Enflo's counterexample to the approximation problem

Remarks

The construction is an explicit but intricate combinatorial one, built from finite-dimensional blocks; it is not short enough to reproduce here. It has two famous consequences. First, $X$ has no Schauder basis, because a space with a Schauder basis has the approximation property. So the basis problem, whether every separable Banach space has a Schauder basis, has a negative answer. Second, by Grothendieck's work on the approximation property, some compact operator into $X$ is not a norm limit of finite-rank operators. Later work showed that the spaces $\ell^p$ with $p \ne 2$ contain closed subspaces without the approximation property.