Is Topology an important class to take before Functional Analysis?

Question

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.

Answer 1

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.

Answer 2

If to put simply topology is needed in functional analysis in order to explain what is convergence for any set. Take a look at the picture:

from Mathematical calculations course you know what is convergence for formulas in number series. But how to prove that something is convergent if in front of you is not number series, but shapes like in the picture?

Answer 3

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.