Since polynomials are dense in \(\mathbb{R}^d\), it follows by (0x6927d9ae) and the orthogonality between spherical harmonics of different degree that
\[ L^2(\mathbb{S}^{d-1}_R) = \bigoplus_{l\ge 1} \mathcal{H}_l. \]Since polynomials are dense in \(\mathbb{R}^d\), it follows by (0x6927d9ae) and the orthogonality between spherical harmonics of different degree that
\[ L^2(\mathbb{S}^{d-1}_R) = \bigoplus_{l\ge 1} \mathcal{H}_l. \]