spectacular applications of functional analysis in resolutions of apparently unrelated problems 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.
 A: 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 200-page memoirs by Appell. (...)
Meanwhile, this assertion follows immediately from the abstract theorem mentioned above. (...)

A: 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.
A: 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).
