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.

Solution 1:

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

Solution 2:

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.

Solution 3:

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.

Solution 4:

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