What are the applications of functional analysis?
Starting from von Neumann and his contribution to economic theory (1937, existence of an optimal equilibrium in the model of economic growth )
The von Neumann model and the early models of general equilibrium
There are lots of applications of functional analysis in Economic theory:
Functional Analysis and Economic Theory
In Financial Mathematics, in the first Fundamental theorem of asset pricing Hahn-Banach Theorem is applied to show that if there is no arbitrage on the financial market then there exists at least one equivalent martingale measure Theorem 1 on page 4, proof on page 6.
More for the financial mathematics: Optimality and Risk - Modern Trends in Mathematical Finance.
Itô stochastic calculus can be nicely introduced through Hilbert spaces, and this approach explains the name Itô isometery, which is indeed an isometry in the sense of Hilbert space operators. It might be worth to have a look at Hilbert Space Methods in Probability and Statistical Inference, Gaussian Hilbert Spaces.
I should also mention Quantum Mechanics. Starting from the postulates of quantum mechanics which use notions such as Hilbert spaces, self-adjoint operators (observables), states etc. You may find Heisenberg picture very interesting. I can recommend Reed and Simon Functional Analysis - Methods of Modern Mathematical Physics books. There are also books entitled Quantum Mechanics for Mathematicians, it might be worth finding those, rather than the ones aimed at Physicists.
There is much more and you may want to google some more books.
The whole field of partial differential equations is an application (and origin of many problems) of functional analysis.
Book: Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis.
Lecture notes: Applied Functional Analysis by H. T. Banks.
And a (big) bit of history: On the origin and early history of functional analysis by Jens Lindström.
Another real world (theoretical physics) application is the Lagrange formalism of classical and modern mechanics which relies on the Euler-Lagrange Equation - which as you properly know is a fundamental result of functional analysis.
A book on the topic: Lagrangian and Hamiltonian Mechanics
Particularly I find that one of the real exciting parts of this theory is Noether’s Theorems which relate symmetries of the action (the integral with respect to time of the Lagrangian of the system) of a system to the conservations laws of the system. This approach is at the very heart of a lot of modern physics especially fields as particle physics.
A book on this topic: The Noether theorems