Spivak, Abbott or Apostol?

Question

I am a beginner who wants to self-study rigorous calculus and real analysis. (I just had elementary high school calculus)

Not a few people recommended me Spivak's calculus, which is known as a great introduction to proof based mathematics.

I was also informed about Abbott's Understanding Analysis, I guess this book would be incredibly enjoyable to work through, nevertheless someone said that it should be a very tough adventure especially with weak calculus background :(

The last one is Apostol's Calculus, but I don't know this book very well.

I am a bit confused since I have no idea which one would be a suitable selection. Thus I'm begging for an advice :0

Answer 1

I don't know Abbott's book. But I strongly recommend Apostol as a first book and Spivak as a second one.

Apostol's book begins from the very basics (the real numbers). He first presents integral calculus, even before the concepts of limits and continuous functions. But he takes a different (but equivalent) approach: The Darboux integral. He never mention it, but it's the real theory as you can see from Wikipedia or other references. Then he presents differential calculus and make the connection via The fundamental theorem of calculus.

The only part I don't like is the treatment of linear algebra and ordinary differential equations on Vol. II. But the rest of the book is excellent.

Then move to Spivak's book. Specially try most of the exercises from that book. This book is a little bit more theoretical than Apostol's. Spivak also takes the Darboux integral as integration theory, but in the Appendix of Riemann sums, he proves (but never mention it) that Riemann and Darboux integrals are the same (at least for continuous functions).

I also recommend Introduction to real analysis by Bartle and Sherbert as an introductory book (after you read Apostol) and little Rudin (Principles of mathematical analysis by Walter Rudin) if you want to go deeper into the subject.

Answer 2

We should be clear about the difference between "calculus" and "real analysis." Based on the candidate texts, it seems you are interested in the former and not the latter. If you have familiarity with high school calculus, then either Spivak or Apostol (Volume 1) will be suitable for an undergraduate-level calculus text. My personal preference is Apostol, but again, either will suffice.

However, if you want to learn about real analysis, neither text is going to be an adequate treatment. To be clear, you will get pieces of this in an undergraduate calculus text--e.g., limits and continuity, the Riemann integral, etc. For an introductory and relatively accessible text in real analysis, I would suggest Walter Rudin's classic text, Principles of Mathematical Analysis. This is a very concentrated treatment of the subject and it is well-regarded precisely because of its brevity and elegance in distilling the material to its essential elements.