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 ℓ 1 -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 ℓ 1 -spreading models.2010 Mathematics Subject Classification. 46B20.

A Banach space X is Grothendieck if the weak and the weak * convergence of sequences in the dual space X * coincide. The space ℓ ∞ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck's result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of L ∞ (µ) for a σ-finite measure µ.2010 Mathematics Subject Classification. 46B26,46B04,46A20.