The never ending question: which books?

[jordi]

I want to study mathematics again (I have a degree in physics). I studied with Mathematical Analysis (Apostol) and a Linear Algebra book of "somebody" in my University (not so good). I studied Naive Set Theory (Halmos) and then some more advanced mathematics (complex analysis, real analysis, functional analysis, groups, ...). However, now I want to study it all back more thoroughly (in my spare time; I work outside the university).

One of my problems is that I want the choose THE optimal set of books. For example, I do not want to study Apostol again if it is already old and there is a "better" book. For example, after some browsing in amazon I have seen a new book by Pugh which is quite well received. Also another one by Koerner (in GMS).

My path is going to be:

1. Set theory (maybe Halmos, but is there another suggestion? I have seen one by Potter that seems quite good).
2. Analysis (Pugh? Koerner? little Rudin? Apostol? something else? + maybe counterexamples in analysis) and Linear algebra (Lang? Roman? Linear Algebra done right? + maybe Halmos problem book)
3. Topology (Munkres + counterexamples in topology) and Complex Analysis (Lang + Lang problem book)
4. Analysis (Lieb and Loss) and something of combinatorics and graph theory?
5. Functional Analysis (Lax and Zeidler + Halmos problem book?)

Later, there are many other things to study (Algebra, probability, number theory, measure, geometry, algebraic geometry, ...) but let us pause with the previous 5 points:

What are the suggestions for this path of study? Would you suggest something better/different? The final purpose of all this study is to study QFT and GR in a "correct" way.

Thank you in advance.

 

[quasar987]

Then you'd better thrown a differential geometry book into the mix.

 

[jordi]

Then you'd better thrown a differential geometry book into the mix.

Definitely, quasar987. I have not included it because my first milestone is to understand better QM, and for this I need mainly functional analysis. When (if?) I do the step towards GR and QFT with non-trivial backgrounds, I will definitely go towards differential geometry (BTW, what people think about Naber books?).