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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.

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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 $1_\Gamma$.

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    $\begingroup$ Product space $\ \mathbb R^A\ $ for uncountable $\ A\ $ works like this too (but my poor brainy malfunctions from time to time, too often). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Oct 1, 2013 at 18:50

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