# For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement:

For a compact Riemannian manifold $$M$$ and the corresponding Laplace-Beltrami operator $$\Delta$$ on $$M$$ we have, that $$L^2(M) = \widehat{\bigoplus_{\lambda}} \operatorname{Eig}\big( \Delta , \lambda \big) .$$ So the $$L^2$$-space of $$M$$ is spanned by the eigenfunctions of the laplacian.

Now the question is: do you know a good book where to look the proof up or from where to cite this result?

Thanks a lot in advance!

• I don't have a good reference at hand, but the argument goes as follows: (1) For every self-adjoint operator with compact resolvent, the underlying Hilbert spaces decomposes as direct sum of its eigenspaces. This is a direct consequence of the spectral theorem for compact operators. (2) The Laplace-Beltrami operator on a compact Riemannian manifold has compact resolvent. This follows from the compact the Rellich–Kondrachov embedding theorem. – MaoWao Mar 20 at 14:39
• Thank you very much! In the meantime I did also find a reference were they proof it for the case of compact hyperbolic surfaces, which is enough for me and might be enough for some other people: The Spectrum of Hyperbolic Surfaces by Nicolas Bergeron – Targon Mar 20 at 17:28
• And I'd say locally in some chart $\Delta_M$ looks like $f\mapsto \Delta_{\Bbb{R}^n} (f \circ \psi)$ for some uniformly smooth $\psi$ so the resolvent is compact for almost every $z$ and the same holds globally because $M$ is compact thus covered by finitely many charts / bounded domains – reuns Mar 21 at 2:55