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> Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development

> Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. Naw! Take another pass through Baby Rudin! Here is some help: With appropriate mild assumptions, a continuous, real value function on a compact set is uniformly continuous. For the finite dimensional space of interest, compact is the same as closed and bounded. That is the main content of the first chapters. First Application: With uniform continuity, get to show that the Riemann and Riemann Stieltjes sums converge so that those integrals do exist. Second Application: The uniform limit of a sequence of continuous functions is continuous. This was a question on my Analysis Ph.D. qualifying exam -- I got it! Thus a certain normed space is complete, that is, is a Banach space -- I'm not sure this remark is in the book. The chapter(s) on sequences and series is used to define ln, sin, cosine so that can do Fourier series. The exterior algebra is essentially separate. That's the main stuff. Don't take the Riemann integral very seriously: The good stuff is Lebesgue's integral (partitions the range instead of the domain) as in the first half of Rudin's Real and Complex Analysis. There also get to see Banach space, Hilbert space, von Neumann's novel proof of the Radon-Nikodym theorem (grown up version of the fundamental theorem of calculus and with astounding power) and the Fourier integral. I won't take off points if you don't read the second half. Rudin's Functional Analysis -- get to see distributions.

siddboots 7y ago
I don't doubt your expertise, and no doubt I still have a lot to learn, but I have to admit I find it off-putting that you immediately assumed a position of talking down to me. It's not at all clear to me what part of my comment you are correcting.
- graycat 7y ago
  "What part"? You wrote > Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. and I removed what was "complex". Many students hear bad things about Baby Rudin and don't try it. Of students who do try it, too many don't finish well. Of the students who do finish, only a small fraction take a second pass where they will get some of the clarity I typed in for you. For my Ph.D., there were five qualifying exams. I did the best on four of them, and one of those was Analysis and fairly close to Baby Rudin. The department didn't offer a course to help students prepare for the Analysis exam. So, I was likely the only student who had taken two passes through Baby Rudin; the first pass was in a course; I did okay; the second pass was on my own and slowly; it was fun! The second pass is the main reason I did the best on that exam. The next year the department tried teaching a course from Baby Rudin: They did that for only one year; I can believe that few or none of the students did well with the course. A friend from another department wanted to learn Baby Rudin, took the course, had a hard time, and dropped it. He went away unhappy, and that was sad and not really necessary. So, my experience with Baby Rudin suggests (i) take a formal course, (ii) use advice such as I gave here to see during the course some clarity in what is going on, how, and why; (iii) also study or glance at one or two other competitive or similar books, e.g., Spivak, Calculus on Manifolds, Fleming, Functions of Several Variables, Kolmogorov and Fomin, Apostol, and more, and (iv) take a second pass through Baby Rudin. Net, Baby Rudin has discouraged a lot of students, too many. To many students, Baby Rudin looks forbiddingly severe and abstract; students can't figure out what the heck is going on, how, or why. E.g., your view was "complex" -- it can be "simple". With some help, e.g., as I typed in here for you, Baby Rudin can be okay.
  - siddboots 7y ago
    You are misunderstanding my use of the word "complex". I'm saying that after encoding the graph structure I found Baby Rudin contained more complexity (i.e. connectedness) than the other texts that I repeated the excersise for.

siddboots 7y ago

I don't doubt your expertise, and no doubt I still have a lot to learn, but I have to admit I find it off-putting that you immediately assumed a position of talking down to me. It's not at all clear to me what part of my comment you are correcting.

graycat 7y ago
"What part"? You wrote > Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. and I removed what was "complex". Many students hear bad things about Baby Rudin and don't try it. Of students who do try it, too many don't finish well. Of the students who do finish, only a small fraction take a second pass where they will get some of the clarity I typed in for you. For my Ph.D., there were five qualifying exams. I did the best on four of them, and one of those was Analysis and fairly close to Baby Rudin. The department didn't offer a course to help students prepare for the Analysis exam. So, I was likely the only student who had taken two passes through Baby Rudin; the first pass was in a course; I did okay; the second pass was on my own and slowly; it was fun! The second pass is the main reason I did the best on that exam. The next year the department tried teaching a course from Baby Rudin: They did that for only one year; I can believe that few or none of the students did well with the course. A friend from another department wanted to learn Baby Rudin, took the course, had a hard time, and dropped it. He went away unhappy, and that was sad and not really necessary. So, my experience with Baby Rudin suggests (i) take a formal course, (ii) use advice such as I gave here to see during the course some clarity in what is going on, how, and why; (iii) also study or glance at one or two other competitive or similar books, e.g., Spivak, Calculus on Manifolds, Fleming, Functions of Several Variables, Kolmogorov and Fomin, Apostol, and more, and (iv) take a second pass through Baby Rudin. Net, Baby Rudin has discouraged a lot of students, too many. To many students, Baby Rudin looks forbiddingly severe and abstract; students can't figure out what the heck is going on, how, or why. E.g., your view was "complex" -- it can be "simple". With some help, e.g., as I typed in here for you, Baby Rudin can be okay.
- siddboots 7y ago
  You are misunderstanding my use of the word "complex". I'm saying that after encoding the graph structure I found Baby Rudin contained more complexity (i.e. connectedness) than the other texts that I repeated the excersise for.