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It seems that the most advanced pure mathematics course most EE engineers take is Fourier analysis, and after that it's basically 'applied' courses. There's probably a good reason for this, but I'm not sure what it is. Functional analysis seems like it pops up all the time in signal processing.
Would exploring functional analysis be worthwhile for an aspiring communications engineer?
It depends on what you want to do. At the graduate level, many electrical engineers in signal processing, communications and control have taken some functional analysis course, and courses based on A.V. Balakrishan's Applied Functional Analysis/ Luenberger's Optimization by Vector Space Methods or similar are pretty common as well as courses based on Naylor & Sell's Linear Operator Theory in Engineering and Science and a lesser extent Young's An Introduction to Hilbert Space (I have been told that undergraduates in EE at Rice use this for a bit - it is really an undergrad book). Kreyszig's Introductory functional analysis with applications is also another decent choice for undergrads. At some point though, the line between "mathematician" and "engineer" does get blurred. Its a useful course thing to know in some cases if you're an engineer whose doing research heavy in probability or control theory, especially.
That being said, those books differ considerably from whats typically offered in a math department (usually start with something like Rudin's Functional Analysis or Conway's A Course In Functional Analysis). In math department courses, you are dealing with operators to study properties of Hilbert/Banach spaces. In contrast, in engineering, we typically have the properties of the vector spaces (usually something nice like $L^p$) and want to study the properties of operators (such as minimizing some functional or something).
All in all though, for most people, if they have to ask the question, I'd say they should probably be looking for something else to take, especially if they tend to the more applied side of things.
There are two questions here: Would studying functional analysis be useful and why don't more engineers study it.
First I will say there's definitely no harm in studying it and if you find it an interesting subject and are of a more mathematical bent its probably quite useful.
Now why don't most engineers study functional analysis. Well in short while its useful its not particularly necessary particularly if studied from a pure mathematics point of view. Mathematicians tend introduce lots unnecessary formalism and proofs that engineers (and anyone else) don't really need to be able to apply the principles.
If I compare how I was taught Fourier analysis (as a Physicist) to how Mathematicians where taught it I would not describe my course as a pure Maths course. It involved little to no formal proofs or theorems.
I suspect this is why all engineers study Fourier but few do more advanced Maths courses such as functional analysis. More advanced courses have less direct applications to engineering and require more mathematical formalism to teach properly but the key principles can still be picked up and applied without formal training where necessary.
It depends on whether you intend to enter academia or industry. Academics write papers and can use all the maths they can get. In industry not so much, unless you intend to enter a research lab, which is like academia.
I think the most important branches of mathematics for telecommunications are probability, statistics, linear algebra, and information theory.
After that it depends on what branch of telecommunications you want to go into. No-one knows it all; everybody has their niche.
