On compact subsets in coechelon spaces of infinite order

Abstract

For coechelon spaces koo (v) of infinite order it is proved that every compact subset of koo (v) is contained in a closed absolutely convex hull of some null sequence if and only if the matrix v is regularly decreasing. In connection with the study of some interesting problems on Montel maps which are closely connected to the classical Grothendieck question on completeness of regular LB-spaces [PB, Problem 13.8.6] and the problem of bornologicity of C (K, E) with E an LB-space [S, Chapter IV], Dierolf and Doman'ski [DD2, Example 3.1] gave an example of a coechelon LB-Montel space of infinite order which has compact sets not contained in closed absolutely convex hulls of any null sequence. Consequently, the well-known characterization of compact sets in Frechet spaces [J, Theorem 9.4.2] turns out to be not generally true in the LB setting. The purpose of this note is to show that for coechelon spaces k, (v) of order oc the condition v regularly decreasing [BMS, Definition 3.1] is necessary and sufficient for every compact set to be contained in a closed absolutely convex hull of some null sequence. For more information on Montel maps and related questions the reader is referred to [DD1], [DD2], [DD3] and [D]. In what follows we recall some notation. Let E be a Frechet space with a fundamental system of seminorms (I1 1In)n; then the inductive dual E' is defined to be indn E', where the En are the completions of the normed spaces (E/ ker II Kln v 11In)* It is known that algebraically E' = E', the inclusion map E4' Eis continuous and E' is the bornological space associated with E , i.e., 4E = (E', 3 (E', E)) [J, Theorem 13.4.2] (E' and EJ denote the topological dual and the strong dual of E, resp.). We also recall that an LB-space E = indnEn is called boundedly retractive (respectively, compactly regular) if, and only if, for each bounded (respectively, compact) subset B of E there is n E N such that B C En and En and E induce the same topology on B (see [PB, Definitions 8.5.32-(ii), -(iii)]). It is clear that a boundedly refractive LB-space is compactly regular. On the other hand, in [N] it is proved that these conditions are also equivalent. Received by the editors August 12, 1997 and, in revised form, April 14, 1998. 1991 Mathematics Subject Classification. Primary 46A45; Secondary 46A50.