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 analog 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 l 1 -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 lcs can be factored through a tame lcs.
We answer an open question due to M. Megrelishvili, and show that for a locally compact group G we have Tame(G) = Tame(L 1 (G)). We also reaffirm a similar known result stating that WAP(G) = WAP(L 1 (G)). This is done by using a generalized Haydon's theorem, the framework of bornological classes and a result due to A.
Abstract. We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem. Using the hyperspace uniformity, we prove a stronger version of this theorem concerning infinitely many ideals in supercomplete, pseudo-valuated rings. Finally we prove two interpolation theorems.
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