Let $X$ be a compact subset of $\mathbb{R}^n$. The classical Stone-Weierstrass theorem describes dense subsets of $C(\mathbb{R}^n,\mathbb{R})$ when it is equipped with the compact-open topology.

Now, suppose that $X$ has a $C^1$-boundary and that $C^1(X,R)$ is equipped with the topology consistent with the following notion of convergence *(in total variation)*:

A sequence of functions in $f^n$ converges to a function $f$ if for every $\epsilon>0$ there is some integer $N$ such that for every $n\geq N$ $$ \|f^n(0)-f(0)\| + \int_{x \in X}\| \nabla (f^n-f)(x)\|dx<\epsilon . $$

**Question:**

Is there a Stone-Weirestrass-type theorem (algebraically) describing when a linear subspace $A$ of $C^1(X,R)$ is dense?

convergence in total variation on compactstopology. $\endgroup$ – AIM_BLB Jun 19 at 14:59