Hereditary approximation property

If X is a Banach space such that the isomorphism constant to lⁿ₂ from n-dimensional subspaces grows sufficiently slowly as n → ∞, then X has the approximation property. A consequence of this is that there is a Ba- nach space X with a symmetric basis but not isomorphic to l₂ 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 sub- space 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.