Abstract
If $X$ is a Banach space such that the isomorphism constant to $\ell_2^n$ from $n$-dimensional subspaces grows sufficiently slowly as $n\to \infty$, then $X$ has the approximation property. A consequence of this is that there is a Banach space $X$ with a symmetric basis but not isomorphic to $\ell_2$ so that all subspaces of $X$ have the approximation property. This answers a problem raised in 1980. An application of the main result is that there is a separable Banach space $X$ that is not isomorphic to a Hilbert space, yet every subspace of $X$ is isomorphic to a complemented subspace of $X$. This contrasts with the classical result of Lindenstrauss and Tzafriri that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space.
Full Article