Approximation property

Summary

A Banach space has the approximation property when the identity operator can be approximated, uniformly on each compact set, by bounded operators of finite rank. Grothendieck showed that this is equivalent to a statement on operators. For every Banach space $Y$, every compact operator from $Y$ into the space is a norm limit of finite-rank operators. Spaces with a Schauder basis, such as $\ell^p$ for $1 \le p < \infty$ and $c_0$, have the property. Enflo's counterexample shows that some separable Banach spaces do not.