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I am in a graduate course of functional analysis (in the abstract setting of topological vector spaces) in which the professor uses his own notes. The notes are very condensed and terse and there is no reference. Classes consist of student presentations in turn and the professor does not lecture.

I struggle with the materials in the notes and try to read related books (e.g., Functional Analysis by Rudin, Topological Vector Spaces by Schaefer, Topological Vector Spaces by Narici and Beckenstein) as much as I can, so that I can somehow understand it. I find this ridiculously time-consuming and painful. The worst thing is that approaches in the notes are usually quite different from the standard references I can find and I am not even familiar with those references.

Is such situation common in the graduate schools in US (especially for math majors)? What shall I do to deal with such difficulties and (maybe) benefit from a class like this?

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    Functional analysis texts frequently are terse. "Time-consuming and painful" sounds very familiar to me in an FA context. Have you discussed this with your classmates and the professors? Oct 29, 2015 at 16:33
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    Welcome to how research is performed? Seriously, that's pretty much how it works, though not every course is taught like this. The vast majority of papers are dense and terse expositions which are perfectly fluid to the authors (one hopes) because they have spent so much time working it all out and reading the literature. You have to put in a similar amount of effort and time to get to a similar level of understanding. Making sense of it all, especially by untangling how to understand it in the at least 8 different ways of stating and doing things, is just a necessary skill for research. Oct 29, 2015 at 20:03
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    Terse doesn't necessarily mean bad. I had a lecturer who managed to condense an entire course onto 4 sides of A4. It was easily understandable; it sounds like your lecturer isn't that good at explaining the material...
    – Ben
    Oct 29, 2015 at 22:01
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    Does your professor hold office hours? Why not go and ask him/her to help you with the concepts you have trouble understanding? Make sure to use up as much of his/her time as is diplomatically possible; maybe that will help drive home the message that the materials your professor is providing you with are not adequate. And if you find the professor's teaching quality (in the broad sense that includes the lack of lecturing and inadequate teaching material) to be sufficiently disturbing, it seems reasonable to complain to your graduate program chair or department chair. Good luck!
    – Dan Romik
    Oct 29, 2015 at 22:27
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    Have you tried Introductory Functional Analysis by Kreyszig? It is one of the more readable books.
    – yoyostein
    Oct 30, 2015 at 0:17

3 Answers 3

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To directly respond to your first question: yes, "density" is quite common in middle-to-top tier grad programs in math in the U.S., once one gets beyond the "required/core" courses. There is a lot of backstory for almost all subjects, and faculty want to catch up to somewhere close to current events. It is also common that one's conception of an incisive, efficient, insightful course is not mirrored accurately in existing textbooks, so one writes notes to match. (This is better than "in the old days", pre-TeX, when most often there were no published notes of any kind, not even hand-written, so note-taking in class was critical.)

About "abstract" functional analysis in particular, yes, I think it is fair to say that the style of many of the standard texts (and of practitioners) has drifted in a certain direction, including a sort of "density". The hilarious and not-really-helpful opposite, from an older time, was the Dunford-Schwartz 3-volume extravaganza, which treated nearly every bit of mathematics as though readers would not have heard of it. Determinants, polynomials, etc. Naturally, this low-density high-volume approach created its own problems, not the least of which was locating the key examples that were not part of "general" mathematics, but specific to the functional-analytic ideas.

The subsequent decades' monographs or texts on "topological vector spaces" have always struck me as peculiarly unhelpful in making connections to other parts of mathematics, though "maybe it's just me". I cannot keep all the possible adjectives straight, and I do not see the point of many of them, nor are examples clear to me. Again, maybe it's just my own limited interest, since I care about "functional analysis" for "applications" (to automorphic forms and related matters) rather than to itself, or to "applications outside mathematics".

A point that was not clear to me for a long time is that the "more exotic" types of topological vector spaces (and families thereof, as in Sobolev-space theory) are not merely "handy", but essential in making many innocent-seeming discussions legitimate. So, not merely Frechet spaces, but LF-spaces (strict inductive limits of Frechet) already appearing as the correct topology on $C^o_c(\mathbb R)$, for example. The notion of quasi-completeness really is necessary, since LF-spaces are never "complete" (unless they're secretly Frechet), but are always quasi-complete. And quasi-completeness is enough for many essential things to still work, like the Gelfand-Pettis (or Bochner) ideas about vector-valued integrals, and Grothendieck's results about holomorphic vector-valued functions, and the strong operator topology (weaker than the Banach-space uniform norm topology on operators on a Hilbert space). Holomorphic distribution-valued functions, and so on. (My on-line functional analysis notes certainly go in such directions.)

In short, examples are critical, both to illustrate adjectives and theorems, and to see the necessity of introducing various seemingly-exotic types of topological vector spaces.

At the same time, I think building up a resistance to random counter-example demands is good, much as with point-set topology. Some counter-examples are significant, others aren't. Much time can be killed, not terribly profitably, by trying to illustrate every possible logical distinction. Don't do it.

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    And as @user37208 notes, there are many ways to organize a "functional analysis" course. The genuine meta-point there is that there is no canonical logical/serial ordering. The same absence is manifest in most more-serious mathematics, despite the affection we may have for trying to impose such an order. Apart from maybe worrying about grades and "what the instructor wants", better to realize the absence of large-scale logical order, and learn to see things as a "reality", and cope with some disorder. Oct 29, 2015 at 21:32
  • Many thanks. (Typos in the second last paragraph?) I've been struggling with the terse notes line by line and almost word by word. I happened to read your online note about the bipolar theorem which is very helpful to me recently by the way. Are you saying that in the US, a course like this is supposed to be beyond the "required/core" courses level which implies that it has to include sort of density?
    – user35
    Oct 29, 2015 at 22:00
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    Oop, yes, a delayed verb never appeared! :) I edited. But, no, it doesn't have to be dense... it's just that there are many forces that will cause it to be dense. After a year or two in grad school, density should be welcomed (up to a point) I think because it is pushing you closer to the present. Easy-going courses all too easily (but not always! appearances can be deceiving!) create a warm fuzzy feeling but dis-serve people attending. That is, there's a lot to do, and not enough time to do it... :) (I'm glad my notes were helpful!) Oct 29, 2015 at 22:09
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    ... and, yes, a serious functional analysis course (as opposed to a too-retro mere introduction of the notions of Hilbert and Banach spaces, for example), whether it emphasizes operator theory or topological vector spaces, is not ordinarily a "first-year grad course", even at nearly-top places, because one needs some grounding in "what can go wrong" that is at least the first part of a fairly serious real analysis course. Oct 29, 2015 at 22:12
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Functional analysis is one of those subjects that can be organized in so many different ways that two randomly-chosen textbooks will almost never have the topics in a similar order, logically or pedagogically. For instance, one textbook might prove Theorem B as a consequence of Theorem A, where another book (or your class notes) might prove Theorem B from scratch, and use it to prove Theorem A. For that reason, your outside references could be confusing the issue more than they're helping. Seeing a different presentation may be very helpful after this class is over, but for now, I think you should double down on understanding the professor's notes, such as they are. Study groups can help tremendously with this, as can asking your professor questions in office hours (which he hopefully has).

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Find a book to supplement the professor's notes? I used J.T. Oden and L. Demkowicz. Functional Analysis. CRC Press. 1996. when I took the course from Prof. Demkowicz in ~2000. I found it useful and still have occasion to reference it from time to time.

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    The OP explicitly mentioned that he is reading related books "as much as he can" - apparently this doesn't help him... Oct 29, 2015 at 16:34
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    Maybe he needs a better book. He didn't say which books he was looking at.
    – Bill Barth
    Oct 29, 2015 at 16:39
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    I was not able to find a book with the title and authors you give. There is a book Applied Functional Analysis by Oden and Demkowicz. That you recommend this book for a functional analysis course "in the abstract setting of topological vector spaces" seems quite curious to me: the book contains the definition of a topological vector space but less than ten pages of material on tvs's which are not Banach spaces. It does not speak of Frechet spaces, completeness in the non-metric context, tensor products, nuclear spaces....It looks like a great book for some course, but not the OP's. Oct 29, 2015 at 16:52
  • Thanks for your comment @Bill Barth. I added the references I've been looking at in the post.
    – user35
    Oct 29, 2015 at 17:55
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    That's it. I blame various distractions for my failure to transcribe the spine correctly. It was in a very applied context.
    – Bill Barth
    Oct 29, 2015 at 17:55

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