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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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    $\begingroup$ To put it a little bluntly: what's your motivation for asking? (see mathoverflow.net/howtoask#motivation) $\endgroup$
    – David Roberts ♦
    Commented Dec 13, 2012 at 3:04
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    $\begingroup$ $c_0$ 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. $\endgroup$
    – Bill Johnson
    Commented Dec 13, 2012 at 3:59
  • $\begingroup$ @Bill That's a nice way to look at it :) $\endgroup$
    – David Roberts ♦
    Commented Dec 13, 2012 at 4:05
  • $\begingroup$ Agree with @David. It's not for answerers to provide motivation! $\endgroup$
    – Anthony Quas
    Commented Dec 13, 2012 at 6:18

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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 $c_0$.

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