One field where functional analysis is brought close to applications is inverse problems.
This is a branch of mathematics concerning indirect measurements.
For a concrete example, consider X-ray tomography.
The physical problem is to find the (position-dependent) attenuation coefficient from measured intensity drop along every line through the object.
(A machine shoots an X-ray through the object and compares the initial and final intensity. This is repeated for a great number of trajectories.)
Using the Beer–Lambert law of attenuation brings us to a mathematical formulation: How to reconstruct a function $\mathbb R^n\to\mathbb R$ from its integrals over all lines?
Is the function even uniquely determined by this data?
This problem becomes more tractable within a functional analytic framework.
We need a space $E$ of functions $\mathbb R^n\to\mathbb R$ and a space $F$ of functions $\Gamma\to\mathbb R$, where $\Gamma$ is the set of all lines in the Euclidean space.
The X-ray transform $I:E\to F$ is defined so that $If(\gamma)$ is the integral of $f$ over $\gamma$.
The mathematical X-ray tomography question can be reformulated: Is the X-ray transform injective?
This leads to a number of questions:
Is $I:E\to F$ continuous?
If it is injective, it has a left inverse.
Is it continuous $F\to E$?
How does this depend on the function spaces $E$ and $F$?
What happens if one only has some kind of partial data, perhaps with errors?
Is there perhaps a good pseudoinverse that is optimal in some way?
How can one define $I$ if $F$ and $E$ are distribution spaces or some other "non-classical objects"?
For example, $E=C_c(\mathbb R^n)$ and $F=C(\Gamma)$ makes $I$ continuous, but the inverse is discontinuous.
The same happens when $E$ and $F$ are $L^2$ spaces.
However, with suitable function spaces (Sobolev spaces) $I$ can indeed be an isomorphism (continuous and continuously invertible).
In many cases it is convenient to study not $I$ directly by the normal operator $I^*I$.
Here $I^*$ is the $L^2$ adjoint, which turns out to be useful even when $I$ is not continuous or even well defined on $L^2$.
A weaker version of the adjoint is needed.
These endeavors can be taken in a number of different directions.
One can study the fine details of stability using microlocal analysis, or extend the theory to geodesics on a manifold (which has applications in seismic imaging, for example), or study converge of numerical approximation schemes, or find a way to get a decent X-ray image with minimal radiation dose, or…
I wrote introductory lecture notes on the topic with very little prerequisites: Analysis and X-ray tomography.
There are a number of books on different aspects of X-ray tomography.
The classics of the mathematical theory are by Helgason and Natterer.
There are still open problems in this field, and even more so in the whole field of inverse problems.