Real Analysis II.
    Fall 2016

    Professor Francis Su
    Office Hours: Olin 1269, Tuesdays 1:30 - 3:30pm or by appointment.
    e-mail: [insert my last name] @ math.hmc.edu

    Graders: Shiyue Li, Max Hlavacek
    e-mail: firstinitial lastname @ g.hmc.edu

This course is a continuation of Math 131. Glad you could join us! There's a lot of interesting and deep ideas in this course that you will enjoy learning about. Topics 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.

Absences, late homework: The learning you are doing in this class takes place in a larger framework of school and life. While I am excited about teaching and I'm sure you are excited about learning, work is not the most important thing, and sometimes life outside the classroom can take precedence. I can be somewhat flexible in accomodating requests for homework extensions and absences for other important events. Please make these requests 24 hours in advance, if possible.

Homeworks, due Wednesdays in box outside my office door by 2pm (Olin 1269).
  • HW #1. (Due Wed Sep 7.) Here is Homework 1 and its LaTeX code. And:

  • HW #2. (Due Sep 14.) Read Chapter 6. Here is HW #2 and the TeX code.

  • HW #3. (Due Sep 21.) Do Chapter 6 (6, R7, 8, 10abc, 11, 15 ).
    Hint on 10a: for a concave up function, its values always lie below the secant line between two endpoints.
    15: yes, that last inequality is strict.

  • HW #4. (Due Sep 28.) Read Chapter 7, theorems 7.1-7.15. Do Chapter 7 ( 1, 2, 3, R4, 5, 6 ).
    In Problem 5, you might notice how this compares with the statement of the M-test.

  • HW #5. (Due Oct 5.) Read the rest of Chapter 7. Do Chapter 7 ( 7, 8, 9, 11, R14, 15 )


    Hints: On 7.8, thm 7.12 can still be useful. On 7.9, use eps/2 argument.

  • Prepare for the midterm, which will be handed out Tue 10/11 and due Friday 10/14 by 5pm (or Saturday if you need it).

    It will be 4 hours (with hour long break for food), closed book, but you're allowed an 8.5x11 sheet of paper (front and back) on which you can have notes of any kind. The exam will cover Chapters 6 and 7. You might want to study old homeworks, and the following problems on equicontinuity:

    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.

    Also try Chapter 7 ( 16, R17, R18, 19, R24). Be sure to really read the 'reading' problems!

    Prepare your note sheet!

  • HW #7. Do the problems on this handout. Due Oct 26.

  • HW #8. Do Chapter 9 ( 3, 5, 6, 8, 9 ). Due Nov 2.

  • HW #9. Do Chapter 9 ( 11, 13, 16, 17 (worth 2 problems) ). Due Nov 9. Assume knowledge of calculus for taking derivatives of things like sin(x), etc.

  • HW #10. Do Chapter 9 ( 18 (worth two problems), 19 (worth 2 problems), 20 ). Due Nov 16.

  • HW #11. Due Wednesday December 30. 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.
    plus potentially others to be assigned on Monday.

    -------------- below this line, homework sets are not finalized -------------------

     

     

     

     

     

     

     
    [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".