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Let $T\colon \ell_\infty \rightarrow \ell_\infty : (a_1,a_2,\ldots) \mapsto (a_1,a_2/2,a_3/3,a_4/4,\ldots) $.

Show that $\operatorname{range}(T)$ is not dense in $\ell_\infty$. I want to ask for a hint or a solution to this problem.

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Hint: If $(x_n)$ is in the range of $T$, then $x_n\rightarrow 0$. Can the sequence $(1,1,\ldots)\in\ell_\infty$ be the limit in norm of a sequence of such $(x_n)$? – David Mitra Apr 10 '12 at 15:18
up vote 6 down vote accepted

Hint: If $(x_n)$ is in the range of $T$, then $x_n\rightarrow 0$. What is a lower bound of the distance in $\ell_\infty$ from the vector $(1,1,\ldots)\in\ell_\infty$ to such an $(x_n)$?

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