Monday, January 13: Started by saying a few things about the course, then continued with section 7.1 in "Spaces", which I more or less completed. We continue with Section 7.2 next time. Notes. Podcast.
Thursday, January 16: Went through section 7.2 on completion of measures. Notes. Podcast.
Monday, January 20: Lectured on section 7.3, and covered it all, although it went a little bit faster than I like towards the end. The contents of the last theorems should be easy to relate to, though: The limit of a sequence of measurable functions is itself measurable. Notes. Podcast.
Thursday, January 23: Covered section 7.4, following the book rather closely. I created some unnecessary confusion by being a little unsystematic in the way I wrote integrals. Integrals with respect to measures are usually written \(\int f\,d\mu\), but \(\int f(x)\,d\mu(x)\) is also used. I even used combinations like \(\int f(x)\,d\mu\), but they all mean the same thing. Notes. Podcast.
Monday, January 27: Lectured on section 7.5 up to an including Fatou's lemma (that part was a bit quick). Remember that the next lecture is on Friday, not Thursday. Notes. Podcast.
Friday, January 31: I first covered section 7.6 and then went back to the comparison between the Riemann integral and the Legesgue integral at the end of section 7.5. I only got halfway and will continue with this topic next time. We will then skip section 7.7 for the time being and go straight to section 7.8. As I forgot to unmute the microphone at the beginning of the lecture, there is no sound for the first minute or so. Notes. Podcast.
Monday, February 3: I first finished the comparison of the Lebesgue and Riemann integrals, and then continued with section 7.8 where I covered everything but Egorov's Theorem. As we had skipped section 7.7, I had to explain in a little more detail what it means to converge in \(L^p\). Next time I start with Egorov's Theorem, then mention the definition of integrals of complex-valued functions from Section 7.9 (but not the rest of section), and then continue to Chapter 8. Notes. Podcast.
Thursday, February 6: I first proved Egorov's Theorem from Section 7.8, and then defined the integral of a complex-valued function and proved Proposition 7.9.1 (we'll pick up the rest of Section 7.9 later). I then continued with Chapter 8 where I covered Section 8.1 and started Section 8.2 where I managed to cover most of Proposition 8.2.3 (the only bit remaining is part (iii). Notes. Podcast.
Monday, February 10: Before the break I finished section 8.2. After the break I started section 8.3 and just managed to prove Carathéodory's Extension Theorem for algebras. I shall continue with the semi-algebra version of the theorem next time. After about one hour of the podcast, a smart board page suddenly disappeared, never to be recovered, leaving a gap in the notes (the gap should have been page 3). Notes. Podcast.
Thursday, February 13: I don't really know what happened with the lecture today, but it was incredibly slow - we only covered the material from Proposition 8.3.4 up to (and including) Lemma 8.3.7. Next time we shall finish section 8.3 and start section 8.4 on construction of the Lebesgue measure. Notes. Podcast.
Monday, February 17: I first completed section 8.3 by proving Lemma 8.3.8 and Carathéodory's Extension Theorem for Semialgebras. I then applied this theorem to prove the existence of Lebesgue measure on \(\mathbb{R}\). I talked a little bit about translation invariance (leaving the proofs to you!) and then turned to the existence of a nonmeasurable set. Here I have proved that the sets \(E\dot{+}q\) form a partition of \([0,1)\), but it remain to see how this clinches the argument. Next time I'll finish 8.4 and then turn to section 8.5. As we haven't studied \(L^p\)-spaces yet, I'll postpone the results that have to do with them till later. Notes. Podcast.
Thursday, February 20: I first finished section 8.4 and then covered all of section 8.5, except that I only proved Theorem 8.5.6 for \(p=1\) as we haven't done \(L^p\)-spaces yet. Tomorrow we'll take a look at product measures (sections 8.7 and 8.8). Notes. Podcast.
Friday, February 21: Went through Section 8.7 and then looked at highlights from Section 8.8 (Fubini's and Tonelli's Theorem) without proofs. For some reason the sound fell out after 18 minutes of the podcast. Notes. Podcast.
Thursday, February 27: Recalled the definition of a normed space and went though most of Section 2.1 from Notes on Elementary Linear Analysis, leading up to but not including the formulation and proof of Theorem 2.1.5 (completeness of Lp-spaces). Notes. Podcast.
Monday, March 2: I first finished section 2.1 by proving the completeness of \(L^p(\mu) \mbox{ for } 1\leq p<\infty\), and then went through section 2.2. The proof I gave for the completeness of \(L^{\infty}(\mu)\) is different from the one in the notes and closer to the proof of the completeness of the other \(L^p\)-spaces. Notes. Podcast (there is a humming noise on the podcast, but at least there is sound!)
Thursday, March 5: The topic of the day was mainly section 3.1 in Notes, but I had to go back to section 1.1 and 1.3 to pick up some background material on equivalent norms and linear operators. I didn't quite make it through all of section 3.1, and had to postpone the stuff on finite rank operators till next time. After we have finished section 3.1, we shall continue with 3.2. Notes. Podcaast.
Monday, March 9: Finished section 3.1 and then continued to section 3.2 where I made it to Proposition 3.2.15, which I formulated, but did not have time to prove. I hope to finish Chapter 3 (and perhaps start Chapter 4) next time. Notes. Podcast.
Thursday, March 12: Finished Chapter 3, following the notes quite closely. As the university buildings are now closed, this is probably the last lecture in the current format. I'll try to find other ways to produce podcasts the next weeks. Notes. Podcast.
Monday, March 16: I have now made videos and notes from the material in section 4.1 (probably a little more than I would have been able to cover in an ordinary lecture). The videos are neither technical nor pedagogical masterpieces, but it is what I have had time for. I have split the material in four parts:
Preliminaries: Notes. Podcast (sorry about the echo, I think I know how to avoid it in the future).
Distance to convex set: Notes. Podcast
Orthogonal projections: Notes part 1, Notes part 2. Podcast
Applications to \(L^2\): Notes. Podcast
Thursday, March 19: The videos are in four parts and cover the material of section 4.2:
Orthonormal bases in Hilbert spaces: Notes. Podcast
Bessel's inequality: Notes. Podcast.
Theorem 4.2.8: Notes. Podcast. (Retake)
Summary and examples: Notes. Podcast
As we are going faster now than we probably would with ordinary lectures, I'll slow down soon, but it seems better to keep the sections together than to split them up.
Monday, March 23: There are three videos covering the theory in section 4.3. I'll make more videos for Thursday, March 26, covering some of the examples in the section.
Linear functionals and Riesz' Representation Theorem: Notes. Podcast.
Adjoint operators: Notes. Podcast.
Images and kernels: Notes. Podcast.
Thursday, March 26: The first three videos deal with Example 4.3.4 in the Notes and show three different aspects of adjoint operators. They are a bit careless, but I hope you can live with that:
Adjoint operators in finite dimensions: Notes. Podcast.
Diagonal operators: Notes. Podcast.
Shift operators: Notes. Podcast.
The next two videos cover section 4.4.
Self-adjoint operators: Notes. Podcast.
The numerical radius: Notes. Podcast
Monday, March 30th: As promised, I am slowing down a little this week (the aim is just to complete section 5.1 before Easter). The three first videos are from section 4.5, and the last is a prelude to section 5.1. It introduces relatively compact sets and proves a result that is not in the notes, but which I think is helpful.
Linear isometries: Notes. Podcast.
Unitary operators: Notes. Podcast.
Hilbert space isomorphisms: Notes. Podcast.
Relatively compact sets. Notes. Podcast. (At the end of this video, I say that a set is bounded because it is bounded. This is correct(!), but what I intended to say is that it is bounded because it is compact.)
Thursday, April 2nd: The theme here is compact operators on general normed spaces as described in section 5.1. The first podcast covers the basic theory, the second deals with the more advanced Theorem 5.1.5, and the last two take a look at some import examples of compact operators.
Compact operators: Notes. Podcast.
Theorem 5.1.5: Notes. Podcast.
Approximation by finite rank operators: Notes. Podcast.
Operators given by integral kernels: Notes part 1, Notes part 2. Podcast. (A small "accident" cut the notes in two).
Thursday, April 16th: These podcasts cover section 5.2. The first two deal with compact operators in general, and the last two with the important class of Hilbert-Schmidt operators.
Approximation by finite rank operators in Hilbert spaces (mainly Theorem 5.2.1): Notes part 1, Notes part 2, Podcast.
Adjoints of compact operators (Corollary 5.2.4 and preparations): Notes. Podcast.
Hilbert-Schmidt operators (mainly Proposition 5.2.8 sliced up into pieces): Notes part 1, Notes part 2. Podcast.
Integral kernels and Hilbert-Schmidt operators (a simplified version of Example 5.2.11): Notes. Podcast.
Monday, April 20th and Thursday 23rd: These videos belong together but as the material is rather advanced and complicated (and the videos long!), I have split them over two ordinary lectures. The topic is the spectral theorem for compact, self-adjoint operators (section 5.3):
Eigenvalues of compact operators (up to and including Lemma 5.3.3): Notes. Podcast.
The Spectral Theorem - main ideas: Notes. Podcast.
The Spectral Theorem - putting in the details: Notes. Podcast. (Be aware that I make a few mistakes in this video that I correct later on. The most significant one is in the Fourier expansion \(Tx=\sum_{k=1}^{\infty}\langle Tx,v_k\rangle v_k\) where I forget the \(T\) on the right hand side at first.)
Singular Value Decomposition for compact operators. Notes part 1, Notes part 2. Podcast.
Monday, April 27th: These videos cover the contents of section 5.4, but in an different order than the Notes:
The equation \((T-\mu I)x=y\). Part 1: \(\mu\) is not an eigenvalue. Notes. Podcast.
The equation \((T-\mu I)x=y\) . Part 2: \(\mu\) is an eigenvalue. Notes. Podcast.
The Fredholm alternative. Notes. Podcast.
Applications of Fredholm theory. Notes part 1, Notes part 2. Podcast.
We have now completed the syllabus. Here are some review lectures that may be of help in preparing for the exam:
Measure spaces (Spaces 7.1-7.2): Notes. Podcast.
Integration (parts of Spaces 7.3-7.6): Notes. Podcast.
Convergence of functions and integrals (Spaces 7.8 and parts of 7.5 and 7.6): Notes. Podcast.
Construction of measures (Spaces 8.1-8.3): Notes. Podcast.
The Lebesgue measure (Spaces 8.4-8.5): Notes. Podcast.
Normed spaces and linear operators (Notes Ch. 1): Notes. Podcast.
\(L^p\)-spaces (Notes Ch. 2): Notes. Podcast.
Finite dimensional spaces (Notes 3.1): Notes. Podcast.
Direct sums (Notes 3.2 and some of 4.1 and 4.3): Notes. Podcast.
Orthonormal bases and Fourier Expansions (Notes 4.2): Notes. Podcast.
Adjoint operators (Notes 4.3): Notes. Podcast.
Self-adjoint operators (Notes 4.4): Notes. Podcast.
Isometries and unitary operators (Notes 4.5): Notes. Podcast.
Compact operators on normed spaces (Notes 5.1 and small bits of 3.1 and 5.2): Notes. Podcast.
Compact operators on Hilbert scales (Notes 5.2): Notes. Podcast.
The spectral theorem for self-adjoint, compact operators (Notes 5.3): Notes. Podcast
The Fredholm alternative (Notes 5.4): Notes. Podcast.