I am currently undergoing a full Apostol Calculus I reading, and with few chapters left I've began planning what other books to read. I was thinking about finishing both Apostol Calculus l and ll, and then read Apostol mathematical analysis. I'm not pretty sure whether these are the best books or if they will provide me with a strong math basis. I've already taken calculus i, ii, iii and linear algebra. The reason I've chosen Aapostol volume 1 and 2 is that I've heard they cover almost all the topics I've seen and will see in the next semester/s. I would appreciate if somebody could give me some advice about this. I really like to have a book that really gives me hard and challenging problems. I don't know whether to read Spivak Calculus book or not.
The reason why I shall never write a Calculus textbook is because Michael Spivak's Calculus is a masterpiece written at a level that I will never be able to attain.
There are some really great books about Calculus and Michael Spivaks Calculus is amongst them. Here I'd like to put the focus on two older books.
Introduction to Calculus and Analysis I,II by Richard Courant.
This two volume classic with first edition from 1965 provides a detailed and comprehensive introduction into Calculus. It was really helpful for me to grasp concepts when they were not only stated using compact formulas but also clearly and thoroughly explained often from different points of view. The language used in the book may be somewhat old-fashioned, but the content and presentation is top.
If you like to work on challenging problems I recommend
Problems and Theorems in Analysis I,II from Pólya and Szegö.
This two volume book is a problem based classic (first edition from 1924). It is a highly inspiring source of many interesting problems in different fields of analysis together with detailed solutions.
R.P. Burn: Numbers and Functions - Steps into Analysis. is a great book requiring you to go deeper into the material than you go when reading a usual text book. I wish more books would have been written like this.