Distribution Theory – Sequential and Topological Duals of Test Function Spaces

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Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test functions with its usual topology, as defined for instance here), I understand that generalised functions may be defined as elements of the topological dual space, in our examples resp. $\mathcal{S}'$ or $\mathcal{D}'$.

$\mathcal{S}(\mathbb{R}^n)$ is a metrisable space, hence sequential. Therefore its topological dual is the same as its sequential dual, by which I mean the space of sequentially continuous functionals on $\mathcal{S}$. $\mathcal{D}$, on the other hand, is not metrisable. I recall having seen somewhere that it is not even first-countable (I would welcome verification). Nevertheless, I have a vague notion that for a functional $f$ to belong in $\mathcal{D}'$, it is sufficient that it be sequentially continuous on $\mathcal{D}$. Hence my following questions:

  1. Is it true that sequentially continuous functionals on $\mathcal{D}$ are the same as the continuous ones? Put differently, do the sequential and continuous duals of $\mathcal{D}$ coincide?

  2. Assuming 1 is true, does it follow that $\mathcal{D}$—in spite of not being first-countable—is a sequential space? In other words, do the notions of continuity and sequential continuity coincide for general mappings from $\mathcal{D}$ to an arbitrary topological space $X$?

  3. For general test function spaces which may not be sequential, which is more appropriate: To define generalised functions as elements of their continuous dual space, or of the sequential dual?

  4. Is 3 even relevant (i.e. can such test function spaces be reasonably conceived), given the many requirements that are normally placed on a test function space, such as nuclearity?

Thanks very much in advance.

Best Answer

A first general remark: A topological vector space is metrizable, if and only if it is first countable. To answer your questions:

  • Yes, Q1 and Q2 are true. Let $K_i$ be a defining sequence of compact sets for the space $\mathcal{D}$. As the topology on $\mathcal{D}$ is the final topology (i.e. the finest such that all injections $\mathcal{D}_{K_i} \to \mathcal{D}$ are continuous), a map $T$ on $\mathcal{D}$ is continuous, iff its restriction to each $\mathcal{D}_{K_i}$ is continuous. And for this, sequential continuity of $T$ is sufficient (assuming the known fact, that a sequence of test functions converges iff the support of the functions are contained in one $K_i$ and the sequence converges in $\mathcal{D}_{K_i}$.
  • Q3 and Q4: Don't know if there exist relevant examples. I would always require continuity.
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