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Let $s$ be the space of rapidly decreasing sequences, i.e. $s=\{\xi=(\xi_j)_j\colon\,\,\sup_j|\xi_j|j^n<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\}$ and $s'$ its topological dual, i.e. $s'=\{\eta=(\eta_j)_j\colon\,\,\sup_j|\eta_j|j^{-n}<+\infty\,\,\text{for some}\,\,n\in\mathbb{N}\}$. Suppose we have a linear operator $T$ which is continuous as a map $s\to s, s'\to s', \ell_2\to\ell_2$. Moreover as a map $\ell_2\to\ell_2$ it is a topological isomorphism. Denote by $e_j\,\,(j\in\mathbb{N})\,$ standard unit vectors. Can we deduce $(Te_j)_j\subset s$ is a basic sequence in $s$?

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