Which Calculus/Real Analysis textbook is recommended for a deeper understanding?

[qntty]

I have already gone through the standard AP Calculus BC curriculum with Stewart and was not fond of the book. I'm looking for an Calculus/Intro to Real Analysis book so I can get a better understanding of the subject. Apostol,Spivak and Courant stand out as the best books for this. Can anyone who has experience with one or more of these books give a recommendation. If you know of any other books that would be appropriate I'm still open to other choices.

 

[integration]

I don't know a lot about Spivak or Apostol, but I definitely don't recommend Courant.

Courant is basically the standard Calc BC from Steward, with less pictures, less applications, *slightly* more abstract and more dense. In terms of insights for a math major, Courant is probably too "easy" to give you a drastically different experience fromt he normal Calc BC, but too dense to be a nice review book.

I would go with a book like "Analysis" by Klichin, a book that probably resembles more to Spivak or Apostol. Be warned, this book is dense and very tough, but introduces a formalism and approach very valuable for math majors.

 

[khemix]

im using spivak and apostol right now as supplements to real analysis.

spivak is the best calculus book bar none. maybe apostol is a superior refrence, but spivak teaches you everything about calculus. he not only proves why results are true, but tells you how the the proofs are designed. my only gripe with spivak is his problems are extremely difficult. at the same time, his difficult problems ensure you walk away with a crystal clear view of how the theory of calculus works. but know what extreme difficulty means: it is very very frustrating, and it will take hours to do the problems.

its not really an intro to analysis text, its pure calculus. but some people call any kind of calculus with rigor real analysis. aside from some continuity theorems and stuff on series, and random problems, the text sticks pretty firmly to calculus as a whole. that means no topology is employed, and it is quite classical.