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I recently started self studying Rudin’s Functional Analysis and man is it slow going when you don’t take any parts of any proof on faith. Should I find a balance? If so, how do I know which clauses of which sentences in which proofs are worth proving and which I can skip?

I’m hoping to be starting a phd in the fall so I’d love to hear some words of wisdom on this.
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I don’t think you verify literally EVERYTHING along the way. That would be impractical and inefficient
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I'm commenting only because I would like to see an answer to this as well.
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I think it's one or the other, and some prudence needs to go in to the choice. For foundational material, I would say don't rush and have it down as solidly as possible.
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I try to read math texts non-linearly and in multiple rounds. First round is a skim to intuitively understand the major themes and the overall "flow" of the logic (e.g. which concepts depend on which other concepts). It's still math so even this skim takes a while. Second round I try to internalize down the definitions and theorems. Third round I verify (or attempt ) the proofs and (if available) tackle the exercises.

In this learning model, you have to be comfortable with not having a full understanding immediately.
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My general advice: For proofs that aren’t incredibly technical, you should try to understand 90% of the details. If you’re having trouble working out details, give it a bit of time, but move on if you’re spending too much* time on it. If this keeps happening, it might be worthwhile to slow down to ensure you’re actually learning more than just definitions and theorems. For me, these strategies have helped ensure I’m learning the key techniques of the field while avoiding spending too much time on technical lemmas whose proofs I’ll never need again.

In an ideal world, you’d know how to distinguish between technical lemmas that you can treat as black boxes versus key results whose proofs illustrate important ideas. But when you’re learning new math, you don’t have a clear sense of what’s important, so you sometimes have to accept that you’ll miss important things on your first pass. I typically go back to the proofs while I’m solving problems to gain inspiration, which usually helps fill in gaps, too.

*Of course, what counts as “too much” time depends on your desired pace, the difficulty of the material, your familiarity with it, etc.
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Could try other books, sometimes the statement/explanation/technique that is easiest to understand is not always in the same book, also YouTube lectures
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It gets easier.  As you see more proofs, whole chains of logic turn into a single chunk in your mind.  After a while, you can verify the “main idea” of a proof and ignore the routine parts.  Then you can remember a multi-page proof as “use Talagrand’s method to limit the tails of the Fourier transform of the long-time limiting distribution” (or whatever).  But to get to that stage, you have to look at a lot of proofs.  I think what you’re doing is quite reasonable.  Just skip verifying anything you find obvious, and you’ll find the obvious segments getting longer and longer.  Keep up the good work.
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Pick shorter textbooks.

Joking aside, I don't really think there is a way. Going through details just takes a long time.

>If so, how do I know which clauses of which sentences in which proofs are worth proving and which I can skip?

IMO it depends on the level of the subject and textbook. I think for simple* enough you should be able to prove all the details. You don't have to actually write out every inequality or diagram chase, but you should be confident that you could if asked.

When I say simple, I mean any theorem that isn't really technical and getting in the weeds (something like Urysohn's lemma, for example) and is generally covered in a first or second course in a topic - so not all the advanced extra material most textbooks have (I don't know how expansive Rudin is, but we covered maybe a third of Lang and Eisenbud for my graduate algebra sequence).
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I'm currently reading through Rudin's "Functional Analysis" while awaiting the start of my PhD in the fall as well. In regard to that specific textbook, I'd recommend reading the first four chapters as thoroughly as possible (perhaps you can skim the section on vector-valued integrals and holomorphic functions in chapter 3). The rest of the book more-or-less covers special topics that you can thoroughly read or skim as needed.      


In general, reading textbooks is hard and time-consuming. Even two semester grad course sequences will rarely cover more than half the textbook. And the level of understanding achieved trails off rapidly the less thoroughly you read them. I'd recommend picking a really great textbook--such as Rudin's "Functional Analysis"--and reading it as thoroughly as you can. Once you start reading into more specialized topics, having a solid foundation in the subject will make it much easier to work through those.
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I think you should allow yourself to move on from a proof if you feel like you get hung up on it for too long. Make sure you understand what the theorem is saying and then skip the proof, but record all the proofs you have skipped. Once you have made it through the book, go back to the proofs you skipped.

You might even want to return even earlier, maybe something later on made something click for you and you think you could give a skipped proof another shot.

There is no point in giving up on an entire book because of a couple of proofs and the likelihood that you will better grasp them on a second round is significantly higher
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