TITLE: Topology of test functions $\mathcal{D}(\mathbb R)$
QUESTION [11 upvotes]: (My motivation for the following question is to understand the distribution theory)
The space of test functions:
$\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb R), \ \text{and support of }\  \phi \ \text{is compact} \}.$
Let us introduce the norms,
$$\|\phi\|_{N}=\text{max} \{|D^{\alpha}\phi(x)|: x\in \mathbb R, |\alpha| \leq N \}$$
for $\phi \in \mathcal{D}(\mathbb R)$ and $N=0,1,2,...$.

My Question: (1) How to use these norms to define locally convex metrizable topology on $\mathcal{D}(\mathbb R)$ ? (2) This topology is not complete; but I don't understand the reason; Pick $\phi \in \mathcal{D}(\mathbb R)$ with support in $[0,1], \phi>0$ in $(0,1),$ and 
  define, 
  $$\psi_{m}(x)=\phi(x-1)+\frac{1}{2}\phi(x-2) +...+\frac{1}{m}\phi(x-m);$$
  so, my sub-question is:
  (a) How to verify $\{\psi_{m}\}$ is a Cauchy sequence in $\mathcal{D}(\mathbb R)$ ? (b) How to verify $\lim \psi_{m}$ does not have a compact support ?

I want to add one more question to it i.e. If $\Omega\subset_{open} \mathbb R^n$ and the norms are defined by
$$\|\phi\|_{N}=\text{max} \{|D^{\alpha}\phi(x)|: x\in \mathbb \Omega, |\alpha| \leq N \}$$ for $\phi\in\text D(\Omega) \ \text{instead of}\ \text D(\mathbb R)$ then how to show that the metrizable topology on $\text D(\Omega)$ is not complete.
Thanks.

REPLY [2 votes]: you can just use the "Baire Category Theorem" as $\mathbb R=\cup_{i=1}^\infty K_i$ where each $K_i$ is compact and $K_i^o\subset K_{i+1}\forall i\in \mathbb N$ . Now each $D_K$ is a closed subspace of $D(\mathbb R)$ and {$D_K^o=\phi\ \forall\  K\subset_{compact}\mathbb R$} and $D(\mathbb R)=\cup_{K=1}^\infty D_K$ and $D(\mathbb R)$ is metrizable hence by Baire Category theorem it cannot be complete.