Right so im reviewing some notes I have from a year or two ago, and im away from my universities library, and theres a few things that are troubling me. Consider a compact Riemanian manifold M, with a metric. I am looking to prove the Hodge decomposition theorem. A heuristic proof is that we want to find a unique representative of the cohomology class by minimizing the norm w, where [w] is the class. Basically (im going to update this post the second I figure out how to get latex working) you see that the minimum of the line w + d sigma forms an affine subspace of sigma ^ k and the minimum will be orthogonal to w + d sigma ^ (k-1), that will subsume the coexact part. The problem im running into is that this is more or less trivial in finite spaces, but we are dealing with infinite dimensional spaces, so I cant see why the harmonic part of sigma needs to be finite. Anyone have a good reference to a place where I can find the proof?