This question is very old, but I'll write an answer anyway for reference for future readers.
Functional analysis is in some sense the "good" infinite-dimensional analogue of linear algebra that you need to do analysis. Namely, if you study functional analysis you will mainly be confronted with various spaces of functions on some topological spaces (classically, open subsets of $\mathbb{R}^n$).
In order to be able to study functional analysis, you will need knowledge of
- Linear algebra: while this is maybe not so fundamental for the subject, it is very important to have strong bases of linear algebra in order to understand the intuition behind many objects and proofs.
- (Real) analysis: you will be studying spaces of functions with various properties. In particular, you will need to be familiar with the concepts of continuity, differentiability, smoothness, integration and maybe most importantly Cauchy sequences and convergence of sequences and series.
- Basic topology: you will be working on metric spaces, so some basic notions of topology are advised.
Moreover, if you want to go deeper into the study of the subject, you will need many more knowledge of other subjects, such as differential geometry (if you want to do PDEs on manifolds) and others.
As for the references:
- Personally, I learned functional analysis on these lecture notes by M. Struwe (they are in German, though).
- Another interesting reference is these notes by Einsiedler and Ward (I haven't read them, but I have been told they are very good; however, I've also heard that they go in some "non-standard" directions and applications).
- Finally, if you are more of a PDEs person, Evans' book is a classic you must read.
I'd definitely recommend baby Rudin for general introductory analysis, his followup textbook is also my favourite analysis book. Fourier analysis is generally very reliant on Lebesgue integration. A book which uses only the Riemann integral (if I recall correctly) is Dietmar's first.
The route I took into harmonic analysis was starting with A (Terse) Introduction to Lebesgue Integration by Franks, which has an online draft here, which introduces the Lebesgue measure/integral in a very rigorous manner before establishing the basic $L^{2}$ treatment of Fourier series on $\Bbb{T}$. After that, the best recommendation that most people interested in harmonic analysis should read is katznelson's book, which covers the standard Fourier transform on $\Bbb{R}$ material very nicely, as well as sketching the locally compact abelian group stuff. From there, there seems to be less of a general consensus. I found Rudin's Fourier Analysis on Groups excellent for the locally compact abelian case, giving nice proofs of several theorems for which I wasn't happy with the proofs given in other books. I also enjoyed Classical Harmonic Analysis and Locally Compact Groups by Reiter and Stegeman as more of a broad introduction to abelian harmonic analysis, although it does ommit quite a few key proofs. I can't offer many references beyond the abelian case, and certainly not beyond the compact case, but the second of Deitmar's books was my favourite general reference for introductory nonabelian harmonic analysis. I've not read much of it, but my favourite treatment of the compact case is that found in Folland's book, which is online here; in particular, I found that its description of the representation theory was much more natural than other treatments.
I'm not American, so I can't relate what I've said to the courses you've listed, but hopefully this will help somewhat. I'd certainly recommend starting with Franks and Katznelson.
As a general aside related to your comments, I'd recommend trying to read as much as possible without asking for help from your professors - even if you ultimately have to ask for some help understanding something, you'll get much, much more from it if you only ask for help once you've beaten your brains out trying to understand it.
Best Answer
As of July 2017, one can find some detailed solutions of some problems from chapters 4-6 in: Herman Jaramillo, "Solution to selected problems of Munkres Analysis on Manifolds Book," at:
http://s3.amazonaws.com/elasticbeanstalk-us-east-1-200981706290/wufu/573279464f6e8
I will attach the file here if Stack permits attachments. Hmm, apparently not.