This is an announcement for the paper "Quantitative Grothendieck property" by Hana Bendova. Abstract: A Banach space $X$ is Grothendieck if the weak and the weak$^*$ convergence of sequences in the dual space $X^*$ coincide. The space $\ell^\infty$ 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 $L^\infty(\mu)$ for a $\sigma$-finite measure $\mu$. Archive classification: math.FA Mathematics Subject Classification: 46B26, 46B04, 46A20 Remarks: 9 pages, 0 figures, submitted to the Journal of Mathematical The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4684 or http://arXiv.org/abs/1309.4684