A property of the embedding of $c_ 0$ in $l_ \infty$

Abstract:

This note proves that if $X$ is an FK space containing $\{ {\delta ^n}\}$ and if $X + {c_0} = {l_\infty }$, then $X = {l_\infty }$. The result is stronger than the fact that ${c_0}$ is not complemented in ${l_\infty }$, and shows that separability can be dropped in a similar theorem of Bennett and Kalton. The proof depends on Schur’s theorem and the fact that ${l_\infty }$ is a GB space to show that $X$ must be barrelled in ${l_\infty }$.