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.

  • $\begingroup$ Related question for which I didn't get answer. $\endgroup$ – Pedro Oct 22 '15 at 0:39

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.

  • $\begingroup$ Thanks for your answer. Let me comment a bit the question. It's quite clear that $\tau$ if finer than the other topology. Having a finer topology is an advantage, beacuse the dual is "bigger", i.e., functionals with worst behaviour at the "boundary" are admitted, beacuse in duality with very regular objects. With respect to $\tau$, locally integrable functions are distributions, and they are not with respect to the other topology. Hence, finer is better. Completeness appears here as an "accident". $\endgroup$ – Federico Oct 21 '15 at 18:09
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    $\begingroup$ I do not completely agree with finer is better. You could endow $\mathscr D(\Omega)$ with the finest locally convex topology so that each linear functional becomes continuous. The dual is thus biggest possible. But then you loose many properties which make Schwartz' distributions meaningful like, e.g., the fact that locally every distribution is the derivative of a continuous function. $\endgroup$ – Jochen Oct 22 '15 at 7:34
  • $\begingroup$ Good point. Actually, I was referring to this specific refinement of the topology, but it is worth to prevent incautious generalizations. $\endgroup$ – Federico Oct 22 '15 at 9:28
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    $\begingroup$ @Federico and Jochen. If we go back one step and ask "which is the advantage to having a topology on $\mathscr{D}$" we can find a Schwartz's answer. Unfortunately, it is precisely like an answer that Federico classified as not acceptable (with Open Mapping theorem replaced by Hanh-Banach Theorem). Initially, Schwartz was not able to define a topology on $\mathscr{D}$ and thus he defined only a notion of convergence which he called pseudo-topology. Some years later he created a topology which yields that notion of convergence (the inductive limit topology). To justify it, he said: $\endgroup$ – Pedro May 3 '17 at 5:47
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    $\begingroup$ "The pseudo-topology is not enough; in order to apply the Hahn-Banach theorem and to study the subspaces of $\mathscr{D}$, you need to work with a real topology" (see here the source). $\endgroup$ – Pedro May 3 '17 at 5:48

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