## Real Analysis II. Fall 2012

Professor Francis Su
Office Hours: Olin 1269, Thursdays 4pm or by appointment.
e-mail: [insert my last name] @ math.hmc.edu

Tutoring Hours: Platt Living Room, Wednesday evenings 6:15 to 7:45pm
e-mail: k[last name, no apostrophe] @ hmc.edu

This course is a continuation of Math 131. Glad you could join us! Topics will include: Riemann-Stieltjes integration, function spaces, equicontinuity, uniform convergence, the inverse and implicit function theorems, differential forms, and an introduction to Lebesgue integration and measure theory.

Text: Rudin's Principles of Mathematical Analysis.

Course webpage: http://www.math.hmc.edu/~su/math132/.

Coursework: Homeworks will be assigned and collected weekly. Lowest homework score will be dropped. There will be one midterm and one final exam. Each component (homework, midterm, final) is worth at least 30% of your final grade, with the "best" component worth 40%.

Honor Code: Cooperation on HW assignments is ﬁne (and in fact encouraged), but appropriate acknowledgements should be given, and you are expected to write up your solutions INDIVIDUALLY, i.e., it should be the case that after said cooperation you have understood the solution well enough to explain it on the homework! It is appropriate to acknowledge the assistance or cooperation of others when given.

 Homeworks, due Fridays at by noon, in box outside my office (Olin 1269). Here is HW #1. Due Fri Sep 7. You'll need these links: to fill out the Info Card and to read the my handout on Good Mathematical Writing. Be sure to read Chapter 6. Here is HW #2 and the TeX code. Due Fri Sep 14. HW #3. Do Chapter 6 (6, R7, 8, 10abc, 11, 15 ). Due Sep 21. Hint on 10a: for a concave up function, its values always lie below the secant line between two endpoints. HW #4. Read Chapter 7, theorems 7.1-7.15. Do Chapter 7 ( 1, 2, 3, R4, 5, 6 ). Due Sep 28. HW #5. Read the rest of Chapter 7. Do Chapter 7 ( 7, 8, 9, 11, R14, 15 ) Due October 5. Hints: On 7.8, thm 7.12 can still be useful. On 7.9, use eps/2 argument. HW #6. Do Problem I below and Chapter 7 ( 16, R17, R18, 19, R24 ). Due October 12. Problem I. (a) Suppose f is an element of the function space C_b(R), the continuous bounded functions on R with the sup norm. What does an open ball around f "look like" in this metric space? (b) Consider the set K, the collection of "spike" functions {f_n} where f_n is 0 except between (n-0.5) and (n+0.5) and where f_n(n)=1. Show that the set K is closed and bounded in C_b(R), but it is not compact: there is an open cover of K with no finite subcover. Start studying for your take-home midterm, which will be available Oct 12. It will cover Chapters 6 and 7. Due October 19 but extension to after Fall Break is possible (just ask). HW #7. Do the problems on this handout. Due Nov 2. HW #8. Do Chapter 9 ( 2, 3, 5, 6, 8 ). Due Nov 9. HW #9. Do Chapter 9 ( 9, 11, 13, 16 ). Due Nov 16. HW #10. Do Chapter 9 ( 17, 18, 19, 20 ). Due Nov 30. HW #11. Due Wednesday December 12. Do Chapter 10 ( 15, 20, R21 ) and Problem A. Prove Theorem 10.20a in your own words. Problem B. Prove Theorem 10.20b in your own words. Problem C. Find a differential 2-form (omega) in R^4 such that (omega)^(omega) is not zero. [Recall "R" means read, but do not do the problem.] p.s. To keep updated by e-mail, be sure to subscribe yourself to "math-132-l".