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?**