Prove the orthogonal decomposition of the space of spherical harmonics

Question

Let $\mathbb{S}^{n}$ denote the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ endowed with the standard metric. For $k=0,1, \cdots$, denote by $\mathscr{H}_{k}^{n}$ the space of spherical harmonics of degree $k$. I wonder if it is possible to prove that \begin{equation} L^{2}\left(\mathbb{S}^{n}\right)=\bigoplus_{k=0}^{\infty} \mathscr{H}_{k}^{n} \end{equation} I think that this follows from the fact that, if $Y^{(k)}$ and $Y^{(l)}$ are spherical harmonics of degrees $k$ and $l$, with $k \neq l$, then $$ \int_{\mathbb{S}^{n}} Y^{(k)}\left(x^{\prime}\right) Y^{(l)}\left(x^{\prime}\right) d x^{\prime}=0 . $$ but I don't know how to proceed. How would I prove the orthogonal decomposition?

Answer 1

I know of a proof that proceeds in three steps.

1. Continuous functions are dense in $L^2(S^n)$.

2. By Stone-Weierstrass, any continuous function on $S^n$ is uniformly approximated by a series of homogeneous polynomial.

3. By a linear algebraic dimension counting argument, the space of homogeneous polynomials is spanned by the spherical harmonics.

Thus, the span of spherical harmonics is dense in $L^2(S^n)$, which is enough to show the Hilbert space direct sum.

A google search shows that Thm 1.3 - 1.6 of https://www.cis.upenn.edu/~cis610/sharmonics.pdf carries out this strategy.