Monday, January 17th: After chatting a bit about the course, the web pages, and the challenges of real analysis, I started lecturing on Section 1.1 about proofs, giving examples of direct proofs, contrapositive proofs, and proofs by contradiction. After the break, I continued with Section 1.2 about sets and set operations, stating and (half-)proving the the Distributive Laws and the Laws of De Morgan. I also had time to state the definition of unions and intersections of families of sets from Setion 1.3.
Wednesday, January 19th: Covered (the rest of) section 1.3, section 1.4 and section 1.6 (section 1.5 is mainly relavant for the chapters (7 and 8) we are skipping). I tried to put the emphasis on what is most important for this course, i.e. the basic properties of functions, inverse images, and countability.
Notes. Podcast (as the microphone fell out about 11 minutes into the first half, there are some 10 minutes without sound).
Monday, January 24th: Talked about selected parts of Chapter 2, including the Triangle Inequality, Definition 2.1.1/2.1.2, Proposition 2.1.3, Definition 2.1.4, Proposition 2.1.5, the Completeness Principle, Theorem 2.2.2, the definition of limsup and liminf, and Proposition 2.2.3. Will start with Cauchy sequences (page 34) next time.
Wednesday, January 26th: I started by talking about Cauchy sequences and the completeness of \(\mathbb{R}^d\), culminating in the proof of Proposition 2.2.6. I then turned to the Bolzano-Weierstrass Theorem (which is a prequel to the important concept of compactness) and proved it. Finally, I proved the Extreme Value Theorem (making typical use of Bolzano-Weierstrass) and the Intermediate Value Theorem. I also mentioned the Mean Value Theorem, but did not have time to prove it. Next time, I'm starting Chapter 3.
Monday, January 31st: (Digital lecture) In the first half, I introduced metric spaces, showed some standard examples, and proved the Inverse Triangle Inequality. In the second lecture, I defined convergence and continuity in metric spaces and proved propositions 3.2.5 and 3.2.6. Next time, we shall start Section 3.3.
Prerecorded videos (replacing the ordinary lectures):
Video 1: Metric spaces. Notes. Video
Video 2: Convergence and continuity: Notes. Video
Wednesday, February 2nd: (Digital lecture) I covered section 3.3 (but forgot to prove that open balls are open, which I leave to you!). Perhaps a little more material than in an ordinary lecture, but we'll compensate for that later.
Prerecorded videos (replacing the ordinary lectures):
Video 1: Open and closed sets. Notes. Video
Video 2: Continuity in terms of open and closed set. Notes. Video.
Video 3: Boolean operations of open and closed sets. Notes. Video.
Monday, February 7th: Covered all of section 3.4 on completeness and started section 3.5 by defining compact sets and proving that all compact sets are closed.
Wednesday, February 9th: Finished the rest of section 3.5. The proof of Theorem 3.5.13 was a bit hurried, but I think we shall leave it at that and continue with section 3.6 next time.
Monday, February 14th: This lecture consists of three prerecorded videos:
Section 3.6: Alternative description of compactness. Notes. Video.
Section 3.7 (just a quick introduction to the main ideas as this is not really part of the syllabus, but good to know about): Notes. Video.
Wednesday, February 16th: I first went through the mandatory assignment, discussing the interpretation of the problems and giving a few hints. I then lectured on Section 4.2, but skipped Example 3 which is so illuminating and helpful, that you should take a look at it yourself. I finished by proving Proposition 4.3.1.
Notes. Podcast. (I lost the sound for a few minutes in the first half)
Monday, February 21st: Finished Section 4.3. Because of a Smart Board incident, the notes got split in two.
Wednesday, February 23rd: Covered Section 4.4. Towards the end I had a real fight with the (not so) Smart Board, and gave up the final steps of the proof of Abel's Theorem. Instead I have made a separate video and separate notes for Abel's Theorem.
Notes. Podcast. Notes for Abel's Theorem. Video for Abel's Theorem
Monday, February 28th: Covered sections 4.5 and 4.6 and even had time for a little preview of section 4.7.
Wednesday, March 2nd: I went through Section 4.7 and then started section 4.8 by introducing dense sets and proving Proposition 4.8.3.
Monday, March 7th: Finished section 4.8 on the Ascoli-Arzela Theorem. definitely one of the toughest in the course.
Wednesday, March 9th. Went through Section 4.9. To simplify the geometric arguments, I only considered the case m=1, i.e. a single, scalar-valued equation
Monday, March 14th: Lectured on the second proof (the "analytic") of Weierstrass Approximation Theorem. Note that section 4.11 is not part of the syllabus. In the last ten minutes, I started chapter 5 by defining norms and proving that every norm induces a metric (Proposition 5.1.3).
Wednesday, March 16th: Covered the rest of section 5.1 and most of section 5.2 (the exception is Proposition 5.2.3 which I shall take next time before I continue with section 5.3).
Notes. Podcast (unfortunately, the microphone seems to have died around 1.19:30).
Monday, March 28th: Finished section 5.2 by proving Proposition 5.2.3 (important for MAT3400) and then covered the first half of section 5.3 (will start on the part called "Abstract Fourier analysis" next time).
Wednesday, March 30th: Finished section 5.3 and started chapter 10, but didn't get any further than general motivation. The serious work with section 10.1 will start next time.
Notes. Podcast. (there is no sound between 53:50 and 58:40 as the microphone fell out)
Monday, April 4th: Covered the rest of section 10.1 and most of 10.2 (still have Lemma 10.2.5 and Theorem 10.2.6 to go).
Wednesday, April 6th: Covered the rest of section 10.2 and all of section 10.3.
Notes. Podcast. (Had some trouble with the microphone just after the break, but think it worked out well).
Wednesday, April 20th: Lectured on section 10.4.
Notes. Podcast (Some information on the Math Olympiad just after the break).
Monday, April 25th: Covered sections 10.5 and 10.6 plus 5.4 up to Theorem 5.4.5.
Wednesday, April 27th: Went rather carefully through the rest of section 5.4 and then took some highlights from section 5.5 (dropped everything about Neumann series, but mentioned Banach's Lemma without proof and even said a few words about the Bounded Inverse Theorem from section 5.7). Finally, I said some (I hope) motivational words about the definition of derivatives in section 6.1.
Monday, May 2nd: Lectured on section 6.1 and didn't get as far as I wanted as the proof of the chain rule took a lot of time. I got up to the definition of directional derivatives (Definition 6.1.10) and will start there next time.
Notes (don't work at the moment, I shall try to rescue them later). Podcast.
Wednesday, May 4th: Defined directional derivatives, proved Proposition 6.1.11, and worked through an example similar to (but simpler than) Example 2 i section 6.2. After the break I covered Section 6.3 on the Mean Value Theorem.
Monday, May 9th: Covered Section 6.4 plus Lemma 6.5.2. I made a mess of the proof of Corollary 6.4.7 (called the "Fundamental Theorem of Calculus, Part II" in the lecture), and it is better to take this part from the book.
Wednesday, May11th: Finished Section 6.5 and got started on 6.7 (we are skipping 6.6) where I formulated the Inverse Function Theorem and said a few words of motivation based on the one-dimensional case.
Monday, May 16th: Proved the Inverse Function Theorem and gave a quick introduction to the Implicit Function Theorem in the form of Corollary 6.8.3. Unfortunately, I didn't have time for examples, but I hope the examples on page 210 and 214 are readable on their own (we shall not attempt more advanced examples). This ends the presentation of new material – from now on the lectures will concentrate on reviewing the syllabus.
Wednesday, May18th: I reviewed Chapter 3 and did problems 2 and 3 from last year's exam along the way. I had to change microphone somewhere in the first half, and probably the sound is missing for a short period.
Monday, May 23rd: Reviewed the relevant parts of chapters 4 and 5.
Wednesday, May 25th: Completed this year's lectures by reviewing the syllabus from Chapters 10 and 6, plus saying a few words at the end about the format of the exam.