Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces. Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is formally weaker that the usual topological continuity in the case of non-metrizable spaces.) Let $L_0\subset L$ be a topologically dense linear subspace. Assume that $u|_{L_0}\equiv 0$.

**QUESTION: Does it follow that $u\equiv 0$?**

I am interested in rather concrete examples of spaces: spaces of generalized functions on smooth manifolds (say $R ^n$) with the wave-front set contained in a given closed set.