What do we know about the space of finite order distributions ?

Question

Hi,

(Question updated)

My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ?

From in the information I gathered, it seems that the natural topology on $\mathcal{D}'_F$ is the inductive limit topology of the spaces $(\mathcal{D}'^m)$ of distributions of order $m$, or equivalently, the dual topology of $\mathcal{D}_F$ [ this space being $\mathcal{D}$ as a set, but with the coarser topology of the projective limit $(\mathcal{D}^m)$ ($C^m$ functions with compact support, this is an inductive limit of Fréchet spaces with obvious semi-norms). Note that $\mathcal{D}_F$ is strictly coarser than $\mathcal{D}$ (and strictly finer than the $\mathcal{S}$), and that $\mathcal{D}'_F$ is strictly finer than $\mathcal{D}'$ (and strictly coarser than $\mathcal{S'}$).

So, the question is: what do we now about this topology on $\mathcal{D}_F$, and its strong dual $\mathcal{D}_F'$ ? It is clearly not Frechet, but is it complete ? Montel? Barrelled ? Nuclear ? Reflexive ? More generally, do we have most of the nice properties of $\mathcal{D}'$ for $\mathcal{D}_F'$ ?

Thanks

Answer 1

Note added in edit: This was an answer to the original question which considered the topology on finite order distributions induced by the topology on all distributions.

I'm assuming that by "finite order" you mean as given on this PlanetMath page.

The answers are: No, Yes, Yes, No, No, No, I Think So.

In more detail:

• $\mathcal{D}_F'$ is dense in $\mathcal{D}'$ because we can approximate an arbitrary distribution by a distribution with compact support and distributions with compact support have finite order. Thus also $\mathcal{D}_F'$ is not closed in $\mathcal{D}'$.

• The obvious topology is to fix a suitable topology on each of the spaces of distributions of fixed finite order and then take the induced topology on the union. I doubt that this is the same as the induced topology, but can't say for sure. It will be the finest sensible topology on this space. (I say "it" but there could be several depending on how you topologise the spaces of fixed order, essentially you need to throw in the order into the semi-norms somehow and there may be several ways to do this.)

• $\mathcal{D}_F'$ is not complete since it is not closed in $\mathcal{D}'$. It is therefore not Montel nor reflexive.

• I'm less confident about Mackey, it having been a while since I looked at these spaces. I suspect "Yes" because it's completion is Mackey and I can't see how not being complete can complicate matters. Perhaps someone more steeped in the lore of LCTVS can help out here.

I'm curious as to the motivation for this question. Could you elaborate?

Answer 2

The following are some night thoughts on finite order distributions which might be of interest since this topic is a rich source of fruitful questions on the relations between abstract functional analysis and spaces of functions and distributions.

I will start with the case of distributions on a compact interval which we can suppose to be $[-1,1]$. This is, of course, trivial in the sense that every distribution there is of finite order but it will serve to introduce some of the questions which arise on the line. Firstly, there is a certain ambivalence in the notion of the order of a distribution. It seems to be used in the sense of the smallest degree of differentiation required to obtain a given distribution from a measure. But there are other versions, e.g., using continuous or integrable functions, which are more appropriate in certain situations. Thus the important example of the dipole can be regarded as the derivative of a measure (the Dirac function, by definition), the second derivative of an integrable function (the Heaviside function) or the second derivative of a continuous function (the absolute value up to a factor). This will play a more important role in the case of distributions on the line.

The very definition displays the space of distributions as a union of a sequence of Banach spaces and it is natural to give it the corresponding inductive limit topology (in the l.c. sense). This leads to several natural questions: is the space complete and do the bounded sets, compacta and convergent sequences arise in the natural way from the component Banach spaces? A further source of questions is whether the space of distributions has a natural representation as the dual of a space of test functions and whether the inductive limit topology has a natural interpretation in terms of this duality. In this situation these questions all have positive answers but on the line this is, as we shall see, less clear.

In this case, the positive properties can be deduced from the fact that the inductive limit is of a special structure--the interconnecting maps between the component Banach spaces are compact (even nuclear) which means that the distribution space is are Silva spaces and these have very special properties (see, e.g., Köthe´s monograph).

In the case of distributions on the line, there are many candidates for the notion of a distribution of finite order and the situation is correspondingly diverse and has scarcely been investigated. In order to illustrate this we shall discuss four important cases--finite derivatives of measures, of continuous functions, of $L^p$-functions and of functions of polynomial growth.

1. Starting with the $L^p$ case. These were already considered by Schwartz. In the reflexive cases, they are so-called weak Silva spaces (defined as for Silva spaces but with weak compact operators). Such spaces were considered in the 60´s of the last century but I have been unable to trace any references.

2. Derivatives of continuous functions. In this case, one obtains a so-called $LF$-space (union of a sequence of Frechet spaces, but not a strict inductive limit). Such spaces were investigated by Grothendieck in his thesis. I don´t think that the answer to any of the questions mentioned in the beginning of this answer is known in this case.

3. The case of repeated derivatives of continuous functions of polynomial growth (obvious definition) is back in familiar territory--this is the space of tempered distributions, perhaps the most important one in mathematical physics. Once again, this is a nuclear Silva space and so all is well.

4. We finish with the case which is the one most used in the term distributions of finite order--repeated derivatives of measures on the line. From the point of view of functional analysis, this is by far the most complicated one. It is an inductive limit of a sequence of so-called $DLF$-spaces (the latter are, in analogy to Grothendieck´s $DF$-space, motivated by the duals of strict $LF$-spaces in the sense of Dieudonne and Schwartz). To my knowledge, nobody has attempted to examine this situation, in particular with respect to the above questions, in detail.