A Subspace of l2(X) without the approximation property

Abstract

The aim of the thesis is to provide support to the following conjecture, which would provide an isomorphic characterization of a Hilbert space in terms of the approximation property: an infinite dimensional Banach space X is isomorphic to l₂ if and only if every subspace of l₂ (X) has the approximation property. We show that if X has cotype 2 and the sequence of Euclidean distances {dn(X *)}n of X * satisfies dn (X *) ≥ α(log2 n )β for all n ≥ 1 and some absolute constants α > 0 and β > 4, then [cursive l] 2 (X ) contains a subspace without the approximation property.