Advanced (graduate level) topics in math with easy problems?

[johnqwertyful]

I find one problem I run into when studying higher maths is that the problems that come along with them are just so difficult sometimes. I find that I spend so much time trying to figure out how to solve the problem that I lose sight of the actual material. One quarter was particularly bad, the second quarter of a graduate class on functional analysis. I really liked the class, but I feel like I got the least out of it compared to all three quarters. The teacher loved clever problems, and it took me easily 20-30 hours a week to do the homework and at the end I had no idea what I had done. I just threw every theorem I had at it. At the end of proving some obscure technical result about weak-* convergence, I would have totally forgot what weak-* convergence even meant. The problem, while seriously stretching my problem solving skills, did nothing to help me understand weak-* convergence. Not blaming the teacher, it was a fun class, I just wish I understood the topics better.

Anyway, I'm looking for books that have easy, straightforward problems in graduate subjects to cement the foundations. Something with true/false questions or simple computations/proofs that follow directly from the definition. Doesn't have to be functional analysis, it could be measure/probability theory, algebraic topology, differential geometry, etc.

 

[BuriedAlive]

Not sure if you've heard of it, but "Introductory Functional Analysis with Applications" by Kreyzig seems like it might be a good fit. My graduate program uses it in our "bridge" analysis course between undergraduate real analysis and the more serious functional analysis course. The proof exercises are pretty straightforward and there is also a smattering of short "theory" questions. Also most odd problems have solutions/hints in the back to help the self-studying student. The last chapter of the book is also entitled "Unbounded Linear Operators in Quantum Mechanics", but since I am not personally a physicist, I have no idea how good/useful that particular material is.

 

[homeomorphic]

I had a similar issue, too. Professors in grad school liked to beat us up so much with difficult problems that we didn't have time to stop and think and reflect on anything. We just had to crank out as many brilliant solutions to the massive onslaught of problems as we could. I would have remembered a bit more if I had a little more breathing room, although I can appreciate the value of having challenging problems to hone your skills. I saw a mathoverflow post of a grad student who said he was doing very well in terms of grades, but he wasn't satisfied because he didn't have time to conceptualize the stuff. I don't believe this sort of scenario is actually a good thing. I think people tend to have sort of a macho attitude that everything needs to be super-difficult, and it's counter-productive. Some things should indeed be super-difficult, but past a certain point, you are going to get diminishing returns because the difficulty is going to prevent you from doing other important things. If professors are concerned that they might not be challenging the top students enough, they should assign bonus problems.

You can always make up your own easy problems to do. I think it may even be assumed by authors/profs that you should do that.

I don't think the problems are the only issue, though. Sometimes, what you need is better motivation and better explanations, which you could take a problem-based approach for, but I don't know of that many books that do that particularly well, since the problems tend to be focused on how to use it more than where it comes from and why it's important, which, it seems not that many people care about, despite the fact that it can be a great aid to understanding and retention. A Radical Approach to Lebesgue's Theory of Integration comes to mind for measure theory, though. If I recall correctly, I thought the problems were not too difficult, and it's very well motivated. Mathematical Physics by Geroch has some really nice sections on functional analysis, and I don't think his problems are terribly hard, but I'm not sure.