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.

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":



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. 


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). 

