For coechelon spaces k∞(v) of infinite order it is proved that every compact subset of k∞(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],

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... lf and Domański [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 Fréchet 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 ∞ 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 Fréchet space with a fundamental system of seminorms ( n ) n ; then the inductive dual E i is defined to be ind n E n , where the E n are the completions of the normed spaces (E/ ker n , n ). It is known that algebraically E i = E , the inclusion map E i → E β is continuous and E i is the bornological space associated with E β , i.e., E i = (E , β (E , E )) [J, Theorem 13.4.2] (E and E β denote the topological dual and the strong dual of E, resp.). We also recall that an LB-space E = ind n E n is called boundedly retractive (respectively, compactly regular) if, and only if, for each bounded (respectively, compact) subset B of E there is n ∈ N such that B ⊂ E n and E n and E induce the same topology on B (see [PB, , -(iii)]). It is clear that a boundedly retractive LB-space is compactly regular. On the other hand, in [N] it is proved that these conditions are also equivalent.