Baccalaureate exam approached Real Analysis (limits, differentiation and integration), Abstract Algebra, Functional Algebra, Linear Algebra, Combinatorics, Complex numbers, Vector Geometry, Analytical Geometry and Trigonometry. My question is what are the books that I should study in order to expand my knowledge in all these subjects and also approach new branches such as Inequalities and Number Theory (they should cross the border of college and get into Olympiad level as well). I need courses to cover the subjects and, if possible, in a natural order in each branch (from the easiest to the hardest) and exercises as well so any suggestions are welcome and I am not afraid of big lists. Thank you!
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Let me start by saying that I will try to explain by the end the level of the material and the order you can read and work through it.
Abstract Algebra, 3rd Edition by Dummit and Foote is a good option. It covers almost all the algebra you will need in your undergraduate and most of the Algebra in your graduate education. A link some question which could be useful here: 1 2. Also, most of its exercises are solved through the net so you can always check your solutions. Some of the exercises are here.
- Number Theory:
Read the answers to this question.
- Real Analysis:
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4ed, does the job fine. You can combine it with David Brannan's A First Course in Mathematical Analysis which is very readable. Some people would point you to Rudin. I haven't read Rudin.
- Elementary Topology:
In order to understand Real Analysis, functional analysis...etc you need some topology. First you only need the basics about Metric and Topological spaces. Introduction to Metric and Topological Spaces, Second Edition, by Wilson A Sutherland does an amazing job presenting the material in easy terms so you can fully understand the basics. Moreover you can legally download some extra materials from the web page and they complement the text very good. Furthermore, it is cheap. I learned lots from this book and in my opinion is a must read. You will simplify lots of proofs of real analysis and also extend some results. Also you will develop your ability proving things.
- The order you can read and work through them:
You can attack them all at the same time. Maybe, if your Real Analysis background is only Calculus you could read a bit of David Brannan's text before Sutherland's one. Dummit and Foote is very long so you just could focus on elementary group and ring theory. In reference to number theory, elementary number theory can be studied alone, then if you want some elementary number theory you will need some algebra.
I am afraid I can not suggest books in the other topics you mentioned. I hope this helps and if you have more questions feel free to ask!
For trigonometry alone, I recommend the old classic Trigonometry , by Hobson, although you may not have time for all of it, and likely won't need all of it. Another great old one: Infinite Sequences And Series , by Bromwich. A topic that becomes a necessary tool in analysis. A good introduction to some aspects and applications of finite combinatorics is Finite Mathematics by Kemeny, Snell, and Thompson.