Let's consider $\mathscr D(\Omega)$, the space of test functions on $\Omega \neq \emptyset \subseteq \mathbb R^n$ as usually defined. For the sake of clareness,
$$\mathscr D(\Omega) = \cup_K \mathscr D_K,$$
$K \subset \Omega$ compact, $\mathscr D_k := \{ f \in C^\infty(\Omega) \mbox{ with support in } K\}$. There are serveral ways to topologize $\mathscr D(\Omega)$. Among them, one is defining the family of norms
$$\| \phi \|_N := \max \{ | D^\alpha \phi(x) | : x \in \Omega, | \alpha | \leq N\},$$
for $\phi \in \mathscr D(\Omega)$ and $N = 0, 1, 2, \dots$. The restrictions of these norms to any fixed $\mathscr D_K \subset \mathscr D(\Omega)$ induce the same topology on $\mathscr D_K$ as do the seminorms $p_N$ defined below:
$$p_N( \phi ) := \max \{ | D^\alpha \phi(x) | : x \in K_N, | \alpha | \leq N\}.$$
Let's denote this topology with $\tau_K$.
It is well known that each $(\mathscr D_K, \tau_K)$ is Fréchet. One can use norms $\| \phi \|_N$ to define a locally convex metrizable topology on $\mathscr D(\Omega)$, but this topology is not complete. Now, suppose we topologize $\mathscr D(\Omega)$ as the inductive limit of $\mathscr D_K$ (see Rudin). Denote this topology with $\tau$. Then $\mathscr D(\Omega)$ is complete ($\tau$-Cauchy sequences do converge). One shows that $(\mathscr D(\Omega), \tau)$ is not metrizable (hence can't be Fréchet). Most important, one shows also that $\mathscr D_K$ inherits from $(\mathscr D(\Omega), \tau)$ the same topology as before, and Theorem 6.5 in Rudin holds.
Rudin says that having an uncomplete topology is a disadvantage. This is taken as a motivation to substitute $\tau_K$ with $\tau$.
Finally, look at continuous linear functionals on $\mathscr D(\Omega)$. Functional like these exist in both cases, because in each case $\mathscr D(\Omega)$ is locally convex. Of course, we would not obtain the same set of linear functionals in the two cases, so we have different sets of distributions. Looking at the above remarks, one could think that these distributions will share a lot of properties. So my question is
Question. Which is the striking advatange to having a complete topology on $\mathscr D(\Omega)$, if our goal is constructing a distribution theory which have the usual well-known properties (cfr. Rudin, beginning Chapter 6)?
Please, note that answers like "completeness it's always better", "in order to apply Open Mapping theorem" or similar will not be accepted, if unsupported by a specific example which shows a "crash" of uncompleteness.
Thanks in advance.
Rudin, Functional Analysis.
One important property of distributions is that for any sequence $u_n\in \mathscr D'(\Omega)$ such that $u(\varphi)=\lim_n u_n(\varphi)$ exists for all test functions the limit is again a distribution. This does not exactly use the completeness of $\tau$ but the fact that it is the topology of an inductive limit of Frechet spaces.