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

Question

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!