Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me to develop a good mental picture of what is going on. Emmanuel Farjoun, in the first lecture of an Algebraic Topology course about a decade ago, described "becoming comfortable with the idea of infinite dimensional manifolds" as being "one of the main conceptual advances in topology in the latter half of the 20th century". But, as I realized in a discussion yesterday, I don't understand whether infinite dimensional spaces are needed, or whether they are merely an intuitive crutch.
Are there situations in which a significant finite dimensional result strictly requires an infinite dimensional construction in order to prove it or in order to properly understand it?
If the answer is no, then at least are there important finite-dimensional theorems for which infinite-dimensional proofs are "clearly" the easiest and the most natural? (for some conceptual reason which you can explain; not just "a finite-dimensional proof isn't known yet").
A closely related question is this, although its focus is somewhat different, and none of the answers there apply here; however Andrew Stacey's answer which argues that infinite-dimensional constructions are usually not strictly necessary, is relevant.
Edit: Your favourite finite-dimensional result proved by infinite-dimensional means answers this question only if you can explain to me why I should not expect a finite-dimensional proof to exist or to be anywhere near as "good".
Best Answer
Any compact Lie group admits an embedding into a large linear group $GL_n (\mathbb{C})$.
I think it is no question that this is a genuinely finite-dimensional and important statement. The proof is via the Peter-Weyl theorem; essentially one has to show that there are enough finite-dimensional representations. How is this done? There is an obvious faithful representation on the Hilbert space $L^2 (G)$. One constructs a compact, $G$-equivariant self-adjoint and injective operator $F$ on $L^2 (G)$. By the spectral theorem, the sum of the eigenspaces of $F$ is dense in $L^2 (G)$, and they are finite-dimensional, so here is a load of finite-dimensional representations! Of course the completeness of the Hilbert space is crucial. The curious thing is that the role of the Hilbert space theory is to cut down infinite dimensional things down to finite dimensions.
The appearance of infinite-dimensional spaces in complex geometry (see Georges answer), Hodge theory, elliptic PDE etc. has of course a similar flavour.
On the other hand, it seems that in algebraic topology, most infinite-dimensional spaces can be kicked out by finite-dimensional approximation, of course at the price of making arguments more cumbersome. The algebraic topologist's infinite dimensional spaces are of the $\mathbb{R}^{\infty}$-type, not of the $\ell^2$-type. Completeness does not play a big role in algebraic topology.