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It's not nearly as easy to work through as SICP is but Principles of Mathematical Analysis by Walter Rudin (sometimes referred to as little Rudin) is a great pl

It's not nearly as easy to work through as SICP is but Principles of Mathematical Analysis by Walter Rudin (sometimes referred to as little Rudin) is a great place to start if you're interested in analysis. It's a hard book, but it's pretty much the standard for undergrad analysis.

kliao 13y ago
Echo the above. Pairing little Rudin with a more user-friendly book like Understanding Analysis by Abbott is a great way to learn the material if you get stuck.
- randomknowledge 13y ago
  Personally I would recommend mathematical analysis by Pugh. Goes into as much depth as Rudin, but more approachable and a great selection of exercises.
  - kliao 13y ago
    I like Pugh too. All three are sitting on my bookshelf :). Coming from very little formal math training, and trying to self-study analysis, I would bounce from Rudin if I got stuck, go to Abbott for a confidence boost, and try out some Pugh for a different viewpoint. Pugh might have been just right in terms of rigor and accessibility.
  - nextos 13y ago
    I was under the impression that Pugh, while more approachable, was also much more difficult. Especially with regards to the exercises.
dominotw 13y ago
Baby Rudin still gives me nightmares.
- jvreeland 13y ago
  Real Analysis by Royden is the source of my nightmares.
  - graycat 13y ago
    Royden is breathtakingly gorgeous. But, it can be good to skip some parts. The main content is just 'measure theory', and that's just freshman calculus grown up. Why? Because near 1900 it became clear that freshman calculus was clumsy for some important progress, especially cases of convergence of functions. Measure theory? Well, first cut, 'measure' is just a grown up version of ordinary area. Simple. Why interested in measure theory? Because want to cook up a new way to do integration, that is, what freshman calculus dues with the Riemann integral. Recall, the Riemann integral partitions the X axis and approximates the area under the curve with tall, thin rectangles. Measure theory partitions the Y axis: At first glance this seems a little clumsy, but in the usual cases get the same number for area under a curve and in bizarre cases, that can get from converging functions, get a nice answer that Riemann integration can't do. Royden likes Littlewood's three principles, and they are cute. So, spend an evening on them. Yes, it's possible to use Littlewood's to do the subject, but there is a better way, also heavily in Royden, roughly called 'monotone class arguments' -- which are gorgeous and turn much of the whole book into something quite simple. So, prove the theorem for indicator functions. Then extend to simple functions by linearity. Then extend to non-negative measurable functions by a monotone sequence. Then extend to integrable functions by linearity. Can knock off much of the book this way, e.g., Fubini's theorem which is just interchange of order of integration grown up. The foundations of this little four step process is Fatou's lemma, the monotone convergence theorem, and the dominated convergence theorem. About 2/3rds of the way through Royden is a single chapter that essentially compresses much of the rest of the book -- I would have to step into my library to find the chapter. For the early exercises on upper and lower semi-continuity, they are a bit much and you likely won't see that topic again. So that exercise can be skipped. Royden is elegant beyond belief; if you still have trouble finding the main themes, then chat for an hour with a good math proof who understand Royden well. Then, don't miss the Radon-Nikodym theorem: It can be seen as a grown up version of the fundamental theorem of freshman calculus but, really, is much, much better. The role of the Radon-Nikodym theorem in 'modern' (i.e., Kolmogorov) probability theory, stochastic processes, Markov processes, martingales, etc. is astounding.

kliao 13y ago

Echo the above. Pairing little Rudin with a more user-friendly book like Understanding Analysis by Abbott is a great way to learn the material if you get stuck.

randomknowledge 13y ago
Personally I would recommend mathematical analysis by Pugh. Goes into as much depth as Rudin, but more approachable and a great selection of exercises.
- kliao 13y ago
  I like Pugh too. All three are sitting on my bookshelf :). Coming from very little formal math training, and trying to self-study analysis, I would bounce from Rudin if I got stuck, go to Abbott for a confidence boost, and try out some Pugh for a different viewpoint. Pugh might have been just right in terms of rigor and accessibility.
- nextos 13y ago
  I was under the impression that Pugh, while more approachable, was also much more difficult. Especially with regards to the exercises.

dominotw 13y ago

Baby Rudin still gives me nightmares.

jvreeland 13y ago
Real Analysis by Royden is the source of my nightmares.
- graycat 13y ago
  Royden is breathtakingly gorgeous. But, it can be good to skip some parts. The main content is just 'measure theory', and that's just freshman calculus grown up. Why? Because near 1900 it became clear that freshman calculus was clumsy for some important progress, especially cases of convergence of functions. Measure theory? Well, first cut, 'measure' is just a grown up version of ordinary area. Simple. Why interested in measure theory? Because want to cook up a new way to do integration, that is, what freshman calculus dues with the Riemann integral. Recall, the Riemann integral partitions the X axis and approximates the area under the curve with tall, thin rectangles. Measure theory partitions the Y axis: At first glance this seems a little clumsy, but in the usual cases get the same number for area under a curve and in bizarre cases, that can get from converging functions, get a nice answer that Riemann integration can't do. Royden likes Littlewood's three principles, and they are cute. So, spend an evening on them. Yes, it's possible to use Littlewood's to do the subject, but there is a better way, also heavily in Royden, roughly called 'monotone class arguments' -- which are gorgeous and turn much of the whole book into something quite simple. So, prove the theorem for indicator functions. Then extend to simple functions by linearity. Then extend to non-negative measurable functions by a monotone sequence. Then extend to integrable functions by linearity. Can knock off much of the book this way, e.g., Fubini's theorem which is just interchange of order of integration grown up. The foundations of this little four step process is Fatou's lemma, the monotone convergence theorem, and the dominated convergence theorem. About 2/3rds of the way through Royden is a single chapter that essentially compresses much of the rest of the book -- I would have to step into my library to find the chapter. For the early exercises on upper and lower semi-continuity, they are a bit much and you likely won't see that topic again. So that exercise can be skipped. Royden is elegant beyond belief; if you still have trouble finding the main themes, then chat for an hour with a good math proof who understand Royden well. Then, don't miss the Radon-Nikodym theorem: It can be seen as a grown up version of the fundamental theorem of freshman calculus but, really, is much, much better. The role of the Radon-Nikodym theorem in 'modern' (i.e., Kolmogorov) probability theory, stochastic processes, Markov processes, martingales, etc. is astounding.