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How to take notes from an Analysis textbook?

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When I took analysis I always made point to really understand any and all definitions or theorems I came across in the book that were relevant to what we would cover in the course. Taking down notes and copying the definitions/theorems by hand made it easier for me to recall them on the fly. For me, getting those down was half the battle in writing proofs for that class, so yeah, I would try to put some effort into taking notes on the material.
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Personally I write down every definition, theorem, and proof. I'm not sure if this is the most efficient approach, since it's quite lengthy, but it aids the understanding and you'll have a neat compact textbook in your notes that you can reference later.
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If you are just trying to get a good head start on an Analysis course you are taking next semester it might be a good idea to just get comfortable doing exercises involving sequential and functional limits. So focus on chapters 2 and 4 in understanding analysis.
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Fill in the gaps, as if you were going to teach the subject. Pretend you were going to give a class on the textbook and you have to explain the intuition behind everything. Add counter examples/examples. See if you can reproduce the proofs/theorems in your own words without checking.

This can sometimes become too time consuming to do for the entire book so you might want to focus on some sections and chapters more than others.

I personally don't find much value in copying anything verbatim but some people find it useful.
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I typically study on my own and while I find that I'm more likely to remember things if I write something down, there's an absurd amount of information in any halfway decent analysis text, even an intro text. This is the advantage of taking an actual course; someone with experience can wittle the info down to the essentials. So my approach is to do something else instead:


As I read through the text, I look for things that make me ask "why?". I'll sit back and think on this and then, either in the margin or on a piece of paper (detailing context in this case) I'll work out the answer. I've found that, for me at least, this works pretty well and is especially helpful in remembering and understanding proofs.


I'll also find examples for certain claims. For instance, if my text claims that for a bijection f:X->Y from an at-most countable set X, the image of X is at-most countable, I might find a specifc bijection that satisfies this claim.


If you want to write down what appear to you to be vital results, try writing them in your own words. Examine what you've written and see if you can find any errors in the way you've written these results and see if you can correct them.
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You should do whatever works for you. That said I encourage you to take pointers from what everybody here has said. Usually when I’m reading a book I actually do very little reading when compared with thinking. I will usually jump to the topic I want to learn and read until I get to something that isn’t obvious to me. At that point I stop and think. If I can figure it out in my head in a minute or two then I keep reading. If it’s trickier and I can’t hold everything in my head or feel like I’m missing part of something, then I stop reading completely and start writing things down. Identify parameters that you can adjust, distinguish hypotheses from conclusions, identify how parameters are quantified (for all or there exists), look for previous results that might be related, try to come up with examples of your own, search for or ask a question on MSE.

The point is that you need to be actively engaged in the material in the sense that you could reproduce it yourself without much reference.
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My personal style is to write the definitions, both technically and "meaningfully".  Write theorems both technically and meaningfully.  And summarize the proofs with just enough detail such that, if I omit any details, I could easily fill them in if I were asked to.
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Copy and write down definitions and statements of theorems. See if you can start with them and figure out the proof on your own. Peek at book’s proof as needed. In the end write down in complete step by step detail your own proof.
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Understanding analysis is a great book. Extremely well written and organized. But it isn't that comprehensive, compared with for example Taos books or Rudin. I think when you read Abbott, you should note down every definition, theorem and proof (sketch of proof at least). And when you are done with the book you should be able to do every single exercise without hints and prove every single theorem.

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