# Reference Request: Stone-Weierstrass on Other Topologies

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?

• This seems too vague. You should be more specific which topologies you are interested in. – Jochen Wengenroth Jun 19 at 13:37
• 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. – AIM_BLB Jun 19 at 14:59
• Could you please describe this topology on $C(X)$? – Dirk Werner Jun 19 at 17:01
• I've updated the question to make things more explicit – AIM_BLB Jun 19 at 17:34
• 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. – Nate Eldredge Jun 19 at 18:48