How can we show that $c_0$ has no closed complement in $l^\infty$. Similarly $C([0,1])$ has no closed complement in $B([0,1])$
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For the first question, see Theorem 2.5.5 in the book of Albiac and Kalton. The second question is immediate from the first and the easy fact that $C[0,1]$ has a complemented subspace isometric to $c_0$. 


$\ell_\infty$
is injective. – Bill Johnson Dec 13 '12 at 3:59