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Christian Remling
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No to both questions. Consider $V=\{ g\in\mathcal S: g(-x)=ig(x) \}$. This space is closed because it is the null space of the continuous operator $R-i$, with $(Rg)(x)=g(-x)$.

Then $T_f=0\in V'$ for every $f\in V$ because $fg$ is odd when $f,g\in V$.

Christian Remling
  • 24.2k
  • 2
  • 48
  • 83