Weakening of a local Bollobás type property and geometry of Banach spaces.

This paper deals with a weaker form of the property so called L o , o for operators, which we call the property weak L o , o for operators. We characterize this property in terms of convergence of approximate norm attainment sets and prove that a pair of Banach spaces (X, Y) satisfies the property weak L o , o for compact operators if and only if X is reflexive. We further investigate the property weak L o , o for bilinear maps and obtain a connection of it with the property weak L o , o for operators. Importantly, we also characterize some geometric properties of Banach spaces with the help of convergence of approximate norm attainment sets. [ABSTRACT FROM AUTHOR]