Functional analysis for signal processing engineers
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.