• Wed, Dec 18. The final exam is tomorrow morning 9am. Remember that I will be bringing copies of the review pages. The homework problem won't be from the last problem set (11) since I have not posted the solutions! Good luck to all you.
• Tues, Dec 17. PS10 solutions posted. (This has a rather elaborate problem on Hermite polynomials, especially the completeness part, that you don't need to study for the final exam.)
• Sun, Dec 15. To simplify my instructions for the final exam: two of the problems will be from the review problem list, one from homework, and one new problem. What I said in class was that there would be two problems from the review sheet, one from the list of theorems, and the other from the list of (6) problems. But if you look at the problems you will see that they include the main theorems. So, to simplify, what you need to prepare is the six problems. (You should know the theorems and use them where appropriate.) Secondly, with your exam, you will get a copy of the review sheet. You do not need to use your own 3 pages double-sided of notes to write down what is on the review pages.
• Sun, Dec 15. Solutions to PS7, PS8 and PS9, just posted. (PS10 and PS11 will be posted later in the week.)
• Sun, Dec 8. Brownian Motion notes just posted.
• Tues, Nov 26. PS11 (last one) just posted. Also, I fixed an error in the last handout, Fourier Integrals 2 (concerning signed measures, which we will discuss tomorrow, Wed Nov 28).
• Thurs, Nov 21, Fourier Integrals 1 notes, just posted.
• Sat, Nov 16. PS10 just posted.
• Fri, Nov 9. PS9 just posted. No lecture notes for now because we are in the middle of Section 3.5.
• Mon, Nov 4. At least two student asked about the misprint on page 137 in the text. There really is a misprint. Please find it, and mention it explicitly on your problem set --- It has to do with a missing function from the list at a very low index. This is essential in order to conclude that the list is complete (parts b and c). Furthermore, you can't leave any functions out when you check orthgonality in part (a).
• Fri, Nov 1. Just posted: PS8 and Fourier series notes covering today's lecture and half of next week: convergence in norm of Cesaro means and applications of Fourier series.
• Mon, Oct 28. Revised PS7 just posted. The last two problems on PS7 are deferred to PS8. In place of that, you are asked to correct your hour test.
• Fri, Oct 25. PS7 just posted. (In Exercise 1, assume that the Fourier coefficients determine an L1 function. We already know this for L2 functions, and will prove it within a week for L1 functions.) Subject of Monday, Oct 28, lecture also just posted under the title Fourier Series 2.
• Wed, Oct 23. Fourier Series 1 notes revised so as to confirm the finite measure property of the simple functions in Step 4 of the last proof. Answers to problems 4 and 5 on the practice hour test posted also in handouts.
• Mon, Oct 21. Fourier Series 1 notes just posted.
• Fri, Oct 18. PS6 solutions have just been posted. Also, the handouts now include the midterm review for the hour test, which will take place 10am, Friday Oct 25.
• Wed, Oct 16. Two sets of lecture notes have just been posted. The first treats completeness of Lp spaces (as discussed in today's lecture). These notes also prove a density statement, to be discussed in lecture next week. The second set of notes covers orthonormal sequences in Hilbert spaces and is very similar to the treatment in Section 3.3 of the text. This will be the lecture topic for Friday, Oct 18.
• Mon, Oct 14. Solutions to PS4 and PS5 just posted.
• Fri, Oct 11. PS6 and solutions to PS1, PS2 and PS3 just posted.
• Fri, Oct 4. PS5 just posted.
• Mon, Sept 16. Just posted: the updated version of PS2 (as promised) with one more problem, namely, to show that the conclusion of the strong law of large numbers implies the conclusion of the weak law of large numbers. This will be discussed in class on Monday, Sept 16, with several important details and hints.
• Wed, Sept 11. Two new postings. PS2 under homework, and remarks on Boolean rings in handouts. The inconsistency between the syllabus and PS1 is resolved on PS2. (There were 2 extra problems from AG 1.3 listed on PS1. If you did not do them for today, you are asked to turn them in next week with PS2 for full credit.) In the handout on Boolean rings, I explain the sense in which a set-theoretic ring, as defined in our text, is an ordinary ring in the sense of algebra. I said this incorrectly in lecture, as one of you generously pointed out after class. (The correct way to define addition in the ring is to use symmetric difference, not union.)
• Wed, Sept 11. One of you asked what reading you should be doing. Most of the time, including now and for several weeks, you should be reading the sections of the book that correspond to the problems assigned on the problem sets. We discussed Section 1.1 of the text, and skipped to Section 1.3. We will be talking about Section 1.3 through Friday, then moving on to Sections 1.2 and 1.4 next week. On Friday, Sept 20, there's a student holiday. On Monday, Sept 23, we will finish up any loose ends in Chapter 1 and start Chapter 2.
• Fri, Sept 6. Three Postings so far: Syllabus. Notes for the introductory lecture given on Wed, Sept 4. Problem Set 1. [The very first problem --- Cantor diagonal argument --- was omitted from an earlier electronic posting of the contents of PS1, but that is a mistake. Do this problem.] As mentioned during the introductory lecture, I will post lecture notes if the lecture deviates substantially from the Adams-Guillemin text, but that won't happen again for several weeks.