Teaching weak topologies

I can’t say that I really have anything like a “teaching philosophy” — maybe because I was lucky and never had to write a teaching statement. However, experimentally, I realize that there are two prominent features in my teaching that I end up feeling fairly strongly about, partly from an impression that they were not emphasized enough when I was myself a student: one is that the development of the course and the underlying theory should be emphasized and motivated enough that it does not feel like the teaching of some disembodied Voice Of God or Deus Ex Machina (even if the motivation is not historical, and indeed the motivations I give are often based on a fair amoung of hindsight); and two, that whenever a theorem of some importance is proved, it should be clear where and how its assumptions are used.

I’ve been thinking a bit about this second point because I’m currently teaching weak topologies in my Functional Analysis class, and in particular I just finished proving the Banach-Alaoglu Theorem that states that the closed unit ball in the dual V’ of a Banach space is compact for the weak-star topology. Here one basic point I wanted to make in class was that the analogue statement is not true (in general) for the closed unit ball of V itself, with the weak topology. I was then surprised to see, in browsing through various textbooks, that none of those I saw explained explicitly why the argument doesn’t work in this situation (it is implicit, of course, in all those that stated, e.g., that this second statement is equivalent with the reflexivity of V, but I think that for a student beginning in Functional Analysis, this may not be sufficient, because the proof of this equivalence is not really obvious). Sometimes, the “broken” step was just glossed over, but sometimes it was simply left as an exercise. In the worst case, the written proof did not even spell out which of the steps of the proof requires that the weak-star topology be used, leaving this also as an exercise (the beginning of the standard proof works just as well with the norm topology)!

[Note: The standard proof of Banach-Alaoglu proceeds by embedding the closed unit ball B in the compact product space

C=\prod_{||v||\leq 1}{\{z\,\mid\, |z|\leq 1\}

by

i(\lambda)=(\lambda(v))_v,\text{ for }\lambda\in B;

then one shows that i is injective, continuous (this works even for the norm topology on B), that the inverse of i, defined on i(B), is also continuous (this requires that the weak-star topology be used, and fails for the norm topology if V is not finite-dimensional); then, finally, one shows that i(B) is closed in C, and this works because we deal with B in V’ instead of the closed ball of V itself — the basic issue is that knowing that (for some sequence in V) we have

\lambda(v_n)\rightarrow w_{\lambda}

for all continuous linear functionals λ on V does not always imply that we can find some w such that

w_{\lambda}=\lambda(w).

A simple example where this fails (i.e., w does not exist), is provided by taking

V=c_0\text{ with sup norm}

(the space of sequences converging to 0) and

v_n=(1,\ldots, 1,0,\ldots )

where there are n ones at the beginning: from the fact that the dual of V can then be identified with the space l1 of absolutely convergent series, one checks that

\lambda(v_n)\rightarrow \sum_{n\geq 1}{u_n}

for all linear functionals λ represented by the sequence (un) in l1. But

\sum_{n\geq 1}{u_n}

is not of the form

\lambda(w)

for any sequence w in c0.]

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

3 thoughts on “Teaching weak topologies”

  1. Dear Emmanuel,
    I consider your blog to be one of the most interesting in the mathematical sphere or even überhaupt.
    The unpredictability of your themes is the feature I like most.
    I also appreciate your impeccable style and your relentlessly candid approch to mathematics, as opposed to the all too common show-off attitude displayed by too many colleagues.
    Although you are not the right person to ask, obviously, I wonder why so few mathematicians comment on your site , while completely boring or frankly idiotic blogs elicit so much electronic traffic. Depressing, n’est-il pas?
    Best wishes, Georges.

  2. i have been reading your blog for a while. I like your eclectic taste
    in math, language and history of math.
    thank you for taking time to write on
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