On the dimension of the l p n l^{n}_{p} -subspaces of Banach spaces, for 1 p < 2 1leq p<2 Academic Article

We give an estimate relating the stable type p p constant of a Banach space X X with the dimension of the l p n l_p^n -subspaces of X X . Precisely, let C C be this constant and assume 1 > p > 2 1 > p > 2 . We show that, for each > 0 , X varepsilon > 0,X must contain a subspace ( 1 + ) (1 + varepsilon ) -isomorphic to l p k l_p^k , for every k k less than ( ) C p delta (varepsilon ){C^{p}} where ( ) > 0 delta (varepsilon ) > 0 is a number depending only on p p and varepsilon .