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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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To put it a little bluntly: what's your motivation for asking? (see – David Roberts Dec 13 '12 at 3:04
$c_0$ is complemented in any separable superspace, David. It is natural to ask if it is complemented in any superspace, which is equivalent to the first question because $\ell_\infty$ is injective. – Bill Johnson Dec 13 '12 at 3:59
@Bill That's a nice way to look at it :) – David Roberts Dec 13 '12 at 4:05
Agree with @David. It's not for answerers to provide motivation! – Anthony Quas Dec 13 '12 at 6:18

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 $c_0$.

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