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Let $1\leqslant p$\ell_p$. Let $B_{L(\ell_q)}$ [was $B_{L(\ell_p)}$ ] be the closed unit ball of $L(\ell_q)$ [was $L(\ell_p)$] considered as a subset of $L(\ell_p)$ [was $L(\ell_p)$]. Is it closed in $L(\ell_p)$ [was $L(\ell_p)$]?

EDIT: I made a correction.

EDIT 2: You are right. Please delete my question.

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    I don't understand how your inclusions are supposed to work, maybe you want to expand on that? We have $\ell^p \subset \ell^q$. Now take $x \in \ell^{q} \smallsetminus \ell^p$ and take a rank one operator on $\ell^q$ of norm one projecting on the span of $x$. How does that give an operator on $\ell^p$ with values in $\ell^p$?2012-05-20

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