Dense subspace of symmetric functions

Question

I do not know much about functional analysis and was hoping to find a reference of the statement that the symmetric power $S^n(C^\infty_c(M))$ of the compactly supported smooth functions on some manifold $M$ is a dense subspace of the symmetric functions $C^\infty_c(M^n)^{S_n}$ on $M^n$. The inclusion $S^n(C^\infty_c(M))\hookrightarrow C^\infty_c(M^n)^{S_n}$ is obtained by $$(\phi_1\cdots\phi_n)(x_1,\dots,x_n)=\sum_{\sigma\in S_n}\phi_{\sigma(1)}(x_1)\cdots\phi_{\sigma(n)}(x_n).$$ This certainly seems as a version of the Stone-Weierstrass theorem.

Answer 1

Consider the linear map $C_C^{\infty}(M^n)\to C_C^{\infty}(M^n)$ that switches two arguments of a function, eg $$\phi \mapsto [(x_1,..., x_n)\mapsto \phi(x_2,x_1,x_3,...,x_n)].$$ Such a linear map is continuous and as such any permutation of the arguments is a continuous map, further any finite sum of such maps is a continuous linear map. It follows that "symmetrization" $$S:C_C^\infty(M^n)\to C_C^\infty(M^n)^S, \quad \phi\mapsto \frac1{n!}\sum_{\sigma\in S_n}\phi\circ \sigma $$ is continuous (and clearly surjective!), so if $D$ is then a dense subset of $C_C^\infty(M^n)$ then $S(D)$ is dense in $C_C^\infty(M^n)^S$. Now the space $S^n(C_C^{\infty}(M))$ is the space of symmetric tensors, so in order to see that it is dense in the space of symmetric functions it is enough to see that the $n$-fold tensor product of functions is dense in $C_C^{\infty}(M^n)$. This is a standard fact that we will now review.

Let $f\in C_C^{\infty}(M^n)$, by choosing a finite partition of unity of the support of $f$ we may assume that $f$ has its domain entirely in a chart. The question then reduces to whether or not the $n$-fold tensor product of $C_C^\infty(\Bbb R^m)$ is dense in $C_C^\infty((\Bbb R^m)^n)$. For this consider a sequence of polynomials $p_k$ so that $\lim_k \partial_\alpha p_k= \partial_\alpha f$ uniformly in some compact neighbourhood of $\mathrm{supp}(f)$ for all $\alpha$. Let $\psi_1,..,\psi_n$ be smooth bump functions on $\Bbb R^m$ so that

$$\psi_1(x_1)\cdot...\cdot\psi_n(x_n)$$ is equal to $1$ on $\mathrm{supp}(f)$ and $0$ outside of the neighbourhood in which the polynomials converge to $f$. Then $\psi_1...\psi_n p_k$ converges to $f$ in $C^\infty$ sense, further it is clearly a sum of products of functions $\Bbb R^m\to \Bbb C$.