Let $\mathcal H_{S^2}:=L^2(S^2)$. The functions $Y_{\ell,m}$, with $\ell\in\mathbb Z_{\ge 0}$ and $-\ell\le m\le \ell$, form an [orthonormal basis](/page/Orthonormal%20Basis) of $\mathcal H_{S^2}$. Consequently, \begin{align*} \mathcal H_{S^2}=\widehat{\bigoplus}_{\ell=0}^{\infty} \operatorname{span}\{Y_{\ell,m}:-\ell\le m\le \ell\}. \end{align*}