Topologies of test functions and distributions

Question

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$:

1. I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my question is: Is $\mathcal D(\Omega)$ sequential?

2. When $\mathcal D'(\Omega)$ is endowed with the weak* topology, is it sequential? (I presume this one is clearly not first countable either).

How about for $\mathcal S$ and $\mathcal S'$?

One of the reasons I ask is that some authors seem to use the theorem:

Given $X$ a sequential space, and $Y$ a topological space, $f:X \rightarrow Y$ is continuous if and only if $f$ is sequentially continuous,

to state that operators of the form $T:\mathcal D'(\Omega)\rightarrow\mathcal D'(\Omega)$ are continuous (like differentiability). However, they never seem to prove or even mention whether $\mathcal D'(\Omega)$ is sequential or not. Are they indeed using the above theorem, some other result, or is this perhaps carelessness?

Answer 1

I do not think that $\mathscr D(\Omega)$ is sequential. On the other hand this is probably not used: By definitionn of the locally convex inductive limit topology of $\mathscr D(\Omega)= \lim X_n$ (where $X_n$ are the Frechet spaces of smooth functions with support in $K_n$ for a compact exhaustion) a linear map with values in any locally convex space is continuous if (and only if) all the restrictions to $X_n$ are continuous and for this sequential continuity is enough since $X_n$ are metrizable.

Concerning linear maps $\mathscr D'\to \mathscr D'$ like partial differential operators, they are almost always defined as transposed maps of continuous operators $\mathscr D\to\mathscr D$ and therefore continuous with respect to the weak* topology.

The situation for $\mathscr S$ and $\mathscr S'$ is simpler because $\mathscr S$ is metrizable.

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EDIT. $\mathscr D(\Omega)$ fails "very much" to be sequential: There are even linear subspaces which are sequentially closed but not closed (see Klaus Floret, Some aspects of the theory of locally convex inductive limits (1980)). Moreover, this is not at all an exotic phenomenon but the hart of the matter when dealing with e.g. linear partial differential operators. If $P:\mathscr D'(\Omega)\to\mathscr D'(\Omega)$ is not surjective but surjective on the space of smooth functions then the range $L$ of its transposed is a closed subspace of $\mathscr D(\Omega)$ such that the relative topology of $\mathscr D(\Omega)$ is different from the inductive limit topology $\lim L\cap X_n$ and these topologies even have different duals. If $u:L\to \mathbb C$ is continuous with respect to the latter but not continuous with respect to the former then its kernel is sequentially closed in $\mathscr D(\Omega)$ (since both topologies have the same convergent sequences) but it is not closed (since otherwise $u$ would be $\mathscr D(\Omega)$-continuous).