I am a Japanese so it is difficult for me to read English and I may make some grammatical mistakes.
I have little experience with mathematics but plan to self-study mathematics by reading mathematics books in English. I would like to know what the best text is in each of the following subjects,
Please provide a rationale behind suggestions. I would prefer texts with many examples (as opposed to formulas), and length and price are not a prohibitive consideration.
Naive Set Theory – Halmos. This is a good starting place for set theory. It is about 100 pages in length. If you find this work interesting, then you can move on to more formal\advanced texts such as Hrbacek & Jech – Introduction to Set Theory.
For Algebra, you will probably get a mix bewteen Fraleigh and Herstein. Both are excellent but I would recommend you get a copy of Fraleigh since he likes to explain everything in minute detail. You can find lectures of Fraleigh (free online at UCCS). If you want something a little bit more advanced, I would recommend Algebra – Artin (there is a free lecture series by Harvard on this text).
For Linear Algebra, ‘Linear Algebra Done Right‘ is very popular. Introduction to Linear Algebra – Strang is also very good. You will be able to find MIT lectures on this book (by the author himself!), which makes it perfect for self study. Even though a bit dated, ‘Matrices and Linear Transformations – Cullen‘ is written very clearly and will guide you to the core of the theory in a very clear way. The notation is just a bit outdated.
Analysis is a difficult one since it largely depends on your ‘mathematical maturity’. Since you say, for self study, I would recommend “Analysis – Steven Lay‘. This book has an excellent introduction to logic and proofs. It also covers set theory and functions and how to prove just about anything relating to sets and functions. Moving on from there, there is of course ‘Rudin’s Principles of Mathematical Analysis‘. Of course, this is not a book that is suited for self study, but you can find free lectures on youtube (Harvey Mudd) which covers about half of the material in this book. The lectures are very good and will enable you to work through this material.