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Abstract

Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^*$ of a Banach space $X$, then $\ell ^1$ embeds complementably in $X$, and $\ell ^\infty $ embeds as a subspace of $X^*$. In this note the Diestel–Faires theorem and techniques of Kalton are used to show that if $X$ is an infinite-dimensional Banach space, $Y$ is an arbitrary Banach space, and $c_0$ embeds in $L(X,Y)$, then $\ell ^\infty $ embeds in $L(X,Y)$, and $\ell ^1$ embeds complementably in $X\otimes _{\gamma } Y^*$. Applications to embeddings of $c_0$ in various spaces of operators are given.