A new characterization of the bounded approximation property

Abstract

We prove that a Banach space $X$ has the bounded approximation property if and only if, for every separable Banach space $Z$ and every injective operator $T$ from $Z$ to $X$, there exists a net $\left(S_{\alpha }\right)$ of finite-rank operators from $Z$ to $X$ with $\Vert S_{\alpha }\Vert \le {\lambda }_T$ such that ${lim\hspace{0.17em}}_{\alpha }\Vert S_{\alpha }z-Tz\Vert =0$ for every $z\in Z$.