This page is for older messages that are no longer important to keep on the main page. On this page, messages are in chronological order.
2004-12-08: Not much to report yet. I will very likely lecture in English; I have already had one foreign student ask about the class, and this class seems to be very popular by the international master students. The first lecture will be on Tuesday, 11 January at 08:15 (ouch!).
2004-12-19: The first lecture will be on Tuesday, 11 January. The topic for the first lecture will be a bit of set theoretic background, in particular transfinite induction and Zorn's lemma.
2005-01-10: Plan for tomorrow's lecture. As mentioned, the topic will be transfinite induction, from chapter 1 of my notes. We will use some of the results later, but I will limit discussion to this one day. What I cannot say in two hours will remain unsaid. I will follow the notes, but not in the same order:
First, definitions (partial order, total order) and examples.
Then, Zorn's lemma: I will state it and use it to prove the Axiom of Choice (AC). (Which seems backward, because Zorn's lemma is hard to prove, while AC seems easy to understand – at least until you learn about its weirder consequences, such as the Banach–Tarski paradox.)
Next, we embark on the proof of Zorn's lemma. We define well-ordering, explore its properties (in particular the principle of induction and definition by induction). This culminates in a proof of the well-ordering theorem which says that all sets can be well-ordered. From this we then prove the Hausdorff maximality principle, from which Zorn's lemma is an easy consequence.
If there is any time left over, I will want to say something about ordinal numbers.
Further plans: In the first weeks of the semester, we cover chapter 2 of my own notes and Kreyszig chapter 4, up to and including 4.7. I will probably switch a bit back and forth between the two, but start with the sequence spaces from my notes. (I have not yet planned much further ahead. More planning will happen next week.)
2005-01-14: Moving the Friday lecture! As agreed during today's lecture, I have investigated the possibility of moving the Friday lectures. After much searching, I found a vacant room. I have placed a reservation on this room, and got it confirmed! So we start immediately, with lectures Monday, 10:15–12:00 in room R59. This room is also known as B2-105 in «realfagbygget» (whatever that is called in English). See map. If you enter the building from the north and walk past the big cafeteria, then turn left and taking the first stair or elevator to the 2nd floor, you will find this room nearby – in the B block.
2005-01-24: This week I am lecturing from Kreyszig ch 4: Starting with the Hahn–Banach theorem (4.2 & 4.3), I will go into the example in 4.4 but skipping a great amount of detail. Sections 4.5 and 4.6 are also on the menu, to the extent that I get to them.
Last week i lectured from ch 2 of my notes up to and including Prop. 22 (the completeness of Lp), and I started on the Hahn–Banach theorem.
I had forgotten to post exercises last week as I had promised. Sorry about that, but now you will find them above.
2005-01-31: This week I keep on lecturing from Kreyszig ch 4. After today, we're pretty much done with section 4.6, and I have just started on section 4.7, with an alternative proof of the Uniform Boundedness theorem. (This proof was lifted from Emmanuele DiBenedetto: Real Analysis. DiBenedetto refers to a paper by Osgood from 1897, thirty years before the paper by Banach and Steinhaus after which this theorem is usually named.)
I have written up this proof as a small note.
Best for on-screen viewing: A5 size (pdf, ps).
Best for printing: A4 size (pdf, ps).
As an extra exercise, I suggested the following problem: If X is a normed space and Y a subspace, prove that the closure of Y is (Y⊥)⊥. (The proof ought to be a straightforward application of the Hahn–Banach theorem or one of its consequences.)
(I am curious to know if your web browser displays this correctly.)
Here, if A⊆X and B⊆X*, we define A⊥ to be the set of f in X* so that f(a)=0 for all a in A, and B⊥ to be the set of x in X so that b(x)=0 for all b in B.
2005-02-08: This week I have roughly speaking finished Kreyszig ch 4. I have skipped section 4.10, but I did section 4.11.
I discovered that I had forgotten an important part in section 4.11: For the convergence proof, it is important to know that the polynomials are dense in C[0,1]. This is the well-known Weierstrass theorem. To make up for my lapse, I wrote up a different proof of the Weierstrass theorem.
Best for on-screen viewing: A5 size (pdf, ps).
Best for printing: A4 size (pdf, ps).
Next week I will first take care of any loose ends from Kreyszig ch 4, and then I will return to my own notes, perhaps picking up some loose ends from ch 2 and proceeding with ch 3, on topological spaces.
2005-02-20: Last week I did not quite finish ch 3 from my notes on topological spaces.
Plan for this week: On Monday, we finish the chapter with the product topology and Tychonov's theorem. Then we start on ch 4, topological vector spaces. After some initial definitions, a bit on locally convex spaces, linear functionals, and weak and weak* topologies, we get to the Banach–Alaoglu theorem, basically just a consequence of Tychonov's theorem.
But the real fun starts when we get to the geometric Hahn–Banach theorem, also called the separation theorem. We actually present two versions of the theorem. This will in turn be used in the study of local convexity with applications to the duals of Lp spaces, but that will probably have to wait until next week.
2005-02-24: This week I finished ch 4 on topological spaces. I did not cover the final section, on Urysohn's lemma. (Nor will I cover it, unless we need it for an example.) I have promised that you will not be asked direct topology questions on the final exam, but you need to know what a topological space is, what compactness is, and what the Tychonov theorem says.
I then covered ch 5 up to and including Banach–Alaoglu, but I skipped Prop 47 (on weak convergence in uniformly convex spaces).
2005-02-24: The midterm test will happen on Friday, 11 March 10:15–12:00 in KJL-24.
Topics covered by the text are Chapter 2 from my notes (up to and including Prop 25) and Chapter 4 from Kreyszig (except 4.10).
The actual test will last only 60 minutes.
Do not forget that there are no lectures in weeks 9 and 10 (starting next Monday, ending on 11 March).
2005-03-14: This week I will start with the geometric Hahn–Banach theorem. After that, we move on to uniform convexity and reflexivity, and – time permitting – we will round off with the proof that Lp is uniform convex and therefore reflexive, and consequently that its dual is Lq. Whew, that is quite a program. I am afraid we won't get through it all.
I will post answers to the midterm as well as previous exercises, and some new exercises. But I don't have the time to do all that before Tuesday evening at the earliest.
2005-03-14: I found a better proof of the uniform convexity of Lp spaces – get the article under Literature above. This will replace 27–29 and simplify the proof of 34 in the notes. Note also the updated errata for the notes.
2005-03-18: Remarks before Easter sets in. First, I remind you all that, because our lectures are on Mondays and Tuesdays, the next lecture is on 4 April. So you will not die from boredom while we're waiting, I have put up a new set of exercises.
I have received a question about what will be on the final exam. While I do not have a complete answer yet, I can at least tell you about the parts I have lectured on so far: My notes: All of chapter 2 will be included, and all of chapter 4 except the final section, on the Krein–Milman theorem. I will not ask any direct questions from chapters 1 and 3. But you will need to understand some of chapter 3 for chapter 4. I have not mentioned the final section, on normal spaces, at all. From Kreyszig we have basically covered chapter 4, except 4.10.
2005-04-13: Since Easter, I have been lecturing from my notes. The first lecture, and a bit of the second, was spent finishing off the parts of chapter 2, by presenting my new proof of uniform convexity for Lp spaces, and then using this to get the basic result on the dual spaces of Lp.
Since then, we started on chapter 5 (still from my notes), spectral theory, which we have now almost finished. What remains is the final section, on functional calculus.
Next week we start on functional calculus, and then the plan is to start on Kreyszig's chapter 9.
2005-04-17: The Weierstrass theorem. We are going to need this theorem, which states that the polynomials are dense in the set of all continuous functions on an interval. I wrote up a brief note containing a proof of the theorem.
2005-04-29: The final exam will be a written one. This is now clear. I apologize for having created the expectation that we will have an oral exam, and moreover for waiting too long to get the question settled. That said, I shall do my utmost in making sure this will not adversely affect the outcome of the exam.
2005-04-29: I will be away after 20 May. I will return late in the evening the day before the exam. So, if you have questions you wish answered, you will need to talk to me before I leave. I will schedule some time and announce it here, but you can also just show up at my office and hope I'll be there, or you can email me and get an appointment. I should be able to receive and send email while I'm gone. (But 30 May I will spend traveling, so no email then.)