[hanzoganz]

Very interesting post. I really like functional analysis because it has such a powerfull tools. Quantum physiscs is based on Hilbert Spaces, so, as you say, we can understand Heisenbergs uncertainty principle by a Fourier transform on a Hilbert space. I work with stability of nonlinear dynamical systems and yes, linearity is much more easier to work with and gives us enough hints about the nonlinear behaviour.
We could say that functional anaysis deals with functionals and operators. the first are mapping from a space into R. the second are mapping from one space to other. I think this field of math uses a lot of imagination. i invite you to read and work a bit on this.

cheers