RA

Real Analysis II (Math 243, McGill, 2005) (Total: 94 pages) It was great fun teaching a semester of real analysis at McGill. I was initially surprised at how elementary the syllabus was -- one variable, no explicit mention of compactness, metric spaces, no Lebesgue integration -- but I ended up not missing any of that stuff much: it meant that almost every result was due to Cauchy, Riemann or Weierstrass. The course text was Russell Gordon's Real Analysis: A First Course, but I ended up writing extensive lecture notes. One nontraditional feature was that the homework problems are directly embedded in the notes -- not even necessarily at the end of each section. So the lecture notes really had to be read, and this worked out rather well. In fact, even the guy who gave me my only frowny-faced rating on you-know-what infamous website -- he called me arrogant and sneaky [sneaky, at least, rings false] -- followed by admitting that I made ``good concise course notes.'' Here they are with some comments.

Series:

• I: Introducing Infinite Series (3 pages) (pdf)
  
• II: Basic Observations About Series (5 pages) (pdf)
  
• III: Series With Non-Negative Terms (Comparisons, Condensation) (8 pages) (pdf)
  
• IV: Ratios and Roots (3 pages) (pdf)
  
• V: Absolute Convergence (4 pages) (pdf)
  
• VI: Nonabsolute Convergence (7 pages) (pdf)
  
• VII: Power Series and Abel's Theorem (6 pages) (pdf)
  
  Comments: It looks like calculus, but many topics were explored in greater depth than in any calculus class I have ever taken or taught. The ratio and root tests are done with lim sup's and lim inf's, and we prove that the root test is stronger. That the ratio and root tests are ultimately linked to geometric series is much emphasized, both in terms of limitations -- the tests are doomed to fail on any series whose terms "decay moderately" -- and merits: any series shown to be convergent using the ratio test (power series!) comes with a ready-made error bound on the partial sums. We look at when the Cauchy product of two convergent series converges to the product of the sums: as long as it converges at all, it converges to the product of the sums; it converges if at least one of the two factor series is absolutely convergent; it need not converge if neither is absolutely convergent. We give a complete treatment of Riemann's theory of rearrangments, using the term ``conditionally convergent'' for a series which converges but can be rearranged to diverge, and showing (in particular) that conditional convergence is the same as non-absolute convergence. We present the beginnings of the theory of power series, Abel's theorem and its relation to the Cauchy product.
  
  Regrettably left out: A more explicit treatment of Abel summation (it appears with little explanation in the middle of some proofs). Applications of Dirichlet's test to convergence of Dirichlet series; more discussion of summability versus convergence.
  

The Riemann-Darboux Integral:

• I. Axiomatic Approach to the Integral; Riemann sums (11 pages) (pdf)
  
• II: Darboux Sums; Further Integration Theorems (10 pages) (pdf)
  
• III: Further Topics in Integration: Improper Integrals; Lebesgue's criterion (7 pages) (pdf)
  
  Comments: Of course to do the Riemann integral in all its glory is quite a production. Reflecting on this before teaching the course, I found it strange that the fundamental theorem of calculus is in fact rather easy to prove: how can this be? I realized that if you write down some simple-looking axioms that the integral of a function should satisfy, then the fundamental theorem follows from these axioms, and moreover it is not hard to check that there is at most one such functional on the space of all continuous functions. (Just a little later I found an almost identical discussion in Lang's Analysis. I am happy to follow in his footsteps.) What is harder is showing that such a functional always exists: I called a particular description of such a functional an integration process. Now there are at least two well-known integration processes which construct this functional: Riemann's approach through convergence, uniformly in the mesh, of arbitrary Riemann sums, and Darboux's approach via upper and lower sums. (Actually there is yet a third approach, which will occur to you when you study topology: the tagged partitions of an interval form a directed set, so one can just require convergence of the Riemann sums in the sense of nets. This condition, a priori intermediate in strength between Riemann and Darboux so ultimately equivalent to both, allows one to define, for instance, non-Archimedean Riemann integrals.) Darboux's approach, later than Riemann's, is technically simpler, and is used without comment (i.e., still called the Riemann integral, which in a sense it is and a sense it isn't) in many modern treatments. Its only drawback: in many applications of the integral in numerical analysis and especially in the sciences, one really wants to be able to compute the integral as a limit of Riemann sums! For instance, in the applications to volumes, surface areas, forces, etc. one meets in calculus, the appeal is clearly to the Riemann, rather than the Darboux, integration process. Even Rudin's Principles of Mathematical Analysis introduces the Darboux(-Stieltjes, but never mind that) process and then later on applies it to Riemann sums: no fair!
  
  Happily, Russell Gordon is an integration theorist, so his text is unusually sympathetic to these issues: reading it carefully one learns about both processes. I decided to be quite heavy-handed about this: first I mentioned the axiomatic Riemann integral, then the Riemann process, then in order to prove some of the stickier theorems on integration of possibly discontinuous functions I switched to the Darboux integral, then I explained that the two were equivalent. (Except that the proof of the equivalence does not appear in the notes!! This will be fixed.) None of this was easy, but actually the repetition of some of the properties of the integral with a second (and easier) definition seemed helpful for the students.
  
  In stating the fundamental theorem, I was careful to emphasize that the Riemann integrability of the derivative is a nonvacuous hypothesis. I mentioned that there is a best integration theory which integrates all derivatives. The Lebesgue integral is not such a theory: the fact that f is Lebesgue-integrable implies |f| is Lebesgue integrable means that Lebesgue integration is not suitably for highly oscillatory functions. Rather an innocuous-looking generalization of the tagged partition leads to the Kurzweil-Henstock integral. Amazingly, it is both simpler and more powerful than the Lebesgue integral, albeit much less general. I have seen passionate letters (e.g. click here) written by integration theorists urging that the Kurzweil-Henstock integral replace the Riemann integral in all courses starting with calculus. Their argument seems to be that almost no calculus student completely understands what a tagged partition is, so their understanding of a tagging which is d-fine with respect to a gauge function d will be about the same. I don't quite buy it at the calculus level but it might be interesting to teach this integral in an undergraduate real analysis course. Indeed, segueing into improper integrals, I found the ad hoc nature of their definitions to be a bit annoying, to the point that I gave a more complicated definition of an improperly integrable function on the whole real line (rather than the ``just pick a point a to break it up; what's that? yes, any point will do'' approach). What if an unbounded function has a complicated set of discontinuities: do we need to know what the set is in order to define the improper integral? (Apparently the Kurzweil-Henstock integral handles improper integrals automatically and thus takes care of this!) I included some discussion of the relationship to infinite series, noting in particular that every infinite series can be viewed as the improper integral of a step function, and noted that the integral test gives good asymptotics for divergent series and error bounds for convergent series. I ended with Lebesgue's criterion for Riemann integrability in terms of the set of discontinuities having measure zero (of course one does not need a full-blown theory of Lebesgue measure to define measure zero), without proof.
  
  Regrettably left out: A better discussion of regulated functions. A proof of Lebesgue's criterion. If you know of a short, self-contained (no measure theory!) one, please let me know.
  

Sequences and Series of Functions:

• I: Pointwise and Uniform Convergence (10 pages) (pdf)
  
• II: Power Series and Taylor Series (7 pages) (pdf)
  
• III: Rigorous Treatment of Elementary Functions (7 pages) (pdf)
  
• IV: The (Stone-)Weierstrass Approximation Theorem (6 pages) (pdf)
  
  Comments: I began with a systematic discussion of pointwise convergence, how many desirable properties of the functions in a pointwise convergent sequence need not be inherited by the limit function, how this can be traced to the fact that limiting operations cannot, in general be interchanged, and that we are not about to take this lying down. This business about the interchange of limit operations points at one of the distinctive charms of real analysis: for many natural questions the answer is both yes and no (``....oohhh; short answer: yes with an if; long answer: no with a but...'' -- Reverend Lovejoy) so we get the fun of both constructing counterexamples and proving the positive results we will later use. (On the other hand, in complex analysis the answer is always ``yes'', and this certainly has its charms as well.) Then we introduce uniform convergence which fixes (almost) everything.
  
  It took the students quite a bit of time to understand the difference between pointwise and uniform convergence; until the end of the course, rather strong ones dropped by to my office hours asking for clarification just on the difference between the definitions. I tried to explain that uniform convergence was the more geometrically natural definition and that a pointwise convergent sequence of functions doesn't have to ``look more and more like'' the limit function in any reasonable sense. Maybe it would have been better to allow the backwards E's and upside-down A's onto center stage and show that their dancing past one another was visibly responsible for both the the difference in the definitions and the different order of limiting operations. The contemporary mathematician's post-formalist distaste for a proliferation of logical symbols comes at the expense of a certain amount of clarity. (How many times have you been to a seminar or colloquium talk in which the quantifiers were missing, ambiguous or incorrect? Deep down, the speaker knows what he really means, but I sometimes don't and in my waning youth I am getting less shy about persistently asking that this be made clear.) I decided to use a different notation for uniform convergence: an arrow with a ``u'' on top, and this was well received. In several places I found that Rudin's book contained stronger results than Gordon's and I faithfully copied several of his proofs. I mention Borel's theorem that any formal power series is the Taylor series of a smooth function.
  
  The course listing (last modified in 19??) sternly stated that the course would close with a rigorous treatment of the elementary functions. This seemed anti-climactic to me, but I was wrong: it's a very nice -- and not overly simple -- application of a lot of the course material to verify that these crazy functions called sine, cosine and exp indeed have the properties the calculus books tell us they do. For the most part I again copied from Rudin's book. But I wanted to end with a bang so I discussed the Stone-Weierstrass theorem in the last (optional) set of notes. I can't say I am especially pleased with the treatment: the strategy to reduce to the case of |x| is well and good, but the proof of this, taken from an exercise in Rudin, is completely opaque to me: what on earth is going on? I noticed that Noam Elkies' webpage sketches a much nicer proof based on the ``identity'' |x| = sqrt(x^2) and the binomial expansion (which I did not cover!); when I get the chance I'll rewrite it according to his hints.
  
  Regrettably left out: So many things. A more systematic treatment of analytic functions. A proof of Borel's theorem. Lebesgue's theorem on differentiability of monotone functions. A nowhere-differentiable continuous function. The fact that a pointwise limit of continuous functions has a dense set of points of continuity. (I hadn't heard of this result until I started flipping through analysis books in preparation for the course. I distinctly remembered that Rudin had an example exhibiting the characteristic function of the rationals as a pointwise limit of continuous functions. Of course I was wrong.) Some discussion of Fourier series would have been nice. In fact if I had the time I would have faithfully lectured from the entire "Sequences and series of functions'' chapter of Rudin, which is surely the book's high point. (Probably the book could have ended here and we would still like it as much; the remaining topics are equally well covered in other texts.) Somehow it seems disappointing to give the Stone-Weierstrass theorem and none of its many applications. One very nice application is a proof of Weyl's criterion for uniform distribution on the unit interval and the consequence that for irrational z, the fractional parts of nz are uniformly distributed. Wouldn't this be a beautiful way to end a course?
  
  There is also the possibility of presenting results which motivate the transition to a more explicitly topological approach, e.g., the Baire Category theorem, the Arzela-Ascoli theorem, a characterization of the subsets of R which are loci of discontinuities of some function.

Final thoughts: The course was packed full, contentwise: I wouldn't want to cover much more material in a single semester. However, one might imagine getting to a bit more by having covered at least some of the material on series in the first semester; indeed, this seems more traditional. I did have time to say once or twice, ``No, not good enough; you need to go back over that material and I'll test you again later," but one should have the luxury of doing this in any undergraduate course. Also the homework was quite an integral part of the course, and the students found it very challenging: I myself led a weekly problem session, in which the vast majority of the solutions were presented by me. It was telling that all in all it was the top half of the class who showed up most consistently for office hours, including some of the very best students. I should also mention that the first semester was taught (not by me) commonly to a group of 60 students who then split up into two sections for the second semester: mine was the non-honors section (although I had a couple of ``ringers.'') If you attended both sections you would certainly be able to tell which was which -- the honors section covered things like metric spaces, and had trickier problems -- but I think that, although my course was easier, I was offering almost as much ``content.'' I say this in part as a warning (and partly to myself, at that): I worked the students long and hard; several of them said that the homework took at least 10 hours a week. In general, I found that Canadian students responded quite well to large work loads (which is not to say that this was the norm in all of their classes; sometimes the course syllabi struck me as old-fashioned and unambitious compared to what I was used to from American universities). I was frankly rather horrified at how little most of the students knew at the beginning of the course (e.g. I assumed that the first batch of convergence tests would be familiar from calculus, but this really seemed not to be the case) and I can honestly say that almost everyone improved steadily and significantly throughout. The final exam I gave was (truly) hard, and though they looked traumatized as they walked out most of them did quite well, in some cases turning in their best performance. All in all this was an outcome I will be trying to replicate the next time I teach such a course.