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I'm currently self-studying functional analysis. My general approach is to read each chapter, make sure I understand all the definitions, and carefully follow every proof. I usually manage this part fairly well but sometimes I need to consult other sources to know why parts of certain proofs work.

I also want to improve my proof writing and problem solving skills, so I try to solve every exercise at the end of the chapter. I'm on chapter 3 of the book now and I've started to really struggle. I often don't even know how to begin the exercise. After spending time trying to work it out and failing, I usually resort to looking up a solution and they often turn out to be simpler than I expected. I can follow the reasoning but I never feel like I would have come up with it on my own.

The first two chapters went better and I at least had some initial ideas and sometimes just needed a hint to finish a proof. But now even after reviewing a solution I find that if I revisit the same problem days later I struggle to reproduce it correctly.

It’s starting to feel like I’m not making much progress if I regularly need to rely on hints or solutions. Is this a sign that there is something specific I should work on? What should I do when I have no idea how to even start an exercise?

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    $\begingroup$ What book are you using? Depending on the book, doing every single exercise will often tire you out, it's much more efficient/beneficial to your learning process to make sure you're minimizing the number of exercises. I know this isn't profound, but proofs are hard. I don't know how much time you're spending on each question, but finding the correct proof will always be difficult. Take as much time as you need on each question (which is why its important choosing the right exercises to work on). I will often choose a problem and think about it for the rest of the day (or even multiple days). $\endgroup$ Commented Jul 21 at 6:36
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    $\begingroup$ sometimes when you’re stuck, it’s because you actually didn’t understand certain concepts so discussing with some people, or evn trying to explain things to yourself from scratch is a good way to start. Review all assumptions and why that might a reasonable or not (the prime example of all of this is our recent conversation about the closed-graph theorem). But, sometimes, problems really are hard (from personal experience, I’ve definitely struggled with several RS problems), and perhaps the reason they’re hard is because the “true” concept underlying the problem will only be introduced later. $\endgroup$ Commented Jul 21 at 6:52
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    $\begingroup$ but regarding simpler solutions: note that sometimes this could just be your failure, but lots of other times, it truly is because people have perfected certain techniques, so they’re able to come up with super slick solutions. But that doesn’t necessarily mean a more long-winded/down-to-earth solution is any worse (in fact, several times you need to fail at a problem to see why a correct solution is the way it is, or why your approach couldn’t possibly have worked). But more concretely, sometimes I like to jump around to differnet problems, and maybe even a different section, then return. $\endgroup$ Commented Jul 21 at 6:55
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    $\begingroup$ Reed/Simon is a compendium of Mathematical Physics, branch Mathematical Foundation of Quantum Field Theory. After 15 years of study and research in Mathematical Physics you buy the four volumes as reference and a source of examples for your students. In order to fast track that field fo a first time, use eg Riesz/Nagy, Bremerman and other more paedagogical introductory texts. Halmos motto at the first page is a warning against its use as an introduction. $\endgroup$ Commented Jul 21 at 7:12
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    $\begingroup$ I have no idea why this question was closed. The reason given above does not apply. Have voted to reopen. $\endgroup$ Commented Jul 21 at 8:01

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