Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ 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 c₀ embeds in L(X,Y), then ${\ell }^{\infty }$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X{\otimes }_{\gamma }Y*$. Applications to embeddings of c₀ in various spaces of operators are given.

We show that a Banach space X has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if X has nontrivial type. If for every Radon probability on X, there is an operator from an $L_p$ space into X whose range has probability one, then X is a quotient of an $L_p$ space. This extends a theorem of Sato’s which dealt with the case p = 2. In any infinite-dimensional Banach space X there is a compact set K so that for...