Wednesday, May 25th: This is the last lecture, and we shall use it to review chapters 10 and 6. As there is less theory that is suitable for exam problems in these two chapters, we shall probably have more time to do old exam problems.
Zoom link: https://uio.zoom.us/j/64796143865?pwd=em9LdVN1aGpzNjNDUUJ4WCtEb3orZz09
Meeting ID: 647 9614 3865 Passcode: 917849
Monday, May 23rd: We shall continue the review, this time with chapters 4 and 5. I shall skip sections 4.4, 4.7, and 4.9 as these are mainly intended to show applications of the theory and are not directly relevant for the exam (at least not this years!)
Zoom link: https://uio.zoom.us/j/66048168059?pwd=SnlTYk9lZHdBeHBFM1FqSTdhSVk2dz09
Meeting ID: 660 4816 8059 Passcode: 096259
Wednesday, May 19th: We'll review Chapter 3 and parts of Chapter 4 (I don't know how far we will get). I'll try to pull things together and emphasize the central ideas rather than the details. I'll also try to find old exam problems that illustrate important techniques and ways to argue.
Zoom link: https://uio.zoom.us/j/62017339806?pwd=YXBTSWpWdjR3UGpvOEtPRDVOV3Axdz09
Meeting ID: 620 1733 9806 Passcode: 029392
Monday, May 16th: This is the last lecture with new material. We have both section 6.7 and 6.8 on the syllabus, but we don't have time for a thorough treatment of both. I have decided to give a full proof of the Inverse Function Theorem in section 6.7, and then see how much time we have left for the Implicit Function Theorem (in the form of Corollary 6.8.3). Here are Ulrik's videos (his proof of the Inverse Function Theorem is organized a little differently from mine).
Section 6.7: The inverse function theorem: Video. Notes.
Section 6.8: The implicit function theorem I (motivation): Video. Notes.
Zoom link: https://uio.zoom.us/j/67911943525?pwd=ZGFBbFVhL3dETFFpdlQ3OXZFVHMzdz09
Meeting ID: 679 1194 3525 Passcode: 260607
Wednesday, May 11th: We are now a bit behind schedule, and as I want to keep my promise of using the three last lectures for review, we have to cut down on something. Here is the new plan: I complete section 6.5 on Taylor's formula, then skip 6.6 on partial derivatives and go straight to section 6.7 on the Inverse Function Theorem (we don't need partial derivatives for this). Then I'll round-off the course by talking about the Implicit Function Theorem in the final dimensional case (i.e. in the form of Corollary 6.8.3). Here are Ulrik's videos for Taylor's formula and the Inverse Function Theorem:
Section 6.5: Multiindices: Video. Notes.
Section 6.5: Taylor's formula (in Rn): Video. Notes.
Section 6.7: The inverse function theorem: Video. Notes.
Zoom link: https://uio.zoom.us/j/68107931483?pwd=cUZkZUd1U0hqK1NTVitUR3UzdWF1UT09
Monday, May 9th: The plan is to cover section 6.4 on Riemann integration of functions taking values in complete normed spaces, and hopefully get started on section 6.5 on Taylor series. Ulrik doesn't have any videos on this material.
Zoom link: https://uio.zoom.us/j/62124126024?pwd=TURrdEZJODh2UWQzUGV3blhaWnIyZz09
Meeting ID: 621 2412 6024 Passcode: 208731
Wednesday, May 4th: The plan is to cover as much as possible of 6.2 and 6.3. I shall first define directional derivatives (Definition 6.1.10) and then show how they can be used to find derivatives and prove that a function is differentiable (through examples like example 2 in section 6.2). Section 6.3 is more theoretical, but provides us with a tool that will be helpful i what follows. Ulrik has two videoes on this material:
Section 6.2: The Gateaux derivative: Video. Notes.
Section 6.3: The mean value theorem: Video. Notes.
Zoom link: https://uio.zoom.us/j/65501822677?pwd=R25Qb3o5d3E2MWRXQWlScGo4N1RwZz09
Meeting ID: 655 0182 2677 Passcode: 654523
Monday, May 2nd: We shall continue with Sections 6.1 and 6.2 on derivatives (6.2 is the practical version of 6.1). It's important stuff, partly because it's easy to make simple exam problems about this material! Ulrik has lots of videos:
Chapter 6 background video: What is a derivative: Video. Notes.
Section 6.1: The Frechet derivative I: Video. Notes.
Section 6.1: The Frechet derivative II: Video. Notes.
Section 6.2: The Gateaux derivative: Video. Notes.
Zoom link: https://uio.zoom.us/j/65300803921?pwd=MG9DOXY0eElUVytIa202K3BUZzZaQT09
Meeting ID: 653 0080 3921 Passcode: 899453
Wednesday, April 27th: We first continue with section 5.4 which I shall cover more or less as in the book. In section 5.5, we have to make some shortcuts to save time. I shall follow the textbook up to to Proposition 5.5.5, and then drop the material on Neumann series and just postulate Banach's Lemma (Theorem 5.5.9). I shall also talk briefly about the Bounded Inverse Theorem 5.7.5, which is definitely outside the syllabus, but helpful to know about in Chapter 6. Hopefully I shall also have some time left to start section 6.1 on derivatives. Ulrik has a number of videos on this material:
Section 5.4: Linear operators I. Video. Notes.
Section 5.4: Linear operators II (boundedness): Video. Notes.
Section 5.4: Linear operators III (spaces of linear operators): Video. Notes.
Section 5.5: Invertible linear operators I: Video. Notes.
Chapter 6 background video: What is a derivative: Video. Notes.
Zoom link:
https://uio.zoom.us/j/67109385919?pwd=cXN3d0VZK29VVWQ4WjlOTUJqcncyQT09
Meeting ID: 671 0938 5919 Passcode: 027488
Monday, April 25th: The aim is to get through Section 10.5 (very short) and 10.6. In 10.6 we shall finally find conditions that guarantee that the Fourier approximations converge to the function. The arguments are more subtle variations of the arguments in 10.4, but now that we have seen one argument of this kind, it's should be possible to follow a more complicated one. If we have some extra time at the end, I'll go back to Section 5.4 and start talking about linear operators. Unfortunately, Ulrik doesn't have a video on 10.6.
Zoom link: https://uio.zoom.u/j/66188913857?pwd=MVR3Wm1zcVg5MEZEaEg3WlRvNXY3dz09
Meeting ID: 661 8891 3857 Passcode: 218481
Wednesday, April 20th: We shall work through section 10.4. I shall skip the proof of Lemma 10.4.2 (self-study), but cover the rest in more detail than Ulrik's introductory video:
Section 10.4: Cesaro convergence: Video. Notes.
Zoom link: https://uio.zoom.us/j/69395539637?pwd=RmJRVE8wcVRzdnFQeXNxeHp3dDJGQT09
Meeting ID: 693 9553 9637 Passcode: 317297
Wednesday, April 6th: We shall finish section 10.2 and hopefully cover all of section 10.3. Perhaps we'll even get started on 10.4. As I forgot to put out videos last time, I link to Ulrik's videos for section 10.2 now instead:
Section 10.2: Convergence of Fourier series I: Video. Notes.
Section 10.2: Convergence of Fourier series II: Video. Notes.
Section 10.3: The Dirichlet kernel: Video. Notes.
Zoom link: https://uio.zoom.us/j/66200279317?pwd=RnF6UmhuT1BzRDRvY3N3eUpWVHMwdz09
Meeting ID: 662 0027 9317 Passcode: 986793
Monday, April 4th: Sections 10.1 and 10.2.
Zoom link: https://uio.zoom.us/j/68863603573?pwd=Q3J6aU1PUnRwSmFUQXBwQjFoR0JRZz09
Meeting ID: 688 6360 3573 Passcode: 777360
Wednesday, March 30th: The plan is to finish section 5.3 (the part called "Abstract Fourier Analysis") and then continue with Section 10.1. Ulrik doesn't have a video specifically about the last part of 5.3, but the one about 10.1 may be worth watching already know:
Section 10.1: Fourier series (introduction and motivation): Video. Notes.
Zoom link: https://uio.zoom.us/j/61606798559?pwd=QnNQdmRNSDlJR3haekRjVmUxWGxZdz09
Meeting ID: 616 0679 8559 Passcode: 392828
Monday, March 28th: (NB: No classes on March 21st and 23rd due to midterm exams in other courses) The plan is to finish section 5.2 by proving Proposition 5.2.3 and then continue with section 5.3. This is a long section we won't be able to finish in one lecture, but it may still be good to know that after 5.3 we shall jump to chapter 10 (the alternative version of Fourier analysis). Here is Ulrik's introduction to section 5.3:
Section 5.3: Inner product spaces: Video. Notes.
Zoom link: https://uio.zoom.us/j/67248705622?pwd=U1BhSEFlcHRDOEs5Q05wcXZiN002dz09
Meeting ID: 672 4870 5622 Passcode: 369729
Wednesday, March 16th: We continue with chapter 5, hoping to cover a much as possible of sections 5.1 and 5.2. Here are Ulrik's introductions:
Section 5.1: Normed vector spaces I (definitions): Video. Notes.
Section 5.1: Normed vector spaces II (equivalent norms): Video. Notes.
Section 5.2: Series and bases: Video. Notes.
Zoom link: https://uio.zoom.us/j/65646783446?pwd=MXFQR3RlYVJNNUtjTEUxdFA1Y1ZZUT09
Meeting ID: 656 4678 3446 Passcode: 127570
Monday, March 14th: The main aim is to get through section 4.10 on Weierstrass' approximation theorem, but I also hope to get started on chapter 5 (section 4.11 is not on the syllabus). The book gives two proofs of Weierstrass' theorem, and I shall only cover the second one (the "analytic* one). As Ulrik gives the first proof in his introduction, I have not linked to his videos this time, but they are well worth watching.
The last weeks have been rather tough, but with chapter 5 we are now heading for calmer waters.
Zoom link: https://uio.zoom.us/j/68325438543?pwd=NjhwcW4xZmxLVTRRa1VkekVuSU1jUT09
Meeting ID: 683 2543 8543 Passcode: 528403
Wednesday, March 9th: We shall use the Arzela-Ascoli Theorem to prove existence of solutions to differential equations under weaker conditions than before. The proof is typical of a class of compactness arguments in differential equation theory. To simplify the geometrical picture, I shall carry out the proof in only one dimension.
Section 4.9: Convergence of Euler's method: Video. Notes.
Zoom link. https://uio.zoom.us/j/62114389762?pwd=S08wbUNXQUpFdjl2ODd4WFlEQ09PQT09
Meeting ID: 621 1438 9762 Passcode: 342661
Monday, March 7th: The goal is to finish section 4.8 on the Arzela-Ascoli theorem. This is one of the longest and toughest arguments in the book and it will be a great advantage to be well prepared. Take a look at Ulrik's notes and video:
Section 4.8: Arzela-Ascoli's Theorem: Video. Notes.
Zoomlenke: https://uio.zoom.us/j/68524269105?pwd=M1RVbE9YMmYwRTR1RHFxaysydGl3UT09
Meeting ID: 685 2426 9105 Passcode: 362022
Wednesday, March 2nd: The main theme is differential equation as presented in section 4.7, but we may also have time to start section 4.8. Ulrik is quite taken with differential equations, and his introductions cover more than is in the book. Still it's a good idea to watch at least the first one.
Section 4.7: Ordinary differential equations I: Video. Notes.
Section 4.7: Ordinary differential equations II (uniqueness): Video. Notes.
Section 4.7: Ordinary differential equations III (existence): Video. Notes.
Zoom link: https://uio.zoom.us/j/66518319878?pwd=cEVBTDdXTkU4UHdwdGxCc0tKUGtkQT09
Meeting ID: 665 1831 9878 Passcode: 728255
Monday, February 28th: The plan is to cover sections 4.5 and 4.6. So far what we have been doing in Chapter 4 has been quite separate from what we did in Chapter 3, but now we are going combine the two by studying metric spaces where the metric is defined in terms of uniform convergence. This will make it possible to apply the results from Chapter 3 to problems concerning functions, e.g. to differential equations. Take a look at Ulrik's introductory videos before the lecture:
Section 4.5: Spaces of bounded function: Videos. Notes.
Section 4.6: Spaces of continuous functions I: Videos. Notes.
Zoom link: https://uio.zoom.us/j/69832651086?pwd=TGpFZDZjTDJXNnV1Uk5HNmdRRnBMZz09
Meeting ID: 698 3265 1086 Passcode: 093292
Wednesday, February 23rd: As we didn't get started on Section 4.4 last time, we shall be covering all of it in this lecture. The section has two parts. The first is on convergence, integration, and differentiation of power series, and the second is on Abel's Theorem which tells us something about what happens at the endpoints of the interval of convergence. The proof of Abel's Theorem shows us an aspect of real analysis that hasn't been emphasized much in the course so far: the art of making estimates. We shall se more of this when we get to the chapter on Fourier analysis. NB: We shall be making important use of limsup in this lecture, and it may be a good idea to review this notion from Section 2.2.
Ulrik hasn't made a video on Abel's Theorem, so we have to stick to the one on power series:
Section 4.4: Power series: Videos. Notes.
Zoom link: https://uio.zoom.us/j/62992685780?pwd=VXBPY3Z6aGZyT0JQUWRreEtFU2xKZz09
Meeting ID: 629 9268 5780 Passcode: 863502
Monday, February 21st: The plan is to finish Section 4.3 and hopefully get started on 4.4. This is still rather concrete material compared to Chapter 3, and deals with convergences of sequences of functions and power series. You'll find Ulrik's introductory videos here:
Section 4.3: Integrating sequences of functions: Videos. Notes.
Section 4.3: Differentiasting sequences of functions: Videos. Notes.
Section 4.4: Power series: Videos. Notes.
Zoom link: https://uio.zoom.us/j/61203951695?pwd=V2pPMXNmZzlXT1pKUHZ3ajBRR3F4Zz09
Meeting ID: 612 0395 1695 Passcode: 479656
Wednesday, February 16th: As there have been a lot of questions about the mandatory assignment, I shall spend some time at the beginning of the lecture discussing (but not solving!) the problems. We then continue with Section 4.2 and will perhaps touch on Section 4.3. Didrik's videos on Section 4.2 and the beginning of Section 4.3 are here:
Section 4.2: Modes of convergence: Video. Notes.
Section 4.3: Integrating sequences of functions: Videos. Notes.
Zoom link: https://uio.zoom.us/j/62343493644?pwd=MExIdlNidkhtd01iZllwZm9UMS9udz09
Meeting ID: 623 4349 3644 Passcode: 597209
Monday, February 14th: This is a prerecorded lecture as I have a hospital appointment that cannot be moved (the videos are now available on the "Reports from the lectures"-page). The main topics are the rather challenging section 3.6 and the much shorter and easier section 4.1. We shall also take a brief look at the main ideas of section 3.7, which may be worth knowing about although they are not part of the official syllabus. You will find Ulrik's videos on Section 3.6 and 4.1 here:
Section 3.6: Alternative description of compactness: Video. Notes.
Section 4.1: Modes of continuity: Video. Notes.
Wednesday, February 9th: We shall continue with section 3.5 on compactness. The stuff on totally bounded sets towards the end of section is quite challenging, and I think you will find it useful to have taken a good look at it beforehand. Here are Ulrik's videos:
Section 3.5: Compactness I: Video. Notes.
Section 3.5: Compactness II: Video. Notes.
Section 3.5: Compactness III: Video. Notes.
Zoom link: https://uio.zoom.us/j/62872841420?pwd=aUVuUCtLMWxheStOalVmMTg4SFRYQT09
Meeting ID: 628 7284 1420 Passcode: 958180
Monday, February 7th. As Auditorium 5 is still closed, the lecture will be in Room 108, NHA. We shall cover Section 3.4 on completeness and probably start 3.5 on compactness. These are extremely central topics for the rest of the course and take some time appreciate. Take a look at Ulrik's introductory lectures:
Section 3.4: Completeness: Video. Notes.
Section 3.4: Banach's fixed point theorem: Video. Notes.
Section 3.5: Compactness I: Video. Notes.
Zoom link: https://uio.zoom.us/j/61453388718?pwd=ZkJRYlg2WjBKSzYycGVwVTk0Y3ZqUT09
Meeting ID: 614 5338 8718 Passcode: 360280
Wednesday, February 2nd. As the Vilhelm Bjerknes building is still closed, this lecture will be replaced by three prerecorded, digital lectures (see the links below). I'll be online at the zoom address at 12.15 for a chat and to answer questions (you may send me the questions in advance by email if you want to, and they may be about any part of the curriculum, not just section 3.3). The lectures will cover section 3.3 on open and closed sets. Before the lecture you are encouraged to take a look at two of Ulrik's introductory lectures:
Section 3.3: Open and closed sets: Video. Notes.
Section 3.3: Continuity in terms of open and closed sets: Video. Notes.
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.
Zoom link: https://uio.zoom.us/j/68680580378?pwd=ZzFSTW95b2h2a2NIc1Y5T2t2MEZDZz09
Meeting ID: 686 8058 0378 Passcode: 901060
Monday, January 31st. IMPORTANT: As the Vilhelm Bjerknes building is closed on Monday and Tuesday, this lecture will consist of two prerecorded, digital lectures (see links below). I'll be online at the zoom address at 12.15 for a chat and to answer questions: We have now come to the heart of the matter: I'll start with Section 3.1 where metric spaces are introduced. This is the setting for the rest of the course, and it is important that you get a feeling for what a metric space is and how we work with them. In Section 3.2 we shall define convergence and continuity in metric spaces in ways that are very similar to what we saw in Chapter 2. As this is new and quite abstract material, you are strongly encouraged to take a look at the introductory videos before the lecture:
Section 3.1: Metric spaces. Video. Notes.
Section 3.2: Convergence. Video. Notes.
Section 3.2: Continuity. Video. Notes.
Prerecorded videos (replacing the ordinary lectures):
Video 1: Metric spaces. Notes. Video
Video 2: Convergence and continuity: Notes. Video
Zoom link: https://uio.zoom.us/j/66285125801?pwd=aFNHQ1JpZHQzREdZcVl3MGplRS9iUT09
Meeting ID: 662 8512 5801 Passcode: 121545
Wednesday, January 26th: The plan is to continue with the rest of chapter 2, starting with the notion of Cauchy sequences on page 34. I may not have time to do all the four theorems in Section 2.3, but I'll make to sure to cover the Bolxano-Weierstrass Theorem as this is the most important one for the first part of the course.
Section 2.2: Completeness. Video. Notes.
Section 2.4: Four important theorems. Video. Notes.
Zoomlenke: https://uio.zoom.us/j/68785827313?pwd=enB0bE5XbW1XV0RGeTZXZmRBNzAzdz09
Meeting ID: 687 8582 7313 Passcode: 620329
Monday, January 24th: The plan for the second week of the course is to cover Chapter 2 of the book. If you have taken MAT1100 and MAT1110 at UiO (and paid attention to the more theoretical parts!), there is little new here, but I'll still cover the parts of calculus that are most relevant for MAT2400. I haven't found any MAT2400 videos that deal with the material in Section 2.1, but there are some videos for MAT1100 that you may want to take a look at (unfortunately, they are in Norwegian):
Section 2.1: Convergence of sequences: Video (in Norwegian)
Section 2.1: Continuity: Video (in Norwegian)
Section 2.2: Inf and sup: Video. Notes.
Section 2.2: liminf and limsup: Video. Notes.
Zoom link: https://uio.zoom.us/j/65423619470?pwd=Y29ia3g3bmlBZnM4VHAwaGcxR1o3Zz09
Meeting ID: 654 2361 9470 Passcode: 119154
Wednesday, January 19th: The plan is to finish Section 1.3 and continue with sections 1.4 and 1.6 (we don't need 1.5 for the present course). This is still background information that we will need throughout the course. I recommend that you take a look at Ulrik's introductory videos before the lecture:
Section 1.4: Functions: Video. Notes.
Section 1.6: Countability: Video. Notes.
Zoom link: https://uio.zoom.us/j/66530356533?pwd=TXVwbTdlK0hNczllSm0vclppa0tBQT09
Meeting ID: 665 3035 6533 Passcode: 728757
Monday, January 17th: The plan is to cover sections 1.1-1.3 in the book. This is background material on proofs, sets, and Boolean operations that you may be more or less familiar with according to your background. In addition to reading the book, you may prepare for the lecture by watching the following videos:
Logic and proofs: Video. Notes.
Operations with sets: Video. Notes.
Zoom link for the lecture:
https://uio.zoom.us/j/65522329668?pwd=MExqQzgxNzNMYlhlcElzR21JRjJlZz09
Meeting ID: 655 2232 9668 Passcode: 028475