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Let $l^{\infty}$ be the Banach space of all bounded real sequences with the $sup$-norm and $c$ the closed subspace of convergent sequences. Is there a continuous linear map $T: l^{\infty} \rightarrow c$ such that $T$ is the identity on $c$?

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This has been asked and answered before on MO somewhere, but I don't have the precise link to hand – Yemon Choi Dec 24 '12 at 2:16
up vote 8 down vote accepted

Such $T$ does not exist because $c_0$ is not complemented in $l_\infty$ but it is complemented in $c$. See for example "Topics in Banach Space Theory" by Kalton and Albiac.

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