Retakh's conditions and regularity properties of (LF)-spaces

摘要

Introduction. Let (En)_(n=N) be an inductive sequence of locally convex spaces and let E:=indEn be its inductive limit. In the sense of Palamodov ([13]), an inductive sequence (En)nsN or its inductive limit E = ind En is called acyclic (resp, weakly acyclic) if the natural map j:En→En, (x_n)_(n=N)→(xn-Xn-1) (x0:=0), is a topological isomorphism (resp. a weak topological isomorphism) onto its range. Retakh ([14]) proved that an (LF)-space E = ind En is acyclic (resp. weakly acyclic) if and only if it satisfies Retakh's condition (M) (resp. Retakh's condition (Mo)), i.e. there exists an increasing sequence (t)neN of absolutely convex neighborhoods Un of 0 in E,, with the following property: for each n, there is m = m(ri) 2: n such that for all k 2: m the (weak) topology of Ek coincides on Un with the (weak) topology of Em, or equivalently E and E, induce the same (weak) topology on Un. Vogt ([16]) investigated (LF)-spaces satisfying Retakh's conditions (M) and (Mo) and the connections between these conditions and completeness, regularity and sequential retractivity. He pointed out that an (LF)-space satisfying Re-takh's condition (M) is complete, regular and sequentially retractive. On the other hand, Fernandez ([4]) proved that every sequentially retractive (LF)-space E = ind En "nearly" satisfies condition (M), i.e. for each ueN there is a neighborhood Un of 0 in E,, and m = m (n) >t n such that Ek and Em induce the same topology on Un for all k m. Recently Wengenroth ([17]) proved that an (LF)-space E = indEn satisfies condition (M) if and only if it is sequentially retractive. However, an (LF)-space satisfying condition (MJ needn't be regular (see [16]). In the present paper we first introduce the notion of weak sequential retractivity and show that weak sequential retractivity implies regular-ity. We shall give a characteristic for an (LF)-space with condition (Mo) E = ind En to be regular. That is, for each n e N, there is m = m (n) k w such that for every bounded set B in En,BEkcEm for all k m. In fact, for those (LF)-spaces, a-regulari-ty > regularity o weak sequential retractivity. As a corollary, we prove that for count-able inductive limits of weakly sequentially complete Frechet spaces, condition (Mo) implies weak sequential retractivity, weak sequential completeness and regularity.