If $K$ and $K'$ are compact such that $K$ is contained in interior of $K'$, then show that $D_K$ is nowhere dense in $D_{K'}$.

Question

Let $D_K$ be the space of all smooth functions on $\mathbb{R}^n$ which are compactly supported in $K$ for a compact $K$. If $K$ and $K'$ are compact subsets of $\mathbb{R}^n$ such that $K$ is contained in the interior of $K'$, then show that $D_K$ is nowhere dense in $D_{K'}$.

It's an old qual problem and I have no idea where to start.

Answer 1

Since $K$ is trivially properly contained in $\mathring{K^\prime}$, we can find $x_0\in\mathring{K^\prime}$ and some $\delta>0$ such that $B_\delta(x_0)\subset(\mathring{K^\prime}\setminus K)\subset (K^\prime\setminus K)$.

Then $$f(x) := \begin{cases}e^{1/\left(\delta^2-\|x-x_0\|^2\right)} & \|x-x_0\|<\delta\\ 0 & \|x-x_0\|\geq\delta\end{cases}$$ is commonly used as a smooth function which vanishes outside $B_\delta(x_0)$.

Note that $\epsilon f\in D_{K^\prime}\setminus D_K$ is arbitrarily small (in the $\infty$ norm, which I presume is used here, but also in any other norm, at least).