# What are the applications of functional analysis?

I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I am not specifically interested in applications of functional analysis on pure branches of mathematics but rather interested in applied mathematics.

Can anyone give a brief on what are the mathematical applications of functional analysis? Also, please suggest some good books for it.

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.