Motivation for test function topologies

I'm a phisicist, who started looking just a little bit into distribution theory, so I can claim to know what I'm doing when throwing about dirac-deltas. Hence I only know two test function spaces: $\mathcal{D}=C^{\infty}_c(\Omega)$ (smooth functions with compact support) and $\mathcal{S} (\Omega)$ (Schwartz-space) where $\Omega\subseteq\mathbb{R}^n$ open. Now I wonder what the motivation is for defining the topologies on these spaces as one does it.

I'm reading "Fundamental Solutions of Partial Differential Operators" by Ortner and Wagner. They avoid actually defining the topologies on these spaces and only talk about convergence of sequences. I'm actually not sure what the exact relationship is between the sequence convergence and the topologies. For Schwarz space the question is irrelevant, since it its topology is metric. However $\mathcal{D}$ is not sequential.

Question 1: Is there a way to characterize the topology of $\mathcal{D}$ with sequences, as in saying "the coarsest topology having that convergence properties for sequences" or something similar? What is the reason most people don't bother talking about the actual topology and seems satisfied with sequences, although the topology is not sequential? I've heared something about that being irrelevant for linear maps, but haven't seen a precise statement.

As far as I know the definitions of sequence convergence are "Uniform convergence of all derivatives on compact sets with supports contained in a compact set" for $\mathcal{D}$ and "uniform convergence of all derivatives" for $\mathcal{S}$ respectively. The rest of my question deals with motivating these definitions.

For Schwarz space "to some extent" the motivation, as far as I know, is that almost everything one needs is continuous on this space and maps back into it. In particular all differential operators, and most particularly the Fourier transform. I'm fairly happy with this definition, although there is surely more to understand there. In particular I would like to know

(Soft) Question 2: Is there a way to characterize Schwarz space as "The subspace $X$ of $C^{\infty}(\Omega)$ where ??? can be defined $?:X\to X$" and the topology (or sequential convergence) is motivated in some way by requiring all the ??? stuff to be continuous? In terms of the ??? stuff I'm thinking of usefull things like derivatives and fourier transforms, not artificial examples making it work out right. [I found a claim that one gets this starting from $L^1(\Omega)$ by taking differentiation and multiplication by polynomials as some kind of closure. Needs clarification and proof though]

Let's turn to $\mathcal{D}$: I'm aware of Why does a convergent sequence of test functions have to be supported in a single compact set?, where the motivation of the convergence criteria of $\mathcal{D}$ is discussed to some extent. In particular it seems to me, that the notion of distribution depends on the topology on $\mathcal{D}$. Hence an answer saying something like "that part doesn't matter for compactly supported distributions" makes no sense to me. I don't really understand the answers and would like more detail. Can something similar to question 2 be answered for $\mathcal{D}$?

(Soft) Question 3: What is the motivation behind the topology for $\mathcal{D}$? Why all that talk about compact sets? Certainly I would be also happy with a motivation for what the topological dual (distribution space) should look like and then looking for what spaces have that as their dual.

In particular, the quoted question confused me on the following: The space $\mathcal{D}$ being locally convex, the topology is given by a family of semi-norms. That would mean we need to reabsorb the criterion "all supports of functions in the sequence (when testing for convergence) lie inside some compact set" into just a set of seminorms. Can this be done? I haven't seen that.

Solution 1:

I'm going to consider a part of your Question 1, namely:

What is the reason most people don't bother talking about the actual topology and seems satisfied with sequences, although the topology is not sequential?

I think that there are (at least) two reasons for this. The first is technical:

1. The topology is not easy to define and it is not easy to manipulate (here "not easy" means "not easy for an introductory course", for example a course with focus in the applications to PDE).

The second is more relevant:

2. The topology doesn't matter for the basic properties of distributions (probably, for the topics of the said introductory courses in which the said topology is not defined).

Sounds unsatisfactory, right? I agree, so let me explain. These are words (not literally) of Laurent Schwartz, who created the theory of distributions. In fact, Schwartz said the following with respect to the time in which he started work with the test functions:

I was unable to put a topology on $\mathcal{D}$, but only what I called a pseudo-topology, i.e. a sequence $(\phi_n)$ converges to $0$ in $\mathcal{D}$ if the $(\phi_n)$ and all their derivatives converge uniformly to $0$, keeping all their supports in a fixed compact set. I only found an adequate topology much later, in Nancy in 1946. But it doesn't matter for the main properties. ([1], p. 229-230).

This quote teach us the following:

• Historically, the notion of convergence of sequences in $\mathcal{D}$ came before the topology of $\mathcal{D}$.

As a consequence, it is natural to begin the study of distribution theory with the notion of convergence (instead of start with the actual topology).

In addition, the quote draw our attention for the following fact:

• There are problems that you can solve in the context of distributions without invoke a topology for $\mathcal{D}$. For some purposes, the usual notion of convergence (which Schwartz called pseudo-topology) is enough.

For example, the fact that the distributional derivative "preserves convergence of sequences" (in $\mathcal{D}'$) is a result that can be obtained and applied to the differential equations without the actual topology of $\mathcal{D}$.

Remark: Sometimes this result is called "continuity" of the distributional derivative, even in the context where the notion of convergence in $\mathcal{D}'$ is defined as the convergence in $\mathcal{D}$: the explicit form of the convergence is given but a topology is not defined. However, it is indeed possible to put a topology on $\mathcal{D}'$ (which implies the said notion of convergence in $\mathcal{D}'$) without put a topology on $\mathcal{D}$. With respect to this topology in $\mathcal{D}'$ the distributional derivative is indeed "continuous" (and thus preserves convergence of sequences). To give a reference for this remark, let me quote what Schwartz said in his treatise:

Nous définissons ainsi sur $\mathcal{D}'$ une topologie (qui, remarquons-le encore, ne nécessite pas la connaissance de la topologie de $\mathcal{D}$, mais seulement de ses ensembles bornés). ([2], p. 71)

Of course, as the quote suggests, Schwartz could define boundedness in $\mathcal{D}$ even in absence of a topology:

I did not have a topology on $\mathcal{D}$, but what I called a pseudo-topology [...]. I could speak without difficulty of a bounded subset of $\mathcal{D}$ [...]. $\mathcal{D}$ was more or less one of the spaces I had studied deeply during that short period [summer of 1943], always with the slight difficulty of the pseudo-topology, which nevertheless did not stop me. ([1], p. 231)

In short, all these things support the fact that is it possible to do (and Schwartz certainly did) many things in the context of the distributions without appeal to the topology of $\mathcal{D}$ (but only with the notion of convergence). In my opinion this justifies the second reason above as a fundamental answer for your "why". Maybe we could just say that people avoid talking about the topology (in some contexts) because it is an efficient strategy (in the context where it is avoided). The point is that the topology was created to yields a prior notion of convergence and allow a deeper development of the theory. The notion of convergence is not a mere simplification to avoid a complicated topology whose origin is a mystery; of course the topology is complicated and people make it seems mysterious (by virtue of an explanation's lack), but the notion of convergence is the cause of the topology and not the converse. Maybe you will agree that, from this point of view, the fact that in some contexts "people don't bother talking about the actual topology" becomes natural and acceptable.

Addendum (details on the creation of the topology). What was the advantage of defining a topology on $\mathcal{D}$? It was to make possible the application of the knows theorems of topological spaces, like the Hanh-Banach Theorem. The last sentence seems vague and sounds like a cliche, right? But it is the truth; it was essentially what Schwartz said:

In Grenoble, I gave an exact definition of the real topology corresponding to the pseudo-topology on $\mathcal{D}$, which later, in 1946, Dieudonne and I took to calling an inductive limit topology. The pseudo-topology is not enough; in order to apply the Hahn-Banach theorem and to study the subspaces of $\mathcal{D}$, you need to work with a real topology. ([1], p. 238)

I carefully defined the neighborhoods of the origin in $\mathcal{D}$, then gave the characteristic property which was precisely that of being an inductive limit, without giving it a name. I only did this for the particular object $\mathcal{D}$, without daring to introduce a general category of objects. Mathematical discovery often takes place in this way. One hesitates to introduce a new class of objects because one needs only one particular one, and one hesitates even more before naming it. It's only later, when the same procedure has to be repeated, that one introduces a class and a name, and then mathematics takes a step forwards. Other inductive limits were introduced, then the theory of sheaves used them massively and homological algebra showed the symmetry of inductive and projective limits. ([1], p. 283)

[1] A Mathematician Grappling with His Century by Laurent Schwartz.
[2] Théorie des distributions by Laurent Schwartz.

Solution 2:

This is just a quick addendum to Pedro's excellent answer. Indeed the topology of $\mathcal{D}$ is rather tricky and that's why most available treatments (e.g., with applications to PDEs) do not really get into it.

As for your very last question about explicit seminorms, the answer is yes. There is such a set of seminorms due (I think) to Horváth. See this MO answer https://mathoverflow.net/questions/234025/why-is-multiplication-on-the-space-of-smooth-functions-with-compact-support-cont/234503#234503

I think another answer to your interrogations is that almost nobody knows these seminorms.

In order to fully understand the topology of $\mathcal{D}$ you need to try your hands first on the space $s_0=\oplus_{\mathbb{N}}\mathbb{R}$ of almost finite sequences. The natural topology is the finest locally convex topology defined by all seminorms one can put on this (algebraic) vector space. This set of seminorm is the same as that of seminorms which are continuous on each $\mathbb{R}$ summand (trivially). A more explicit set of seminorms is $$ ||x||_{\omega}=\sum_{n\in \mathbb{N}}\omega_n |x_n| $$ for all $\omega\in [0,\infty)^{\mathbb{N}}$.

The elements of a convergent sequence in this space form a bounded set and this last property implies that there is a common finite support. This space $s_0$ is like a discrete analogue of $\mathcal{D}$ where the domain $\Omega$ is replaced by $\mathbb{N}$.

Then you can quickly visit the space of rapidly decreasing sequences $s$ with obvious definitions. Finally, you can upgrade to $$ \oplus_{\mathbb{N}} s $$ which is the analogue of $s_0$ where scalars in $\mathbb{R}$ are replaced by sequences in $s$. It is a little known fact due to Valdivia and Vogt that $$ \mathcal{D}(\Omega)\simeq \oplus_{\mathbb{N}} s $$ as topological vector spaces.

PS: I have no idea what $\mathcal{S}(\Omega)$ is for general open sets. The notion of Schwartz space is not a purely differential notion. It involves algebra if only to make sense of "polynomial" in polynomial growth. Schwartz's original definition is the set of elements in $\mathcal{D}$ which are restrictions of distributions on the sphere seen as the one-point compactification of Euclidean space. There are generalizations but they involve some algebraic structure like the notion of Nash manifold as in this article. There are also other references on this issue on the MO page: https://mathoverflow.net/questions/80094/the-schwartz-space-on-a-manifold