Mapping Properties of ${\displaystyle c_0}$

Bessaga and Pełczyński showed that if ${\displaystyle c_0}$ embeds in the dual ${\displaystyle X^{\cdot }}$ of a Banach space X, then ${\displaystyle {\ell }^1}$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\displaystyle {\ell }^1}$ contains a copy of ${\displaystyle {\ell }^1}$ that is complemented in ${\displaystyle {\ell }^1}$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of ${\displaystyle L^1[0,1]}$ contains a copy of ${\displaystyle {\ell }^1}$ that is complemented in ${\displaystyle L^1[0,1]}$. In this note a traditional sliding hump argument is used to establish a simple mapping property of ${\displaystyle c_0}$ which simultaneously yields extensions of the preceding theorems as corollaries. Additional classical mapping properties of ${\displaystyle c_0}$ are briefly discussed and applications are given.