# 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?

• 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