Quantitative Grothendieck Property

Abstract

A Banach space $X$ is Grothendieck if the weak and the weak${}^{\ast }$ convergence of sequences in the dual space $X^{\ast }$ 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 $\mu$.

arXiv comment9 pages, 0 figures, submitted to the Journal of Mathematical Analysis and Applications