I'm doing the second half of the following exercise in my lecture notes:
"Let $C_c(R)$ be the vector space of continuous functions $f : R \to R$ with $\mathrm{supp}(f)=\overline{ \{x \in R \mid f(x)\neq 0 \} }$ compact, with the norm $\|\cdot\|_\infty$. Show that this space is not complete, and find a Banach space containing $C_c(R)$ so that the induced norm obtained by restriction is $\|\cdot\|_\infty$. Can you do the same for the norm $\|f\|_\Psi =\|f \Psi \|_\infty$ where $\Psi : R \to R_{>0}$ is a fixed continuous function (for example, $\Psi (x) = e^{x^2}$ )?"
Assuming "do the same" means that I want to show that
1) $C_c$ is incomplete with respect to $\|\cdot\|_\Psi$ and then
2) find a Banach space containing it:
1) For a Cauchy sequence $f_n$ in $(C_c, \|\cdot\|_\infty)$ with limit not in $C_c$ we get a Cauchy sequence $g_n$ in $(C_c, \|\cdot\|_\Psi)$ with limit not in $C_c$ by taking $g_n = f/\Psi$. I guess one should argue that $\|f\|_\Psi$ defines a norm on $C_c$: but that's clear since the product of continuous functions is continuous and the support of $f \Psi$ is $\mathrm{supp}(f) \cap \mathrm{supp}(\Psi) \subset \mathrm{supp}(f)$.
Unfortunately, $\|\cdot\|_\Psi$ does not define a norm on $B(X)$, the space of bounded functions because $\Psi$ is not bounded. So we are looking for a smaller space: $C_0(X)$ is probably the space we're looking for. It contains $C_c$, $\|\cdot\|_\Psi$ defines a norm on it and it's complete w.r.t. $\|\cdot\|_\Psi$: if $g_n$ is Cauchy w.r.t. $\|\cdot\|_\Psi$ we can use that $\|g_n\Psi\|_\infty \geq \|g_n\|_\infty$ so $g_n$ is Cauchy w.r.t. the sup norm hence its uniform limit $g$ is in $C_0$ and hence $g \Psi$ is, too.
Apparently the construction $g_n = f_n \Psi$ only works if $\Psi$ doesn't have compact support. If it does, $f \Psi$ will have compact support and hence $\lim_{n \to \infty} g_n$ will be in $C_c$ and hence $C_c$ will already be complete w.r.t. $\|\cdot\|_\Psi$ in this case. So a necessary condition on $\Psi$ is that it has non-compact support.
What's a sufficient condition on $\Psi$ so that $C_c$ is incomplete w.r.t. ?