This page is for messages which are no longer important to keep on the main page. Whereas the main page has messages in inverse chronological order, I keep them in plain chronological order here.
2005-11-29: I will be teaching the class once more. The first lecture, on Thursday 12 January, will be based on the first part of my notes: Primarily, transfinite induction and Zorn's lemma.
2005-12-30: No lecture on Friday, 13 January. I will be gone to Oslo for a meeting that day. The second lecture will therefore be on Thursday, 19 January. There will be a few more irregularities in the schedule later.
2006-01-23: After the first two weeks (only three lectures total), we have covered some material from the notes: Chapter 1 in the first lecture, then we started on chapter 3. We have covered sequence spaces and started on L^{p} spaces, starting with the Hölder and Minkowski inequalities.
The next topic will be the completeness of L^{p}, and the uniform convexity of these spaces (Clarkson's theorem). But then we shall have to leave the notes behind and develop some general theory for Banach spaces: Kreyszig's chapter 4. Only after that can we return to the notes and complete the proof of the reflexivity of L^{p} spaces.
2006-01-27: The web page for the notes now contains links to errata for the notes. Let me know of any misprints and errors you spot; I will add them to the errata.
Today I explained a bit about the closed unit ball of a normed vector space, and proved Clarkson's theorem (31 in the notes), that the L^{p} spaces are uniformly convex. The one bit missing is the part of the proof of Lemma 30 estimating the integral of |u−v|^{p} over the set where v=0. I will cover that on Thursday. It shouldn't take more than a few minutes.
After that, we need more general theory, so we go to Kreyszig's book and start on Chapter 4, with the Hahn–Banach theorem (4.2 & 4.3). I'll look briefly at the example in 4.4, and then it's onto 4.5 (the adjoint operator) and 4.6 (reflexive spaces).
After that, it will be time for the Uniform boundedness theorem: This is section 4.7 in the book, but I prefer the proof that is in chapter 2 of the notes. We're not too likely to find time for this in the coming week, though?
2006-02-09: Last week, after finishing the proof of Clarkson's theorem we covered the Hahn–Banach theorem (Kreyszig 4.2–4.3), with the application in 4.4.
The plan for today is sections 4.5 and 4.6, on the adjoint operator, the canonical embedding of X in X**, and a word about reflexive spaces. I hope to get to the uniform boundedness principle, also known as the Banach–Steinhaustheorem (chapter 2 in my notes).
Note that I am using somewhat different notation than Kreyszig: X* instead of X′ for the dual space, (I have no interest in the algebraic dual) and A* instead of A^{×} for the adjoint map. The latter introduces an ambiguity when dealing with Hilbert spaces, so care is needed.
2006-02-14: Last week, I did get to the uniform boundedness principle as expected, talked a bit about weak* convergence (Kreyszig 4.9), and worked through the example of Fourier series (Kreyszig 4.7-5). I also proved the Baire category theorem (Kreyszig; beginning of 4.7) and stated and proved the Open mapping theorem (Kreyszig 4.12).
My proofs came from Kreyszig, but were slightly differently organized. I wrote up a note with even more different organisation: (pdf: A5, A4).
This week the plan is first to investigate a consequence of the open mapping theorem: The closed graph theorem. I will also return to 4.11 on numerical integration.
Afterwards, the next thing to attack is chapters 3 and 4 in my notes, on topology and topological vector spaces.
2006-03-02: There are no lectures this week and next (weeks 9 & 10).
The midterm exam will be on Friday, 17 March during the regular lecture period (location to be announced).
I know, I have been slow to put up the promised exercises. Of course, the smart self-guided student tries exercises from the book on his own. Come ask me if you're stuck on one.
2006-03-16: Tomorrow's midterm exam will be in the usual auditorium (F3). Be there at 10:15. The exam is only 60 minutes.
I today's lecture, I finished the topology chapter with the Tychonov theorem. Then I started on topological vector spaces, and stated and (more or less) proved the Banach–Alaoglu theorem. Main topic for next week will be the Hahn–Banach separation theorems.
2006-03-30: Today I rounded off Chapter 5 of my notes with the section on uniform convexity and reflexivity, proving the Milman–Pettis theorem that uniformly convex spaces are reflexive. If I had ten more minutes, I could also have polished off the much desired consequence, Theorem 37 on the dual space of L^{p}, at least for the case 1<p<∞ by applying Lemma 35 to X=L^{p} and Y=L^{q}. But I didn't have ten more minutes. Before all that, we did the Hahn–Banach separation theorems which, when applied to locally convex topological vector spaces (LCTVSs), implies such vitally important facts that any closed, convex set is an intersection of closed half spaces (defined using continuous linear functionals).
There has been no time for the section on the Krein–Milman theorem. It remains to see if we have time to look at it later.
Tomorrow I have to go to a meeting in Bergen, and I shall spend the next week in Bath. So Magnus Landstad has promised to step in and do these three lectures for me. He will go through chapter 6 of my notes, on spectral theory. There is a huge amount of overlap with Kreyszig's chapter 7 plus the first bit of chapter 9. In the (unlikely?) event that he gets through all that, I suppose the next thing will be Kreyszig section 9.5, more or less, followed by the rest of Kreyszig chapter 9. But that will take much of what remains of the semester.
I was going to put up new problems today, but there wasn't enough time. Preparing for tomorrow's meeting ate up all the time I had left for the day. So I shall have to postpone it to tomorrow, or Saturday at the very latest. (There aren't very many flights to Bergen, so I have to get out of bed at 04:30 in the morning. Ouch!)
2006-04-18: Magnus Landstad has given three lectures before Easter, in which he covered most of Chapter 7 from Kreyszig, as supplemented by my notes (that would be Ch 6 of my notes except the final two subsections).
My plan, then, is to lecture on the last two subsections of Chapter 6 from ny notes, on spectral properties of self-adjoint operators on a Hilbert space and the functional calculus for these operators. This will replace much of 9.1–9.4 of Kreyszig. (Kreyszig doesn't do the full functional calculus until after he has proved the spectral theorem. But he needs a little bit of functional calculus, mainly the existence of the square root of a positive operator. I think my approach is more natural, in that we don't need to spend so much time on what is after all a very special case.)
After finishing Ch 6 from ny notes, we return to Kreyszig for a bit of work with projection operators (9.5–9.6), after which we will be ready to tackle the spectral theorem just in time for the end of the semester.
2006-05-11: One final update to my notes (well, final for this term anyway) has taken place. All I did was add some new material to the end of the chapter on the spectral theorem. If you printed out the A4 version, you can print pages 37–43 (if you print double sided) or pages 38–43 (if you print single sided) and replace the pages from the 2006-04-25 version. (The changes start on p. 71 in the A5 version.) The added material is just part of what I lectured at the end of the term. Nothing new, though the proofs may have improved a bit.
I shall post a detailed syllabus («pensumliste») during the weekend.
2006-05-13: More information about the final exam is found above. The topics will be assigned at random and announced on Monday, 22 May. At the same time, the times and place for the exam will be announced. Please, if you wish to withdraw or you find yourself unable to attend the exam, let me know as soon as possible. Also let me know if you prefer to have the exam on either Saturday 27 May or Monday 29 May. (For the majority of you, it will be on Friday, 26 May as announced.) I cannot make firm promises yet, but the earlier I know your wishes the easier the planning will be.
2006-05-16: The final examination: It turns out that we can squeeze the whole event into a single day after all. So nobody needs to take it later.