Quantification of the Banach–Saks property

We investigate possible quantifications of the Banach–Saks property and the weak Banach–Saks property. We prove quantitative versions of relationships of the Banach–Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach–Saks property of a set using uniform weak convergence and ℓ1ℓ1-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for ℓ1ℓ1-spreading models.