Are there any problem book series on different topics with proper solutions? I have already found some in analysis (Problems in Mathematical Analysis 1-3) but not any in less popular topics.
Try Berkeley Problems in Mathematics, a collection of Berkeley's preliminary exams for graduate students.
This is a problem book for complex analysis, and I know there is one for real analysis, titled A Problem Book in Real Analysis. I'm fairly certain that parallel ones exist for Introductory Abstract Algebra
Try "Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991" - it's a book that contains problems (and their solutions) from the hardest higher mathematics competition (harder than the Putnam and the IMC). The contestants are given a week to solve the problems, and they can consult literature.
The topics "range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, topology, to set theory". Beautiful and engaging problems, written by Hungrian mathematicians, some of them well-known.
I don't know if this is the type of thing you are looking for, but T. Y. Lam has a problem book associated with his A First Course in Noncommutative Rings, which contains all the solutions to the problems in the "First Course."
Have you tried looking at the Schaum series? They have many books with lots of problems in complex analysis, topology, set theory, probability and more.
Murty and Esmonde, Problems in Algebraic Number Theory, Springer Graduate Texts in Mathematics, Vol. 190, 2nd ed., 2005.
I am very fond of the book 1001 Problems in Classical Number Theory by Jean-Marie De Koninck and Armel Mercier. As suggested by the title, the book consists of 1001 problems in number theory, broken up into the following sections:
- Mathematical Induction and Combinatorics
- Divisibility, Prime Numbers
- Representation of Numbers
- Primality Tests and Factorization Algorithms
- Integer Parts
- Arithmetical Functions
- Solving Equations Involving Arithmetical Functions
- Diophantine Equations
- Quadratic Reciprocity
- Continues Fractions
- Classification of Real Numbers
The book has more than 200 pages of solutions, which often times provide a rough outline rather than a full solution, giving one a path to follow in order to fill in the details oneself.
I've always been fond of "250 Problems in Elementary Number Theory," by Waclaw Sierpinski. It provides full solutions for all the problems as well.
Ian Adamson has 2 excellent problem courses with complete solutions in basic set theory and general topology, A Set Theory Workbook and A General Topology Workbook. Both are very good indeed-I'm using the latter to review for my Master's qualifying exam.
Some of the older classic textbooks for graduate students are essentially problem courses,despite not being labeled that way. John Kelley's General Topology and Paul Halmos' Measure Theory are both essentially problem courses in each of their respective subjects.
There is also a series of problem texts for undergraduate algebra, Algebra Through Practice,in 5 volumes,coauthored by T.S. Blyth. They look quite good, although I haven't used them myself.
Lastly,there's one of my favorites: Paul Halmos' A Linear Algebra Problem Book. It develops an entire linear algebra course through problems and it has 3 sections: problems, hints and full solutions. Me and a number of the honors math students at Queens College of CUNY were joyously challenged by this book several years ago while reviewing and attempting to master linear algebra. For the good students that get stuck,the hints will get them unstuck most of the time-they don't need to look at the full solutions. Even so,they'll be happy they're there.
I most highly recommend it!
Springer-Verlag has a series of advanced problem books. The series is edited by Peter Winkler, and the books which are currently available are listed here: http://www.springer.com/series/714?detailsPage=titles