Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of is greater than or equal to c√(n/log³n).
Banach space theory splits into several subtheories. On the one hand, there are an isometric and an isomorphic part; on the other hand, we speak of global and local aspects. While the concepts of isometry and isomorphy are clear, everybody seems to have its own interpretation of what "local theory" means. In this essay we analyze this situation and propose rigorous definitions, which are based on new concepts of local representability of operators.