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With the usual notation of distribution theory, let $(\phi_j)_{j=1}^{\infty}$ be a sequence in $\mathscr{D}(\mathbb{R}^n)$ such that for every $T \in \mathscr{D'}(\mathbb{R}^n)$, we have $T(\phi_j) \rightarrow 0$ (which is to say that $\phi_j \rightarrow 0$ in the weak topology of $\mathscr{D}(\mathbb{R}^n)$). Does we have $\phi_j \rightarrow 0$ in the original topology of $\mathscr{D}(\mathbb{R}^n)$?

Analogously, let $(\phi_j)_{j=1}^{\infty}$ be a sequence in $\mathscr{S}(\mathbb{R}^n)$ such that for every $T \in \mathscr{S'}(\mathbb{R}^n)$, we have $T(\phi_j) \rightarrow 0$ (which is to say that $\phi_j \rightarrow 0$ in the weak topology of $\mathscr{S}(\mathbb{R}^n)$). Does we have $\phi_j \rightarrow 0$ in the original topology of $\mathscr{S}(\mathbb{R}^n)$?

I think the answers to these two questions are negative, but I could not find any counter-example.

Thank you very much for your attention.

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    You will not find counterexamples, the answer is positive. – 2017-01-20
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    Hint: $\mathcal{D}$ and $\mathcal{S}$ are Montel spaces as are their duals. On bounded subsets of such spaces, weak and strong topologies coincide, so every weakly convergent sequence is strongly convergent. – 2017-01-20

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Theorem: Let $E$ be an locally convex space, where every bounded set is relative compact (a so-called Semi-Montel-space). On a bounded subset $B$ of $E$ the weak topology $\sigma(E,E')$ and the original topology coincide.

Proof: Let $C$ be the closure of $B$ (with respect to the original topology). Then $C$ is a compact topological hausdorff space, so its topology is minimal, i.e. does not admit a strictly coarser hausdorff topology. As the weak topology is coarser than the original topology, they must coincide on $B\subseteq C$.

As $\mathcal{D}$ and $\mathcal{S}$ are Montel spaces you may apply the theorem on $B=\{\phi_j; j\in\mathbb{N}\}\cup \{0\}$ for your specific question.

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    Thank you very very much for your invaluable help. I could never answer the question by myself. I am a beginner in distribution theory and functional analysis. I have to study hard!! – 2017-01-20
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    @MaurizioBarbato You are asking very good questions, and you seem to like functional analysis. Go ahead and stay motivated. I left the university in the mid-90's and still like this subject. – 2017-01-21
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    Thank you very much for your encouraging words. I like very much mathematics in general: it is la wonderful land with incredibly fascinating landscapes, where I can find Beauty. My congratulations for being still so trained after os many years! – 2017-01-21
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    I add here just some reference for the results you used in the proof for those who don't know them. For the minimality property of the topology of a compact Hausdorff space see [Compact Hausdorff Topology](compact hausdorff minimal topology). For a proof that $\mathscr{D}$ and $\mathscr{S}$ are Montel spaces, see e.g. Treves, Topological Vector Spaces, Distributions and Kernels, p. 357 or Horvath, Topological Vector Spaces and Distributions, pp. 239-241. Finally, the Theorem proved by Vobo is a generalization of Proposition (36.9) in Treves, cit., p. 376. – 2017-01-23