In the common Hodge theory books, the authors usually cite other sources for the theory of elliptic operators, like in the Book about Hodge Theory of Claire Voisin, where you find on page 128 the Theorem 5.22:
Let P : E → F be an elliptic differential operator on a compact manifold. Assume that E and F are of the same rank, and are equipped with metrics. Then Ker P is a finite-dimensional subspace of the smooth sections of E and Im P is a closed subspace of finite codimension of the smooth sections of F and we have an L²-orthogonal direct sum decomposition:
smooth sections of E = Ker P ⊕ Im P*
(where P* is the formal L²-adjoint of P)
In the case of Hodge Theory, we consider the elliptic self-adjoint operator Δ, the Laplacian (or: Hodge-Laplacian, or Laplace-Beltrami operator).
A proof for this theorem is in Wells' "Differential Analysis on Complex Manifolds", Chapter IV - but it takes about 40 pages, which is quite some effort!
Now that I'm learning the theory of elliptic operators (in part, because I want to patch this gap in my understanding of Hodge Theory), I wonder if this "functional analysis" is really always necessary.
Do you know of any class of complex manifolds (most likely some restricted class of complex projective varieties) where you can get the theorem above without using the theory of elliptic operators (or at least, where you can simplify the proofs that much that you don't notice you're working with elliptic operators)? Maybe the general theorem really requires functional analysis (I think so), but the Hodge decomposition might follow from other arguments.
I would be very happy to see some arguments proving special cases of Hodge decomposition on, say, Riemann surfaces. I would be even happier to hear why this is implausible (this would motivate me to learn more about these fascinating elliptic differential operators).
If this ends up being argumentative and subjective, feel free to use the community wiki hammer.