Projection onto the subspace of spherical harmonics with the same degree

Recently, I've been interested in expanding an angular function over the spherical harmonics, and particularly in retrieving the amplitude of the part corresponding to a given degree \(\ell\). More precisely, let \(F(\Omega) = F(\theta,\phi) =\sum_{\ell} \sum_{m} Y_{\ell m} (\Omega)\). The projection of \(F\) onto the subspace spanned by the harmonics with a given degree \(\ell\) (I believe this space is generally denoted by \(\mathcal{H}_{\ell}\)) is:
\begin{equation}
\label{eq:proj1}
\operatorname{Proj}_{\ell} \left [F \right ] (\Omega) = \sum_{m= - \ell}^{\ell} c_{\ell m} Y_{\ell m} (\Omega)
\end{equation}