The spherical harmonics of degree $k$ in $n$ dimensions are the restriction to the sphere $\mathbb S^{n-1}$ of harmonic polynomials homogeneous of degree $k$ in $n$ variables. It is a classical fact of analysis that an Hilbertian basis of $L^2(\mathbb S^{n-1})$ can be made with spherical harmonics and that the Laplace operator on $L^2(\mathbb S^{n-1})$ can be written as $$ -\Delta_{\mathbb S^{n-1}}=\sum_{k\ge 0}k(k+n-2)\mathbb P_k, \qquad \text{Id}_{L^2(\mathbb S^{n-1})}=\sum_{k\ge 0}\mathbb P_k, $$ where $\mathbb P_k$ is the orthogonal projection onto the space of spherical harmonics of degree $k$.

My Question : The proofs that I know of the second identity above are not so elementary and I would like to know if there is a simple proof of the fact that the orthogonal (in $L^2(\mathbb S^{n-1})$) of the space of all spherical harmonics is reduced to $\{0\}$.