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.

**Answer**

Take $c(\Gamma)$ with $\Gamma$ uncountable under the topology of pointwise convergence. $c_0(\Gamma)$ is dense but not sequentially dense. Let $u$ be the linear functional that vanishes on $c_0(\Gamma)$ and is one at $1_\Gamma$.

**Attribution***Source : Link , Question Author : makt , Answer Author : Bill Johnson*