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Firstly I'd like to apologize if this is too much of an opinion-based question. I do not know where else would be appropriate for such a post, if there is such a place please let me know.

I am very interested in mathematics, particularly mathematical physics, but I currently do not have access to an institution or any professors. As a result I have mainly been self-teaching myself through textbooks. My usual routine is to very carefully go through a chapter in a book, making sure I understand every step in every proof, and then move onto the exercises. Going through the proofs this way helps me measure which theorems/results are more important, and also helps me learn common methods of proof. This part of my studying is going fine and I'm able to finish the chapter and understand most of the material, sometimes with the help of other books or online resources such as math stackexchange.

The problem I have been having is with the exercises. I use these as a way to reinforce the material and test my understanding, but also as a way to develop my problem solving skills. I very rarely am able to solve a problem in any graduate math textbook I have read (which I have already finished 2-3), and have trouble even getting started with them. Eventually if I make no progress I open a question on stackexchange, or try to find a solution online which I usually am able to follow but I would not have thought of myself. I want to get to a point where I can actually do these exercises and not just be able to understand the solutions. I am beginning to question if I'm just not cut out for it or if my approach is wrong.

In summary, I am able to read about these topics and understand proofs written by others, but I struggle on actively doing the math myself. I am looking for some advice on how to develop these problem solving skills and do better on the exercises. I have taken a look at similar questions on this site such as:

The advice given in the intersection of all these posts seems to be more practice and to get help/hints from your professor. As someone who is self-studying and often gets stuck on where to even begin, I am not sure how to apply this advice and I would love to hear what working mathematicians would advise. For example, when I don't know how to start or I am stuck what should I do? I often think about the problem for some time but get nowhere, only to find a clever trick or small lemma should have been applied.

For some reference on what kind of books I am studying, I am currently going through the following:

  • Methods of Modern Mathematical Physics Vol 1. (Functional Analysis), by Reed & Simon
  • Mathematical Gauge Theory by Hamilton
  • Partial Differential Equations by Evans
CBBAM
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    Generally questions about the content of specific fields are outside the scope of Academia.SE, and I think this question basically boils down to "how do I learn math" which is too broad for any Stack Exchange site. It is however very normal for problems in graduate-level textbooks to be extremely difficult, so do not be discouraged! – Nate Eldredge Jun 17 '23 at 20:03
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    One general tip though - when you do solve problems, with or without help, make sure you are learning not only the result but also the techniques. When you see a "clever trick" for the first time, think about how and why it works, how it generalizes, and try to develop intuition for it. Then you will have a better chance of seeing where to use it next time. – Nate Eldredge Jun 17 '23 at 20:06
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    Compared to those books you mentioned, how is your experience with solving (proof-based) problems in 'less advanced' topics such as real analysis or linear algebra? – Jochen Glueck Jun 17 '23 at 20:21
  • @NateEldredge Thank you for your advice. Should I remove this question? – CBBAM Jun 17 '23 at 20:27
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    @JochenGlueck I learned both fields in a very unconventional way which was over time and as I went. For example, I did not read a real analysis or linear algebra book in a focused manner and at one time as I'm doing with the graduate books I mentioned. As a result I'm not sure how I would describe my experience in those topics, but I'm sure there are a good amount of questions in Rudin's PMA that I would struggle with. – CBBAM Jun 17 '23 at 20:32
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    When I read "making sure I understand every step in every proof," I got the impression that you might be looking so intensely at details that you're missing a bigger picture. It's good to understand all the details, but I'd suggest that, after carefully reading a chapter as you've been doing, and before moving on to the exercises, you should go through the chapter again to get a broader picture of what's going on. Perhaps just review the theorems (skipping the proofs) and try to understand how they fit together and what the key ideas are. – Andreas Blass Jun 18 '23 at 00:37
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    @AndreasBlass Yes that is correct, I focus on the details so that I'm not fooling myself in understanding a proof, but I've noticed at times I do end up losing sight of why we're doing a certain step to begin with. A good example of this is in functional analysis or PDE when a string of inequalities appears. I try to work out for myself why each step is justified and I usually end up figuring it out on my own, but I sometimes forget why these inequalities appeared in the first place. As a result I have a hard time coming up with those kinds of steps on my own. – CBBAM Jun 18 '23 at 00:53
  • @CBBAM: Thanks for your response. Your sentence "I'm sure there are a good amount of questions in Rudin's PMA that I would struggle with" is quite hypothetical, though. No matter whether you studied real analysis and linear algebra in an unconventional way, it is important (in order to give an answer that fits your situation) to know whether you actually did do a good amount of problems in those topics and, if yes, how well that went. – Jochen Glueck Jun 18 '23 at 08:09
  • @JochenGlueck Thank you for your response as well. I wouldn't say I've done a lot of problems directly in real analysis, and even less so in linear algebra (though I do feel more comfortable with the latter). Most of the problems I've come across in those fields have been in the context of other subjects. For example, I really got a good grasp on much of the theory behind linear algebra once I learned about $L^p$ and Banach spaces. Likewise, I did not fully utilize concepts and techniques from real analysis until I started studying measure theory (though I had learned about them before). – CBBAM Jun 18 '23 at 08:20
  • @JochenGlueck With that said, should I revisit those subjects and do problems or would it be best to continue learning what I need as I go in these more advanced topics? – CBBAM Jun 18 '23 at 08:21
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    @CBBAM: "For example, I really got a good grasp on much of the theory behind linear algebra once I learned about $L^p$ and Banach spaces. Likewise, I did not fully utilize concepts and techniques from real analysis until I started studying measure theory" Ok, just to be sure I understand correctly: Did you (try to) solve a significant amount of problems when you studied $L^p$- and Banach spaces and measure theory? Or is your study of the graduate books mentioned in your question the first time that you engage in solving (or trying to solve) a large amount of problems? – Jochen Glueck Jun 18 '23 at 09:55
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    As other contributors suggest, it is extremely important that you are able to write a proof for any considerably less advanced statements / exercises. In my opinion, there is no point in studying functional analysis if you can't provide epsilon-delta proofs when half asleep. – Thomas Bakx Jun 18 '23 at 10:15
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    @JochenGlueck Yes I did try to solve exercises when studying those topics, but I often ran into the same problem of not being able to do the exercises on my own. While I did study the solutions until I was able to understand every step, I'm not sure how helpful that was. If I may be hypothetical again, if I were to revisit those $L^p$ or measure theory problems I think I would do better now but I'm sure I would still struggle with a large number of them. For reference, the book I was using was Folland's Real Analysis. – CBBAM Jun 18 '23 at 17:50
  • @ThomasBakx Thank you for your comment. I am able to write proofs for some very easy problems, for example showing that the limit of uniform limit of a sequence of continuous functions is also continuous. A step above that is about where I begin to struggle. Should I start revisiting analysis/linear-algebra problems as part of my studying? If so, what is the suggested way I should do this since I'm already familiar with most of analysis on a conceptual level. – CBBAM Jun 18 '23 at 17:55
  • @CBBAM I see, so in that case it seems there is a gap between the subject you're trying to grasp and the ones you already truly grasp (which, as you acknowledge, is very different from understanding proofs that are already presented to you). Do any of the books you read state formal prerequisites? Maybe we can take it from there. – Thomas Bakx Jun 18 '23 at 18:59
  • @ThomasBakx Yes they state prerequisites but they are very general, for example they might say familiarity with manifolds or $L^p$ spaces. What makes it hard for me is since I already know the subjects conceptually I don't know if I should read an undergraduate book from page one and start my math journey over again or if I should just do problems. My main goal is to develop problem solving skills. – CBBAM Jun 18 '23 at 19:18
  • @CBBAM If your main goal is to develop problem solving skills, then there's your answer. You say you know the prerequisites 'conceptually', so, for example: can you give me a definition of the tangent bundle of a smooth manifold without looking it up? Forgive me if it sounds like an elementary question. Can you state the inverse function theorem? Do you know the definition of a differential form of degree p? If the answer to any of these questions is not an immediate 'yes', then you won't be able to solve problems that refer to these subjects. – Thomas Bakx Jun 18 '23 at 19:40
  • @ThomasBakx I am able to state the definition/theorem of all the concepts you mentioned, and I can even have a discussion about them on an intuitive level and state some of their properties. The part I have trouble with is if you asked me to prove a theorem about them. – CBBAM Jun 18 '23 at 19:50
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    Have you tried opening a question on SE Mathematics giving one or two examples of such exercises asking for how to go on finding a proof for such a statement rather than asking for a solution? – Christian Hennig Jun 18 '23 at 23:24
  • @ChristianHennig Thank you for your suggestion, that is a good idea. – CBBAM Jun 19 '23 at 07:27

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I expect that what you lack is appropriate mathematical experience and maturity. Graduate-level textbooks often assume a level of mathematical maturity that is different than the "logical prerequisites". You're also hampered by the lack of mentors/classmates to discuss things with.

My suggestion is to begin by going through easier (e.g., undergraduate) books in your area of interest, where you can do many of the problems on your own. That will help you naturally improve your problem solving abilities, and make it easier for you to understand more challenging books later.

You can also look on Math StackExchange (e.g., questions with the [book-recommendation] tag) for recommendations about books for either your particular interests, problem solving, proof writing, etc. And if you don't find exactly the the kinds of suggestions you want, ask a new question there.

Kimball
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    I'd strongly second the observations of this first paragraph: it's not just having the "formal, logical" prerequisites, but also having enough experience with those prerequisites so that they are now "second nature", rather than requiring conscious thought or recollection. Somewhat parallel to the difference between knowing the rules to a sport, and playing it... much less playing it well. :) – paul garrett Jun 19 '23 at 17:26
  • Thank you for your answer and @paulgarrett for his comment. Based on your suggestions, I took a look at that tag and came across the book Berkeley Problems in Mathematics, which according to the book description, is a collection of problems to help students review undergraduate mathematics in preparation for preliminary exams. This seems to be exactly what I need. I'll try doing a few problems from it everyday as I also study the more advanced material. – CBBAM Jun 19 '23 at 19:30
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    @CBBAM I get a bit of a sense from your comment that you're attempting to "skip" undergraduate-level material. An undergraduate level of understanding can't simply be rushed past, and you should make sure to work through dedicated undergrad textbooks in a subject first. You can't do an end run on mathematics this way. Your problem is a symptom of this strategy. You do not lack knowledge per se, but the aforementioned mathematical maturity. Cramming with that Berkley book is to continue with the same flawed methodology. I will gently suggest that you have not yet internalized Kimball's answer. – Jerome Jun 21 '23 at 19:56
  • @Jerome Thank you for your comment. I agree with your assessment that I do not have the mathematical maturity I need to tackle tough problems and this is what I would like to develop. For more on my background, I actually have done my undergraduate in mathematics but it wasn't a great program and left me with a very shallow foundation. This is why I only truly learned things such as real analysis and linear algebra while self-studying more advanced topics. – CBBAM Jun 21 '23 at 22:50
  • @Jerome So far I have only gone through the first 10 problems of the Berkeley book and I have been able to do about 7 of them without too much trouble. Should I instead go through an undergraduate textbook such as Rudin and treat it as if I've never seen the subject before? Would that be better than directly tackling problems from the Berkeley book? – CBBAM Jun 21 '23 at 22:51