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Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?

by hsikka 7y ago

throwawaymath 7y ago
Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course targeted at applied math majors to do much other than that. The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.
- hsikka 7y ago
  Fantastic, thank you for the tip! I'll probably prepend Real Analysis to the list!
- umanwizard 7y ago
  > You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R. The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most. This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
ghufran_syed 7y ago
As someone who also went back to study math after not doing it for a long time, I would recommend doing a book that focusses on teaching how to prove things, using the math you already know. Most upper div math courses either assume you know how to prove things, or try and teach it in passing while also teaching the topic itself (classically linear algebra and calc 3 were taught this way, though now many places now have separate "proof" classes instead) I would strongly recommend Chartrand and Zhang's "Intro to mathematical proofs", once you know how to prove things, every other more advanced math class you take will be much easier than it would have been otherwise. [Edit] In particular, do the book, including the chapters on "proofs in calculus" and "proofs with real and complex numbers" before doing real analysis - you'll enjoy it much more that way! [/Edit] I went through most of this book a few years ago, now about to finish my MS in math :)
- dorchadas 7y ago
  Agreed. I'm trying to move from being a secondary mathe teacher to actually learning pure math, and proofs are the killer. I feel like I understand topics, but creating proofs kills me.
chestervonwinch 7y ago
Analysis requires a certain level of mathematical maturity to motivate, follow, understand, and appreciate. This level is roughly acquired after having taken undergraduate coursework in ODEs, linear algebra, or abstract algebra.

Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?

by hsikka 7y ago

Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?

Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course targeted at applied math majors to do much other than that. The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.

hsikka 7y ago
Fantastic, thank you for the tip! I'll probably prepend Real Analysis to the list!
umanwizard 7y ago
> You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R. The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most. This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.

ghufran_syed 7y ago

As someone who also went back to study math after not doing it for a long time, I would recommend doing a book that focusses on teaching how to prove things, using the math you already know. Most upper div math courses either assume you know how to prove things, or try and teach it in passing while also teaching the topic itself (classically linear algebra and calc 3 were taught this way, though now many places now have separate "proof" classes instead) I would strongly recommend Chartrand and Zhang's "Intro to mathematical proofs", once you know how to prove things, every other more advanced math class you take will be much easier than it would have been otherwise. [Edit] In particular, do the book, including the chapters on "proofs in calculus" and "proofs with real and complex numbers" before doing real analysis - you'll enjoy it much more that way! [/Edit] I went through most of this book a few years ago, now about to finish my MS in math :)

dorchadas 7y ago
Agreed. I'm trying to move from being a secondary mathe teacher to actually learning pure math, and proofs are the killer. I feel like I understand topics, but creating proofs kills me.

chestervonwinch 7y ago

Analysis requires a certain level of mathematical maturity to motivate, follow, understand, and appreciate. This level is roughly acquired after having taken undergraduate coursework in ODEs, linear algebra, or abstract algebra.