2022/03/04 by Matan Komisarchik, Michael Megrelishvili, Komisarchik, Matan +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #37Bxx #46A03 #46A17 #46B22 #54Hxx #Advanced Banach Space Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2203.02368
openalex publication_date 2022/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: \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* is the strong closed convex hull of its extreme points iff \rmco ^w*(K) = \rmco (K) for every weak-star compact equicontinuous subset K of E*; \bullet E is tame iff there is no bounded sequence equivalent to the generalized l1-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 \colon E → F between a lcs E and a Banach space F can be factored through a tame (i.e., Rosenthal) Banach space.