Math 424 F Home Page

This will be the home page for Math 424 F -- "Honors Real Analysis" This class meets for the Fall 2019 semester at MWF 2:00-2:50 in 345 Altgeld.
The mathematical links page will be here. Suggest a new one!

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

Tu 9/17 -- Here is the Third Homework. Paper copies will be distributed in class tomorrow.

M 9/16 -- The Second Homework Solutions were distributed. In addition, I presented at the board the solution to the ungraded problems and talked more about compact sets in metric spaces. On Wednesday, I will give a special case of Heini-Borel: proving that [0,1] is a compact set, before doing the full proof. Also, HW1 should be returned on W.

Homework 2 comments:

  • On problem 5, we should have a0 = 4 (not 2). My apologies for the typo.
  • Also, recall that the absolute value appears in the definition of limit and | |x| | = |x|
  • (-1)^n is often useful in describing x_n when the behavior is different for even n and odd n.

    F 9/13 -- Completion of the proof that R is complete and the extension to R^n. A beginning of the discussion of compactness, which is a strange definition. I'm aiming to get HW1 back on Monday; HW2 is due (see comments above) and HW3 will be distributed. The phone/email list (not linked) was distributed.

    W 9/11 -- Monotone sequences, complete spaces, and the reals are complete. The proof I used involved limsup and liminf which are barely in the book, and in a weird formulation. Once I write some notes on them, I'll talk about them some more.

    M 9/9 -- The First Homework Solutions were distributed. The Second Homework was also passed out. In terms of new material, we are progressing through the definition of convergence and some of its properties. Next stop: monotone sequences.

    F 9/6 -- Much more topology, up to the end of section 3.2. I talked about comparable metrics, and when I write up notes with the 1-norm and 2-norm and infinity-norm, I'll put that definition in as well; for now, it means that if d and d' are two metrics on the same set X, and there exist positive r and s so that r*d(x,y) \le d'(x,y) \le s*d(x,y), then d and d' are called comparable and the theorem is that they determine the same topology: a set is open in (X,d) if and only if it is open in (X,d'). Monday: HW 1 due, HW1 solutions distributed, HW2 put out and the phone number / email list will be circulated one last time.

    HW 1 comments:

  • On #5, d_1 and d_2 are just two distances, not necessarily the "L_1" and "L_2" ones we were discussing in class.
  • I intended in #5 that r > 0!

    W 9/4 -- More examples of metric spaces: to wit, R^n with the "1-norm" and "infinity norm", as well as the metric space of n-tuples from an alphabet, where the distance of two n-tuples is the number of places in which they differ. The relatively familiar definitions of an open set in a metric space. More examples on Friday. If you want notes on any of the unfamiliar material, please let me know.

    M 9/2 -- Labor Day, so no class, but here is the First Homework, which is due 9/9. I will distribute these in class on W 9/4.

    F 8/30 -- I started with a "close reading" of the book's proof of the existence of square roots, with a handout (not given here) of two pages of the book, and then an exposition of what was missing. Moving to chapter three, we defined metric spaces and gave some examples. The real challenge was proving Cauchy-Schwarz, using what "quadratic polynomial approach" (found in the book), and also the "sum of squares" approach.

    W 8/28 -- Observation: many questions you were asking me after class could have been asked in class. I'd prefer that. I WANT you to interrupt if what I am saying is incorrect or unclear. For example, At one point, I wrote "-N < |x| < N" when I should have written "-N < x < N". Tell me this. Class should be interactive. I don't want to be replaced by a videotape!
    Homework 1 is postponed to be due M 9/9, because there won't be enough covered material for an assignment otherwise. We talked a bit more about the properties of |x| and then moved on the the LUB axiom: every set in R that is bounded above has a least upper bound. We then took implications of this (all in the book).

    M 8/26 -- Introductions. Handouts are: Course Organization, How to Solve It guide, Class questionnaire. and information about the links page (see above.) We covered, basically, pp.1-22 of the text. Please ask if you have questions. I did some "enrichment" material on ordered fields, which won't be on the exams. We agreed to have homework due on Mondays, though with Labor Day next week, HW1 will be due W 9/4 (and available on 8/28) [see correction above]. For more on ordered fields see my short expository article: What does `>' really mean?