Analysis – Difficulty Level of Courant’s Book

Question

I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John. I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the problems provided. After reading that book, should I go for one of the two above or should I study something else like Rudin? My focus is on being rigorous and also adept at problem solving.

Answer 1

I think Courant and John's book is the richest of the three textbooks you mention: it essentially contains the other two.
Spivak is the most rigorous (and is very, very aesthetic) but I think that if you want rigour, it would be boring to apply it to material you already know: better start learning more advanced analysis.

1. Rudin is of course good but a bit dry.

2. John and Barbara Hubbard wrote a very geometric book,Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach whose center of gravity is the subject traditionally called "calculus of several variables".
   It is probably the best book in that category, because of the rigour and depth of the presentation, its modernity (everything done in the language of manifolds and differential forms), the wealth of material presented (including linear algebra and integration) and, last not least, the interesting (mostly historical) vignettes in the margins.
   There are many, many friendly easy little calculations made in the text; moreover the exercises are carefully thought out, starting from trivial computations and aiming for more difficult results, provided with hints.

3. Lang wrote a very impressive book, Real and Functional Analysis, which contains an amazing lot of mathematics packed in less than 600 pages.
   As always with Lang his forte is not in explicit, down-to-earth calculations, but the book is an amazing synthesis of basic analysis: you will be introduced to topology, functional analysis, integration theory ( in $\mathbb R^n$ but also on locally compact spaces), manifolds, differential forms,...
   And the book profits from Lang's vast culture: it must be the only textbook which explains the role that Hironaka's resolution of singularities (one of the most difficult theorems in algebraic geometry) might play in the statement of Stokes' theorem!

2 1/2. As a stepping stone to 3), you might use Lang's more elementary Undergraduate Analysis, which you could consider a review of Courant-John in more modern language, and an ideal preparation for the more ambitious 3).