I am looking for references where I can find proofs for the following statements. Also, if somebody could point out how these statements relate to each other, I would be very grateful. Let $M$ be a smooth manifold.
- $C^\infty(M)^*=\mathcal{D}_c(M)$ the space of compactly supported distributions.
- If $M$ has a metric, then the space of compactly supported smooth functions $C^\infty_c(M)$ can be identified as a dense subset of $\mathcal{D}_c(M)$. In here it is understood that the action of a $\phi\in C^\infty_c(M)$ on $f\in C^\infty(M)$ is given by $\phi(f):=\int \text{d}\text{vol}_g \phi f$.
- The symmetric tensor product $S^n(C^\infty_c(M))$ is a dense subspace of $C^\infty_c(M^n)^{S_n}$, the symmetric smooth functions of $n$ variables on $M$. In here, a symmetric tensor product $\phi_1\cdots\phi_n\in S^n(C^\infty_c(M))$ is understood as an element of $C^\infty_c(M^n)$ via $(\phi_1\cdots\phi_n)(x_1,\dots,x_n)=\phi_1(x_1)\cdots\phi_n(x_n)$.
- Forgetting whether $M$ has a metric or not, $\Omega^{n-1}_c(M)$ lies densely on $\Omega^1(M)^*$, where an element $A^*\in\Omega^{n-1}_c(M)$ acts on $A\in\Omega^1(M)$ by $A^*(A)=\int A\wedge A^*$.
Let me make some commments. I know Rudin proofs $\mathcal{D}_c(M)\subseteq C^\infty(M)^*$ in Theorem 6.24 of his book Functional Analysis. Is the other inclusion obvious? On the other hand, 2. and 4. seem to be intimately related. Namely, proving 2. is equivalent to proving that $\Omega^n(M)$ lies densely on $C^\infty(M)^*$. Finally, 3. looks a lot like some sort of Stone-Weierstrass theorem.
All of these assertions are made in Factorization Algebras in Quantum Field Theory of Costello and Gwilliam. My background in functional analysis is Part 1 of Functional Analysis of Rudin and bits and pieces of theory of bounded operators on a Hilbert space. Any help is much appreciated!