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 $c$?



split of s.e.s. of Banach spaces
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.


