Tameness and Rosenthal type locally convex spaces

Motivated by Rosenthal’s famous l¹ -dichotomy in Banach spaces, Haydon’s theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analogue of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results: extending Haydon’s characterization of Rosenthal Banach spaces, by showing that a lcs E is tame iff every weak-star compact, equicontinuous convex subset of E^∗ is the strong closed convex hull of its extreme points iff co¯w∗(K)=co¯(K) for every weak-star compact equicontinuous subset K of E^∗ ; E is tame iff there is no bounded sequence equivalent to the generalized l¹ -sequence;strengthening some results of W.M. Ruess about Rosenthal’s dichotomy;applying the Davis–Figiel–Johnson–Pelczyński (DFJP) technique one may show that every tame operator T: E→ F between a lcs E and a Banach space F can be factored through a tame (i.e., Rosenthal) Banach space.