Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let $\mathcal{D}(X)$ be the space of Schwartz distributions equipped with the $w^\*$-topology, namely the weak topology induced by the natural pairing $C^\infty(X)\times \mathcal{D}(X)\to \mathbb{C}$.


Let $F\colon \mathcal{D}(X)\to \mathbb{C}$ be a linear functional which is sequentially continuous in the weak topology, namely it maps weakly convergent sequences to convergent ones.


**Question: Is $F$ continuous in the weak topology?**



When sequentially continuous linear functional is continuous?
Thanks for the two previous answers. Both are very interesting. Meantime I have found another general result which seems to be particularly convenient for my purposes.


**Theorem. Let $E$ be a separable complete locally convex space. Then any sequentially continuous (in the weak\* topology) linear functional on $E'$ is weak\* continuous.**


See Schaefer's book "Topological vector spaces", Ch. IV, $\S$ 6, Corollary 3.


In our situation $E=C^\infty(X)$ is obviously complete and separable when $X$ is a compact manifold. Furthermore the above theorem is applicable when $E=\mathcal{D}(X)$ is the space of infinitely smooth compactly supported functions on $X$ when $X$ is a countable at infinity manifold (i.e. $X$ is representable as a countable union of compact subsets). Indeed $\mathcal{D}(X)$ is a strict inductive limit of a sequence of complete locally convex (even Frechet) spaces which is complete by Ch. II, $\S$ 7, (6.6) of Schaefer's book. The separability of $\mathcal{D}(X)$ is obvious.


