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.
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.