Suppose I am interested in really learning / thoroughly reviewing some subject (e.g. the basic theorems of infinite Galois theory, or the classification of compact Lie groups). One approach I might consider would be to write up a set of notes for myself going through all the main definitions, notation, and proofs of the main theorems in the subject, with appropriate references. I can think of several good reasons to do this:
- I can't find a single textbook that does everything in quite the order I want it done in, or proves things quite the way I want them proven, and I want to synthesize material from multiple sources.
- Just the act of writing down all the proofs in my own words will force me to understand the flow of ideas better.
- I can revise the notes as my understanding of the subject improves.
- I can put the notes online for others to benefit from.
But suppose one day I find a textbook that does things nearly perfectly (e.g. Neukirch for the basic theorems of algebraic number theory), and I am having trouble seeing the point of "competing," as it were. Is it still worthwhile to write up the notes that I was planning? Should I just do exercises instead?
(Feel free to interpret this question more generally, e.g. to ignore the body and concentrate on the title. CW because this question is mildly subjective and I'd like to encourage a broad range of answers.)