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.
 A: 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.
A: 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.
A: 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.
A: In prime number theory, density of primes in some interval has prime importance since primes generate the composites most efficiently. Therefore, it is a key metric.  And density functional analysis involving primes is a fruitful and fertile research topic in pure mathematics.  
Although the following links considers density functional analysis from a physical perspective, I believe they will help you gain valuable insights on some  key ideas of functional analysis...  Good luck!
Reference links: 'A Primer in Density Functional Theory', http://link.springer.com/book/10.1007%2F3-540-37072-2;
'Density functional theory for beginners', http://newton.ex.ac.uk/research/qsystems/people/coomer/dft_intro.html;
'Functional Analysis for Quantum Mechanics', https://www2.mathematik.hu-berlin.de/~berg/Functional_Analysis_Seminar_2010_03_29.pdf;
https://physics.stackexchange.com/questions/42017/is-the-density-operator-a-mathematical-convenience-or-a-fundamental-aspect-of.
A: 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
A: Much of the theory of probability can be considered as a branch of functional analysis (although some probabilists might object to this statement).  For example, the Strong Law of Large Numbers can be considered as a special case of Birkhoff's Ergodic Theorem.
A: You could also go the more numerical route with functional analysis. In the study of Finite Element Methods (abbr. FEM) for example, some results of functional analysis that have applications in PDE's have natural applications as well, the Riesz representation theorem and the Lax-Milgram lemma being two popular ones. The applications in question tend to be proving existence, convergence and uniqueness results of the numerical solutions you can achieve.
Moreover, there are applications throughout numerical analysis in general, but I am not too familiar with those at the moment (I believe Kreyzig's book mentions something, but I don't have the book close by at the moment).
A: Other important applications are signal analysis and data compression. The theory of wavelets is of particular interest for the latter domain (and a brilliant mathematical achievement). See for example this course http://www.ima.umn.edu/~miller/waveletsnotes.pdf 
A: The book "History of Functional Analysis" by J. Dieudonne connects functional analysis to its roots in differential equations, harmonic analysis and distribution theory. It connects many of the standard theorems in functional analysis to the applications that inspired them.
"An Introduction to Linear Analysis" by Kreider, Kuller, Ostberg and Perkins gives a functional analysis viewpoint of methods of solving differential equations. 
A: To complete one of the above answers, the entire realm of quantum field theory, with the calculations of Feynman propagators, gauge theories, scattering amplitudes and all the rest is entirely based on functional analysis. 
A: Check out these notes for some application to optimization problem
http://personal.lse.ac.uk/sasane/ma412.pdf
A: To expand on the answer by Martín-Blas Pérez Pinilla, the formal theory of distributions, which was largely developed to give solutions to partial differential equations (which have no solutions in terms of classical functions) requires some heavy functional-analytic machinery. An excellent book to look at is Topological Vector Spaces, Distributions, and Kernels, by François Trèves, which gives a very thorough treatment of the functional analysis that goes into the development of distribution theory. It also reminds everyone that Functional Analysis is about more than just Hilbert and Banach spaces. An additional resource that would be helpful if you're interested specifically in distributions is Distributions: Theory and Applications, by Duistermaat and Kolk.
