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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?

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    $\begingroup$ This seems too vague. You should be more specific which topologies you are interested in. $\endgroup$ – Jochen Wengenroth Jun 19 at 13:37
  • $\begingroup$ Well I was hopeing for something general, but I was most interested (at the end of the day) in the convergence in total variation on compacts topology. $\endgroup$ – AIM_BLB Jun 19 at 14:59
  • $\begingroup$ Could you please describe this topology on $C(X)$? $\endgroup$ – Dirk Werner Jun 19 at 17:01
  • $\begingroup$ I've updated the question to make things more explicit $\endgroup$ – AIM_BLB Jun 19 at 17:34
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    $\begingroup$ The ideas used in math.stackexchange.com/questions/78311/… may give you a start. Basically, you try to approximate $\partial_{x_1} \dots \partial_{x_n} f$ by polynomials, then integrate those polynomials with respect to all variables to get your approximating sequence. Boundary terms may cause some difficulties in this case, though. $\endgroup$ – Nate Eldredge Jun 19 at 18:48

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