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Right so I am reviewing some notes I have from a year or two ago, and I am away from my universities library, and there's 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 I am running into is that this is more or less trivial in finite spaces, but we are dealing with infinite dimensional spaces, so I can't 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?
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 I am running into is that this is more or less trivial in finite spaces, but we are dealing with infinite dimensional spaces, so I can't 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?