Best way to profit from a book whitout exercises

Question

Sometimes, I find notes or books that has examples and/or theorems but no exercises. I know that exercises are a fundamental part of math learning, so I don't know how to proceed in this case in order to really get a grip on the material. What is good practice in this case?

Answer 1

The best approach, and this goes for when there are exercises in the book as well, is to treat each proof presented as its own exercise. And this holds true at all levels of learning, or researching, really.

When you get to a theorem/lemma/proposition/corollary read it through, then take a few minutes to think about it. What are the key ideas for proving it? Where would you start, what do the hypotheses set up for you that you know you're going to want to make use of? Write them down (that's actually really important). See how far you can get with the proof by yourself.

For major theorems, if you only get the basic idea but not the details, that's excellent -- those are the things that took people time and trouble to get right in the first place. For a corollary, you should probably be able to get all of the details right as well.

And then when you're done and you've read through the proof and considered it (and maybe posted here if your solution is different and you can't see why one way is better than the other), go back and check you understand why each hypothesis is needed. Because a good theorem only sets up the minimum requirements for the result to hold, so try and find a counterexample for everything where a hypothesis is dropped (e.g. if the set of interest needs to be compact, work out what goes wrong, where the proof fails if you drop compactness).