Tameness and Rosenthal type locally convex spaces

Abstract

Motivated by Rosenthal's famous $l^1$-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: $\bullet$ 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 $E^{\ast }$ is the strong closed convex hull of its extreme points iff ${\overline{\mathrm{co}}}^{w^{\ast }}(K)=\overline{\mathrm{co}}(K)$ for every weak-star compact equicontinuous subset $K$ of $E^{\ast }$; $\bullet$ $E$ is tame iff there is no bounded sequence equivalent to the generalized $l^1$-sequence; $\bullet$ strengthening some results of W.M. Ruess about Rosenthal's dichotomy; $\bullet$ applying the Davis-Figiel-Johnson-Pelczyński (DFJP) technique one may show that every tame operator $T:E\to F$ between a lcs $E$ and a Banach space $F$ can be factored through a tame (i.e., Rosenthal) Banach space.

43 pages