Quantitative Grothendieck property

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 Grothendieckʼs 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.