Real Analysis II, Spring 2018

• Course Information
• Announcements
• Problem Set

Sun 5/13:

• Problem 5 on Taylor’s theorem has been corrected in the Final Exam Problems. The remainder term had the wrong sign and was missing a 1/k! term.
• Office hours this week will be on Monday from 3:00-4:30.
• The final exam is Tuesday from 8:30-11:30 in Pritzker’s LETRA classroom (the room with all the aquarium tanks).

Wed 5/9:

• Please take a moment to (anonymously) fill out the exit survey on the Moore method. I am very interested in your feedback on how to improve it for future iterations of the class!
• Also, Problem 5 of the final exam problems on Taylor’s Theorem has been updated to specialize to the case of 2 dimensions, so you don’t have to worry about the multiindex notation.

Tue 5/8:

• Final Exam is on Tuesday, May 15, 8:30-11:30am in the LETRA room in Pritzker (the room with all the fish tanks).
• The final will consist of a selection of these Final Exam Problems.

Sun 5/6:

• More problems on integration theory added to the Problem Set, including sets of measure zero and the Dominated Convergence Theorem. Some additional problems will be added before Thursday.

Wed 5/2:

• More problems on integration theory added to the Problem Set.

Tue 5/1:

• There will be no office hour today, Tuesday 5/1. Please feel free to email me with questions or to set up another time to meet.

Mon 4/23:

• Thanks to everyone who reported on the class activities last week. It sounds like the class was productive.
• Despite what I said earlier, I have decided not to hold class on Monday, 4/23, during bacc week.
• I will hold my usual office hours (M,T,Th 3:30-4:30)
• Class will resume on Thursday, 4/26 with Lebesgue integration. I will spend some time lecturing on this material during class, but not the whole time. You should prepare some of the problems (especially the motivating examples) as well.

Mon 4/16:

• The class will meet without the instructor this week.
• If you attend class, please send me a brief email with a short digest of what happened in class (e.g. Person A proved Theorm X. Person B gave an example for Example Y. Person C started to prove Theorem Z but got ran into a problem with Q. The class discussed the issue and came up with solution R. Etc.)
• After you finish with the inverse and implicit function theorem material, you can get started with the new Lebesgue integration material (Chapter 6).

Thu 4/12:

• Added some problems on the inverse and implicit function theorems (5.6.8-5.6.11).

Mon 4/9:

• Fixed hypotheses in the Picard Existence theorem. (The delta ball in which F satisfies the Lipschitz condition should be closed, so that the continuous functions into it form a complete metric space.)

Mon 4/2:

• New problems added to problem set, in particular:
  • (New) problem 5.4.4 on explicit computations of the total derivative.
  • (New) problem 5.4.5 on an inequality verson of the MVT for multivariable functions.
  • New remark at the end of section 5.4 on total second derivatives.
  • Section 5.5 on the contraction mapping principle and application to ODE existence
  • Section 5.6 on the inverse function theorem for multivariable functions.

Mon 3/26:

• Added a section on gradient (5.3) and continuity of derivatives (5.4) in the problem set. I will discuss these on Monday, and you can prepare them for Thursday.

Mon 3/12:

• Added a few more problems to the multivariable derivatives sections (5.1, 5.2) in the problem set to be prepared for Thursday. Please look at and prepare these problems!

Fri 3/9:

• Added a few more problems to the multivariable derivatives sections (5.1, 5.2) in the problem set to be prepared for Monday.

Thu 3/8:

• Corrected a typo in the problem set. The definition of the operator norm (5.1.1) should use a sup, not an inf. Thanks to Zach Halladay for the catch.

Mon 3/5:

• New Chapter 5 on multivariable derivatives. Please prepare some of this material for Thursday’s class.

Mon 2/26:

• Assigned a homework problem (4.6.9) from 4.6; added 4.7 on Weierstrass approximation as discussed in lecture today. Note that 4.7.6 is also an assigned homework problem for Thursday.

Fri 2/23:

• New section posted 4.6 in the notes. Please try and prepare this material for Monday’s class.

Mon 2/19:

• On Monday we will do student presentations on section 4.3 (also see new problem 4.3.8).
• Then Prof. Kottke will lecture on new sections 4.4 (trancendental functions) and 4.5 (the space of continuous functions).

Sat 2/10:

• Sections 4.2 (on series of functions) and 4.3 (power series) added to the Problem Set.

Sat 2/3:

• A few new problems added to section 4.1. See 4.1.11-4.1.13 in the Problem Set.
• Also, please write up and hand in 4.1.9 and 4.1.12 this week.

Sat 2/3:

• New section (4.1) added to the Problem Set.
• On Monday we will finish the proofs from sections 1.6 and 1.7, then I will lecture on 4.1.

Tue 1/30:

• Corrections to the series problems. As Yonathan pointed out, the p-series (1.6.8) converge if p < 1 and diverge if p >= 1. As Daniel pointed out, I missed a hypothesis in the alternating series theorem (1.6.13), which should be that the a_n are decreaasing. The Problem Set has been updated accordingly.

Mon 1/29:

• Material on series added to the Problem Set. See the new sections 1.6 and 1.7.