What are some of the spectacular applications of functional analysis to apparently unrelated problems.One that comes to my mind is Per Enflo's resolution of Hilbert's 5th problem.There are also reformulations of RH in Hilbert Space.I would be happly to hear about it's nice applications in geometry,number theory,topology,etc. specially in context to solving conjectures.

3$\begingroup$ I don't know what you are looking for. For example, Peter Weyl theorem is an application of compact operator theory and yileds the consequence that every compact Lie group is linear. The proof of Hodge's theorem on harmonic forms representing cohomology also uses functional analysis. At various points, that in a representation space of semisimple lie groups, analytic vectors exist in profusion (HarishChandra theorem) uses elliptic operators. $\endgroup$ – Venkataramana Feb 24 '13 at 13:50

2$\begingroup$ I should have added. A large part of the work of margulis is on structure and algebraic properties of lattices (arithmeticity, normal subgroup theorem etc). However the proofs use ergodic theory, which you may think of as part of functional analysis. $\endgroup$ – Venkataramana Feb 24 '13 at 13:58

1$\begingroup$ hodge's theorem is in geometry (see warnIt is basic to er's book on differential geometry) and not in harmonic analysis. it is basic to algebraic geometry, for example. $\endgroup$ – Venkataramana Feb 24 '13 at 14:31

2$\begingroup$ Should be a CW question. $\endgroup$ – Benjamin Steinberg Feb 24 '13 at 16:01

1$\begingroup$ Yes, Cleft, I could write a book on transference of knowledge, results, and techniques from modern Banach space theory to other areas, but just as striking is merely listing people who made big reputations through research in other fields based to a large extent on what they learned and did in Banach space theory, including, among many others, Bourgain, Casazza, Gowers, Ghoussoub, Grothendieck, Milman, Pisier, Rudelson, Talagrand, Vershynin, ... $\endgroup$ – Bill Johnson Feb 24 '13 at 16:50
One should probably mention Gelfand's proof of Wiener's theorem that, if a nowhere zero periodic $f$ has absolutely convergent Fourier series, then so does $1/f$.
There is also Kantorovich's famous note "On a problem of Monge":
Monge, in his memoirs of 1781, considering the problem of the most rational ways of transporting earth from an embankment to an excavation, proposed the following problem: divide two equal volumes into infinitesimal particles and associate them one to another so that the sum of the path lengths multiplied by the volumes of the particles be minimum possible.
In connection with this problem, Monge created the geometrical theory of congruences. As to the problem itself, he conjectured, but did not proved rigorously, that the paths of the mass translocation form a family of normals to a certain family of surfaces.
The same problem was studied later by Dupin, but a rigorous proof of the Monge theorem was given only a century later, in 1884, in a 200page memoirs by Appell. (...)
Meanwhile, this assertion follows immediately from the abstract theorem mentioned above. (...)
Here is one application, which may not seem spectacular to the modern mathematician, but it has many profound applications.
Suppose that $X$ is a reflexive Banach space $E: X\to (\infty, \infty]$ a convex function such that
$$ \lim_{\Vert x\Vert\to\infty} E(X)=\infty, $$
and
$$ E(x)\leq \liminf_{y\to x} E(x). \;\;\forall x\in X. $$
Then there exists $x_0\in X$ such that
$$ E(x_0)\leq E(x),\;\;\forall x\in X. $$
For example, one can use this to settle the so called Dirichlet principle which generated many debates in the 19th century.


$\begingroup$ Look at Proposition 10.3.20 in www3.nd.edu/~lnicolae/Lectures.pdf or Corolllary 3.23 in H. Brezis' book "Functional Abalysis, Sobolev Spaces and Partial Differential Equations'' $\endgroup$ – Liviu Nicolaescu Feb 25 '13 at 9:53
For every smooth function $g$ the linear partial differential equation with constant coefficients $P(D)f=g$ is solvable in convex sets. Although the statement has nothing to do with functional analysis Malgrange's proof heavily relied on Frechet space theory (and, of course, Fourier transformation).
The same holds if $g$ is a distribution (but one may object that distribution theory is part of functional analysis).