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For anyone who started from no mathematical maturity to grandpa rudin, how did you do it?

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You do not need to go through the entire trifecta before applying to grad school: that’s what you do during the first two years of your grad school. Abbott+Baby Rudin (or Tao 1,2) is more than enough. Also you are applying to statistics programs not pure math, so while analysis/measure theory is important for you, the level of rigor you need eventually is probably not mastery level for all three. So I’d say keep feeling good about yourself; self-studying from scratch is hard and you need every motivation you can get. There is plenty of time to feel inadequate in grad school /s
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I guess Fourier Analysis on Groups is the relative no one talks about.
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I'm still only a beginner in mathematics, but I don't think you can read the Rudin trilogy linearly like how you  planned. Baby Rudin is meant for undergraduate while Papa Rudin is for graduate students. A normal mathematics students gain maturity through other courses like abstract and linear algebra, topology, etc. throughout his undergraduate studies, so by the time he gets to Papa Rudin, he has gained much more maturity than someone who only read Baby Rudin
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The answer is to be patient, and do a LOT of the exercises (preferably all), this is espacially true for papa and grandpa rudin. The exercises there are very difficult and time consuming, but they are really build you up mathematically and force you to reach far more mastery than what university courses on the subject usally offer. Once you reach a point where you have solved most exercises in a chapter is when I would personally move forward.

I had the pleasure of self studying the rudin trilogy on my own before being accpeted into any university, so I reached grandpa rudin without any prior experience in linear algebra or topology beyond what papa rudin taught. wheter this is good or bad is debatable (I definetely think learning some linear algebra could have been helpful, but I actually think learning topology through natural uses of it and not as a standalone was actually a very good decision) but the point is the books are pretty self contained so you dont need much more to move from one to the other than just making sure you master each one. Conquering the rudin trilogy is a long and difficult journey, but once you get to the point where you are able to finish most exercises in papa and grandpa rudin, I would definetely consider this as a \*very\* high level of analysis mastery
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In my case it's mostly a question of waiting :)
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During my undergraduate, I spent many months struggling through Rudin's principles of mathematical analysis (baby Rudin) which is mainly what gave me confidence in analysis and proofs. I assumed that all it would take to read Rudin's book is time and effort.

When I took a course in complex analysis I tried to read Rudin's real and complex analysis (papa Rudin). I found the book very difficult to follow and felt quite discouraged, especially since the course moved at a swift pace and I didn't have enough time to work through the book slowly. I switched to using Stein and Shakarchi's book for the course. I revisited Rudin's book at the end of the semester, and I felt I could really appreciate it and understand why people say the presentation is so elegant. I learnt complex analysis well when I relearnt it from Rudin. Armed with this experience, when learning measure theory, I read a couple of chapters from Royden's book before reading the chapters on measure theory in Rudin's real and complex analysis. Having a bit of familiarity with measures really made it easier to read Rudin's book. I had a similar experience with Rudin's functional analysis (grandpa Rudin) when learning functional analysis.

I think Rudin's books are an excellent source if you have the time to think through and work out the arguments yourself. I am not the most mathematically mature or sharp reader, and so I found it difficult to use as a first introduction to the subject. But they were excellent second books to read in a subject, and they supplied with with a majority of the intuition I have in these subjects.
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Just because some guy on stack exchange said it doesn’t make it true. If you’re going for a Statistics PhD then most of the material in those books will not even be relevant to you. Even for probability theory you’d be better off learning whatever functional analysis you need from a less abstract treatment. There’s no reason for you to read those books unless they personally interest you. In that case, just open them up and start reading, which is what I did at one point. Otherwise, don’t.
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Controversial opinion to some perhaps : just learn what you need to/want to. A lot of mathematics was invented to formalize intuitions and investigate certain problems/examples. Often simply learning topics of interest to you will give you a context with which to understand otherwise abstruse topics. Math books don’t need to be read linearly and almost no one does this for very long.

You need to master fundamentals - linear algebra, basic analysis, basic algebra or whatever but after that just learn whatever you wish/need to.
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So far I still haven't finished Part II of his Functional Analysis because I have not that much motivation to study more of differential equations, but other than that let me put down what I can think about about this trilogy, which may answers your question, partially.

* A hidden motivation of all three books should be, explaining his *Fourier Analysis on Groups*. After studying some parts of first two books you should be able to give this book a try.
* It is not necessary and not realistic to read all three books linearly. For example, the prerequisite of R&C is PMA's first 7 or 8 chapters. R&C's final chapters are somewhat technical and you may want to read them when absolutely needed. By the way chapter 18, on Banach Algebra, is a lightweight version of the Chapter 10 in his Functional Analysis (you may simply skip it if you will go into part 3 of Functional Analysis; a lot of duplication). For his Functional Analysis, he arranged a map of logic chains. If you see it, you will realise that after first 4 chapters (and chapter 5 for some specific examples) you can jump to part 2 or part 3 as you wish, because these two parts are not that related. Trying to do everything linearly will burn yourself down real quick.
* As I am studying number theory, I'm really grateful about Rudin's setting in his books. In his R&C he put everything into "locally compact Hausdorff space" whenever possible, meanwhile from what I've learnt, many objects in number theory may enjoy this property. As a result I have less things to worry about whatsoever.
* Rudin did not teach you how to compute things, but he did not think it was not necessary. Instead, he thought you could handle them yourself. Therefore you may need some extra help.
* Exercises. Some exercises are classic (counter) examples, some ask you to fill the gap in main text, some are asking you to prove other key theorems, some are fairly technical things, asking you to reproduce theorems proved by other mathematicians (maybe from a 1909 German paper and it's nowhere to be found on the Internet).  I think first three are really important.
* When I was studying chapter 6 of R&C, I felt lost because I had no idea why this should be that. But when I was studying chapter 11 or 12 of Functional Analysis those in R&C came up naturally and everything connected. I suppose many have to go through this kind of journey.
* If you don't have motivation to study complex analysis anyway, at the very least you should finish chapter 10 of R&C. This is the so-called bread and butter.
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Dont just learn analysis. I gained more mathematical maturity from also taking topology, algebra, and geometry classes

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