counterexample for $C^1(U)$ not complete in any dimension

Question

Cleary $C^1[a,b]$ is not complete with $\|\cdot\|_{\sup}$.

I am looking for a counterexample which is working in any dimension, i.e. $C^1(U)$ is not complete for any open $U\subseteq \mathbb R^n$ open with supremums norm.

All I did find were counterexamples in one dimension, anybody an idea for a counterexample which fits to any dimension?

Answer 1

Suppose $a\in U.$ For $k\in \mathbb N $ define

$$f_k(x) = \frac{((x_1-a_1)^2 + 1/k)^{1/2}}{1+|x|^2}.$$

Then each $f_k \in C^1(U),$ but

$$f_k(x) \to \frac{|x_1-a_1|}{1+|x|^2}$$

uniformly on $U.$ The last function does not belong to $C^1(U).$

Answer 2

We first note that $U$ is locally compact, so we might be able to use Weierstrass to show that $C^1(U)$ is not closed.

Consider $C = C^1(U) \cap C_0(U) \subset C_0(U)$. Then is $C$ a subalgebra of $C_0(U)$. So if we show that $C$ does not vanish and separates the points, we know that $C$ is dense in $C_0(U)$ w.r.t. the supremum norm. And as $C_0(U)$ contains non-differentiable functions, we know that $C$ is not closed, and so $C^1(U)$ is not closed.

To show that $C$ does not vanish, fix $x \in U$ and $r>0$ such that $B(x,r) \subset U$. Then we can consider the characteristic function $f$ of $B(x ,\frac{1}{2})$. If we consider the mollifier $$\eta(x) = \begin{cases} C\exp( 1/(\|x|^2 -1)) & |x| < 1 \\ 0 & |x| \ge 1\end{cases}$$ and $\eta_\epsilon (x)= \frac{1}{\epsilon^n} \eta (\frac{x}{\epsilon})$.
Then the convolution $g$ of $f$ and $\eta_{\frac{r}{4}}$ lies in $C^\infty_0(U)$, and in particular in $C$. Furthermore, we have $g(x) \neq 0$, so $C$ does not vanish.
In a similar fashion can we show that $C$ separates the points of $U$, as we can find two sufficient small balls around $x$ and $y$ and again take the characteristic function of one of those balls, and then again a convolution with a mollifier, which gives $g$ such that $g(x)=1$ and $g(y)=0$.
Thus with Weierstrass, we find that $C$ is dense.

So we can find a net in $C^1(U)$ which converges w.r.t. the supremum norm to a non-differentiable function.