Can I Start Analysis? Seeking Your Advice on My Journey to Mathematics! I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the mathematics and its phenomenal applications.  I will be taking the theoretical linear algebra and multi-variable calculus on upcoiming semester and proceed with introductory analysis (Rudin's PMA) on next Fall along with the theoretical differential equations.  I used the calculus textbooks called "Calculus and Analytic Geometry" by George Simmons and "A First Course in Calculus" by Serge Lang to pass the single-variable calculus.  The theoretical linear algebra course (one I will be taking on next semester), teaches the proof-based linear algebra and the proof-methodology; its required textbook is one written by Friedberg & Insel & Spence.  The multi-variable calculus is heavily on computational and use my university's course notes...In this case, should I self-study the Spivak/Apostol/Courant before advancing into the analysis like Rudin's PMA?  I own the Apostol's two volumes and also Lang's Basic Mathematics but I do not think I will be able to study them fully befoe the end of Summer.  Is it okay to jump toward the math analysis without completing the Apostol or Spivak?  My current research utilizes a lot of differential equations, linear algebra, and numerical analysis.  Is it okay to study for numerical analysis without taking the analysis courses like real, complex, and functional analysis?  If so, could you recommend a good introductory/elementary textbook on numerical analysis?
Also about the theoretical linear algebra textbooks, which among those books complement well with Friedberg?  Hoffman & Kunze, Axler, Lang's LA, or Lang's Introduction to LA?  If Hoffman & Kunze better for a starter than Friedberg?  I always love to use at least two textbooks on given topic.  Also are books called "How to Prove It" by Velleman and "How to Solve It" by Polya good for learning the proof methodology and techniques?
Thank you very much for your time, and I apologize for asking too many questions especially on the books.  I think it is important in mathematics to choose the right books to study for.   I look forward to your advice!
Sincerely,
MSK
 A: There are several questions here.
The first is whether you can start with Rudin's analysis book now. I would say that for most people this would be difficult after only Lang's calculus book, although Lang is far better than most calculus books in use these days.
I don't recommend reading Spivak or Apostol (Vol. 1) now. It will take too long, as they're really intended for people who don't know the routine part of calculus yet. Instead, I would suggest reading an analysis book at an intermediate level of difficulty, such as Apostol's Mathematical Analysis, or even the easier analysis book by Ross. Once you've read several chapters of one of those, as a byproduct you'll probably find that you can just jump in anywhere in those calculus books and understand.
The second issue raised by your question is about how to approach multi-variable calculus at this point. I would say that without having studied either analysis or single-variable calculus at the level of Volume 1 of Apostol's Calculus, it will be difficult for you to follow much of the theory in Volume 2. In this situation, since you need multivariable calculus immediately, it's okay to study it at a slightly lower level and fill in the gaps later. (Parts of the theoretical material are so difficult that even Apostol is forced to omit some proofs in Volume 2 anyway.) Lang's book on multivariable calculus would be good in this situation. The amount of theory is intermediate in it.
For linear algebra books, I don't really know which is best. I imagine one would be sufficient. The two introductory chapters of Lang's Introduction to Linear Algebra could be good preparation if you haven't worked with vectors or matrices before.
On numerical analysis, I'm afraid I don't know.
Finally, you mentioned some books about proofs. Not everybody needs a book that teaches them how to write proofs specifically; some people just pick it up when they learn math with proofs for the first time. But if you would like to read a book like that, I would recommend that it be something that actually teaches you interesting math as you go along, rather than being too narrowly focused on proof techniques. My favourite in this regard is Journey into Mathematics by Rotman, which covers several interesting topics. Even if you don't need a book on proofs, you won't feel like you've wasted your time. It has a complete solution manual too.
