The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of $k$-forms and $G$ is the Green's Operator for the Laplacian $\Delta=d\delta +\delta d$. (http://en.wikipedia.org/wiki/Hodge_theory)
It isn't clear to me how this relates to the following "Hodge theorem" from http://math.bu.edu/people/sr/articles/book.pdf:
Let $(M,g)$ be a compact, connected oriented Riemannian manifold. Then there exist an orthonormal basis of eigenfunctions (eigenforms) for $L^2(M,g)$ (or $L^2\Lambda^k(M,g)$) of the Laplacian. All eigenvalues are nonnegative, accumulate at infinity, and have finite multiplicities.
In particular, I don't see the connection to the orthonormal basis of eigenfunctions part: how does this follow from the classical version?
This is cross-posted from https://math.stackexchange.com/questions/617557/relation-of-hodge-theorem-to-eigenfunction-basis-of-laplacian. References would be appreciated!