Intro Modern Analysis II · Mathematics
Power Series. Definition of outer measure and Lebesgue measure. Measurable sets and measurable functions. Step functions and simple functions. Egorov Theorem and Lusin Theorem. Fubini Theorem and Tonelli Theorem. Fourier Series and Transforms. Multivariable Analysis.
We covered Chapters 7-11 of Rudin's Principles of Mathematical Analysis (1976). Topics include sequences and series of functions, differentiation and integration of functions of several variables, special functions, the integration of differential forms, and Lebesgue integration/measure theory.
Fourier Analysis, Measure theory, Lebesgue integration, functional analysis, and some differential topology.
Rigorous proof-based mathematical analysis.
Sequences/Series of Functions, Uniform Continuity, Weierstrass Theorem Fourier Series, Derivative in higher dimensions, Lebesgue measure
Weekly homework (11 total), 2 midterms, Final
Quite a bit. We had homeworks every week, which took a while but varied (5-10 hours, maybe). We also had two midterms, which were incredibly stressful for me, at least. And of course a final. Grade weights respectively 25%, 2 x 20%, and 35%.
Two midterms, one final, around semi-weekly problem sets (6 total this semester). Exams: Each exam consists of 60% memorization, i.e. 20% definitions (given in advance), 20% proofs of specific theorems (also given in advance), 20% homework problems (solutions given in advance). On one exam, an easy problem replaced the HW question. All in all, if you feel you can memorize things well, you should be able to start at 55% for each exam (assuming a ~5 point margin of error for miswriting a definition or something). If you can do around half of the doable question and steal a few points off the last, you'll do fine. If you can actually solve the doable question, you'll do great. Note: whatever you do, write everything you can possibly think up for the last two questions, including random verbal explanations and such. She's very willing to give partial credit. Homework: do in groups, go to office hours, look up solutions as a last resort. You should be fine, since averages will be very low.
Homeworks: - His homeworks were pretty hard, but short -- just 4 problems. About half of the class would regularly go to the TA's office hours for help, and a lot of the time it would be us doing the problems together with the TA, since the TA didn't know how to do them immediately. Probably spent 4-5 hours on homework per week.
We covered measure theory on R^n—we constructed the Lesbegue measure and integral, and proved facts about the Lesbegue integral like the dominated convergence theorem and Fatou's lemma. After this, we discussed the basic theory of Hilbert and Banach spaces (including a class on the L^2 convergence of Fourier series), before moving onto multivariate calculus, proving the chain rule, and thereoms like Clairaut's, inverse function, and implicit function. Note that most classes do multivariate calculus before measure theory.