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I am starting a graduate degree in math pretty soon and I am planning to take a course in Functional Analysis and Spectral Theory. Topology is being offered next semester as well but I don't think it is required. Am I doing myself a major disservice if I decide not to take it anyway?

Update: As for my background: I have had two extremely challenging upper division Linear Algebra courses, and my Analysis professors covered a lot of material on toplogical and metric spaces, but for example the only stuff I know about homotopy is stuff I have read on my own.

Here is the stated content of Functional Analysis:

Topologische und metrische Räume, Konvergenz, Kompaktheit, Separabilität, Vollständigkeit, stetige Funktionen, Lemma von Arzela-Ascoli, Satz von Baire und das Prinzip der gleichmäßigen Beschränktheit, normierte Räume, Hilberträume, Satz von Hahn und Banach, Fortsetzungs- und Trennungssätze, duale Räume, Reflexivität, Prinzip der offenen Abbildung und Satz vom abgeschlossenen Graphen, schwache Topologien, Eigenschaften der Lebesgue-Räume, verschiedene Arten der Konvergenz von Funktionenfolgen, Dualräume von Funktionenräumen, Spektrum linearer Operatoren, Spektrum und Resolvente, kompakte Operatoren.

And the professor posted the following additional text:

Funktionalanalysis ist die Theorie unendlichdimensionaler Vektorräume. Schlagworte aus dem Inhalt: Banachräume, Bairscher Kategoriensatz, Lineare Funktionale und reflexive Räume, Hilberträume, Distributionen, $L^p$- und Sobolevräume, kompakte Opratoren und Fredholmtheorie.

Here is the same for Topology:

Grundkonzepte der allgemeinen Topologie (metrische Räume, Konvergenz, topologische Räume, stetige Abbildungen, Unterräume, Summe und Produkt, Quotientenräume, Trennungsaxiome, Zusammenhang, Kompaktheit), Homöomorphie und Homotopie, simpliziale Komplexe und simpliziale Approximation, Euler-Charakteristik, Gruppen und Homomorphismen, Präsentation einer Gruppe durch Erzeuger und Relationen, Fundamentalgruppe, Überlagerungen, geometrische Anwendungen, Klassifikation der geschlossenen Flächen.

There isn't anything posted yet for Spectral Theory. If any of the German needs explaining, let me know, but most of the words are pretty similar to their English equivalents.

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Yes. Topology is assumed in nearly all later math courses. Functional analysis uses a great deal of topological notions. –  Fredrik Meyer Sep 8 '11 at 20:50
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Separate grad courses in Point Set Topology are far less universal than they used to be. The parts essential for Analysis are often more or less covered elsewhere. –  André Nicolas Sep 8 '11 at 21:03
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Does the "topology" course mean algebraic topology? Even though that is an important course, it is not required before Functional Analysis. –  GEdgar Sep 8 '11 at 21:36
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This is very dependent upon your background and the instructor for the functional analysis course. I would ask him first (I would say "him/her" but it's just so clumsy, and no one uses Spivak pronouns!). –  Dylan Moreland Sep 8 '11 at 21:40
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This looks like I suspected: For FA you don't need anything of what's mentioned after the parenthesis in the description of the topology course, except homeomorphisms. The functional analysis course seems to be designed so as not to use a lot of topology, most likely nothing that you don't know already. By the way, I disagree with the opinion that you need a lot of topology. It's only some basic notions and usually they are explained anyway. Since you read German, I heartily recommend Werner's book Funktionalanalysis‌​. –  t.b. Sep 10 '11 at 9:54

2 Answers 2

up vote 11 down vote accepted

From the brief description of your background (and knowing a bit about how the German system works) I think it is safe to say that you should be well prepared to follow that functional analysis course. The description of contents indicates that the topological concepts are recapitulated. You'll likely know most of these basic things already.

Assuming you know the Arzelà-Ascoli theorem already, probably the only serious piece of topology you'll see in your functional analysis course is Baire's category theorem (that often isn't taught in basic topology courses) which I like to think of as a spiced up version of the nested intervals theorem (Intervallschachtelungsprinzip). It takes some time of getting used to and its applications are so magic that they often seem "too good to be true", but you'll see that in your course. Here's one of my favorites: If a continuous function $f: \mathbb(0,\infty) \to \mathbb{R}$ has the property that for each $t \gt 0$ we have $\lim\limits_{n \to \infty} f(nt) = 0$ then $\lim\limits_{x\to \infty} f(x) = 0$. But there are many more, among which three of the four basic theorems in functional analysis: the open mapping theorem, the closed graph theorem and the uniform boundedness principle. The fourth (more or less unrelated) basic result I have in mind is the Hahn-Banach theorem. But I digress.

Anyway, I liked my own functional analysis course since it encompassed both analysis and linear algebra in a clean axiomatic setup. If you follow that course closely, you'll likely be revisiting and rethinking your courses on analysis, linear algebra and measure theory (if you had one). I found this immensely helpful for developing and deepening my understanding. As for prerequisites, I think the outline of the course speaks for itself:

Topological and metric spaces along with their most important basic properties will be revisited so acquaintance with them will help but is probably not strictly needed. All in all, it looks like a solid course is being offered: Nothing too fancy and all the standard topics will be treated. Whatever is added to the course outline regarding spectral theory won't change the above assessment, I believe.

As for your specific question, I don't think the topology course will help you very much for the functional analysis course. As I said in a comment, only the topics you mention before homeomorphism and homotopy will be directly useful.

On the other hand, as I stated in the comments already, I think the topology course covers topics that every mathematician should know about. Therefore I strongly recommend that you visit it if possible. Whether you take the exam or not is of secondary importance, I think (I'm referring to your comment addressed at me). It is very likely that sooner or later you'll be needing material covered in that course and having some acquaintance with a topic is always helpful, so yes, I think there is the risk of doing yourself a disservice by not visiting it.

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As a follow up: All of the lectures in my question ended up being completely self-contained, and none of them required more topology than one learns in an undergraduate analysis course. That said, The more lectures I visit in other subjects, the more topology keeps seeping in. Anyway, if I could give you a second +1, I would. –  Tim Seguine Jan 17 '13 at 21:09
    
Thanks for the feedback, that's very nice to hear and I really appreciate it. Good luck with your further studies! –  t.b. Feb 3 '13 at 9:14

It depends on what kind of background in real analysis you have. If you've had a strong real variables course that contains a great deal of basic point-set topology-such as one based on Apostol's Mathematical Analysis or Pugh's Real Mathematical Analysis-then you can get by without a graduate topology course. What's more important to a functional analysis course is that you have a very strong background in linear algebra. You need to be virtually automatic on computations of matrices, basic vector spaces,basis theory, linear transformations, orthogonality and inner products and their spaces. Indeed-Peter Lax commented after writing his book on the subject that in his experience,the biggest problem most graduate students in mathematics in the US have is a fundamental weakness in linear algebra-they either have forgotten most of it from their undergraduate courses or didn't learn it very well to begin with. He wrote his book largely to collect in one book everything on the subject he thought graduate students needed to know before beginning a careful study of function spaces.

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Hm. This is a bizarre answer; "a great deal of basic point-set topology" in Pugh? You must have read a different book than I. Also, your points on linear algebra seem a bit off the mark. Yes, you need linear algebra for virtually everything in modern mathematics, but I wouldn't deem that "a very strong background". What you mention is nothing but the bare minimum every student beyond the first year should master without any difficulty. –  t.b. Sep 10 '11 at 10:03
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@Theo: I don't really agree with that one needs a very strong background in linear algebra either. Sure, one needs to know many of those concepts and they can be handy when thinking about infinite dimensional spaces, but most things that work in finite dimensional spaces fail in infinite dimensional spaces. –  Jonas Teuwen Sep 10 '11 at 12:39
    
@Theo,Jonas:Compared to books like Ross,Mattuck or Abbott, there is quite a bit of point set topology in Pugh's book-especially in chapter 2. Exactly what is covered in a basic undergraduate topology course is undergoing a dramatic rethinking the last few years at most universities, but a full discussion of this fascinating development really belongs in another thread... (continued) –  Mathemagician1234 Sep 10 '11 at 16:46
    
@Theo,Jonas cont:I FULLY agree with both of you that for modern math,a strong background in linear algebra is really essential and it's importance tends to be underscored in most places.But I think it'd be VERY difficult to effectively learn functional analysis without not only the basic concepts of finite dimensional vector spaces,but the TECHNIQUES one learns in those spaces.For example,orthogonality and inner product space theory are both critical for functional analysis.And in my experience,students who are weak in LA tend to do very poorly in these classes. –  Mathemagician1234 Sep 10 '11 at 16:49

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