Assume that $X$ and $Y$ are two Banach spaces and $T:X\to Y$ is a bounded surjective linear operator.


A consequence of the Michael selection theorem is that:"There is a continuous function $g:Y\to X$ such that $T\circ g=Id\_{Y}$".



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> Can we always find a linear map $g$ as above? 
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A linear consequence of the Michael selection theorem
Answer No. Just to add that the example that there is no bounded linear 1-1 mapping 
$\ell^\infty/c\_0 \to \ell^\infty$ is due to Phillips in 1940 (as commented in the link provided by Yemon Choi above, discussing the complementary subspace problem, 
also credited to Phillips in the link to pdf I found and posted). 


