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.