In my homework about heat equation the professor want me to prove the following:
Let $S$ e a compact subset of $\mathbb{R}^n$, show that the restrictions of polynomials to $S$ is dense in $C(S)$ in the uniform norm.
I know this is a consequence of Stone–Weierstrass theorem. But, this questions might be tricky.
Is there some magic relation between the density of polynomials and the theory of the heat equation? A hint to see the magic would be appreciated.
Thanks!