but they were introduced in such a way that I always felt the solutions were within my reach.

I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.

• Pinter, "A Book of Abstract Algebra"
• Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"

Marian Fecko: Differential Geometry and Lie Groups for Physicists is written in such a style. As you read the book you develop the theory by solving exercises scattered throughout the text. I hope you count mathematical physics as other field of math (the book is mostly math though).

The author has two chapters on his webpage so you can check the style if that is what you are looking for: Chapter 2 and Chapter 15

Convex Optimization. Stephen Boyd and Lieven Vandenberghe. Cambridge University Press.

For algebraic topology, in Homotopical Topology the set of exercises is dense and has quite a high cardinality. In general, I consider this book much better than Hatcher.

For abstract nonsense, Emily Reihl also makes you work your way through in Category Theory in Context, Categorical homotopy theory, and Elements of ∞-Category Theory.