Countable dense subset of $C_c^\infty(\mathbb R^n)$

Question

$(C_c^\infty(\mathbb R^n), ||.||_{L^p})$ is separable as a subset of $L^p(\mathbb R^n)$, which is itself separable ($1<p<\infty$).

Hence, what are the countable dense subsets of $C_c^\infty(\mathbb R^n)$?

I am not looking for all countable dense subsets, but rather, what are the well-known sets of functions that separate $C_c^\infty(\mathbb R^n)$?

Answer 1

Great question.

Let me give you a taste of how to prove it. Let $\mathrm{X}$ be a metrisable, separable, locally compact space. Then, there exists an increasing family $\mathrm{K}_n$ of compact subsets of $\mathrm{X}$ such that $\mathrm{K}_n \subset \mathring{\mathrm{K}}_{n + 1}.$

Denote by $\mathscr{K}(\mathrm{X}; \mathrm{K})$ the set of real-valued function with support contained in the compact set $\mathrm{K}.$ If you find a dense subsequence for each $\mathscr{K}(\mathrm{X}; \mathrm{K}_n)$ ($n \in \mathbf{N}$), then you are done.

Canonically, $\mathscr{K}(\mathrm{X}; \mathrm{K}_n)$ is identified with the subspace of $\mathscr{C}(\mathrm{K}_{n + 1})$ of functions vanishing outside $\mathrm{K}_n,$ where $\mathscr{C}(\mathrm{L})$ is the the space of real-valued continuous function defined on $\mathrm{L}.$ Finally, a common application of Stone-Weierstrass theorem gives a dense subset of $\mathscr{C}(\mathrm{L}).$