Quantitative Grothendieck property

A Banach space X is Grothendieck if the weak and the weak^? convergence of sequences in the dual space X^?$X^{?}$ coincide. The space ?^∞${?}^{\infty }$ 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)$L^{\infty }(\mu )$ for a σ-finite measure μ.