Hereditary Approximation Property

If X is a Banach space such that the isomorphism constant to `2 from n dimensional subspaces grows sufficiently slowly as n → ∞, 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 `2 so that all subspaces of X have the approximation property. This answers a problem raised in 1980 [8]. 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 [15] that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space. Dedicated to the memory of Joram Lindenstrauss