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:
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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?
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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 $X$?
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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?
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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: