I'm starting a master's program in mathematics in September and I would like to do some review over summer, before I start, so I am looking for book recommendations. My background is in engineering (electrical and physics), but I took some pure mathematics courses at university. However, this was a while ago, since it has been a couple of years since I graduated.
Analysis
My analysis class used Rudin's Principles of Mathematical Analysis. We read most of the book except the final two chapters on differential forms and Lebesgue theory, and only parts of the chapter on special functions. Overall I liked the book, though the material on special functions and functions of multiple variables was perhaps too compact; I still feel somewhat weak in these areas. (I recall the proof of the implicit function theorem to be particularly difficult.) To review analysis I am considering either to:
Go back to Rudin and do more exercises. I did most of the book exercises in the early chapters (1-2), but not very many in the later chapters when I took the course (because the homework problems written by the teacher took most of my time).
Instead go on to Rudin's Real and Complex Analysis. Perhaps it is more useful to take the next step, instead of dwelling on the previous?
Read some other author, to get a complementary viewpoint on the material. I have considered Tao's Analysis I and II, though I am a little worried that they may be too verbose (especially for review!), requiring two whole volumes.
What would you recommend?
Abstract algebra
My abstract algebra class used Judson's Abstract Algebra: Theory and Applications, but the course emphasis was certainly on the core theory rather than the applications. My experience with this book was not great, but that is likely largely due to mathematical immaturity; this was the first rigorous mathematics course I took, even before real analysis. I am considering using Hungerford's Algebra (not Abstract Algebra: An Introduction) for review. It is more compact than the book by Dummit and Foote, which I have also considered.
Is Hungerford's Algebra a good choice? Are there better alternatives?
Update 1: I started reading Hungerford and found that he requires proper subsets to be nonempty. Similarly he excludes the trivial group from being a proper subgroup. This is annoying, and turned me off somewhat.
Update 2: A commenter suggested Aluffi's Algebra: Chapter 0, which seems promising, if perhaps a bit too conversational (and lengthy!). I like that it seems to offer quite a different perspective than Judson's text.
Other topics
Maybe it would also be useful to study/review some other topics before starting. For example, while I have used a lot of linear algebra (including some advanced topics) in theoretical physics, I have not properly studied linear algebra beyond the basic engineering level. Perhaps some review of discrete mathematics would also be good; I took one somewhat rigorous course in the subject, though it was aimed at engineers. For topology I have been reading Lee's Introduction to Topological Manifolds lately, and though I have only completed the beginning chapters, I believe this is enough to prepare me for the master's program.
Any good recommendations for linear algebra and discrete mathematics, for a student with some mathematical maturity?