(LF)-空间的正则性与完备性

REGULARITY AND COMPLETENESS OF (LF)-SPACES

设(E,t)=ind(E_n,t_n)为(LF)-空间,我们证明了下述结果: (i)(E,t)为正则当且仅当存在(E_n,t_n)中○的圆凸领域U_n,使U_1 U_2 …且(SP[U_n^E],η_n)为速完备,这里η_n是以{εU_n^E∩U:ε>0,U∈}为○-邻域基的局部凸拓扑,而为(E,t)中○-邻域基; (ii)若对于任意n∈N,存在(E_n,t_n)中○的圆凸邻域U_n及m=m(n)≥n,使U_n^E E_m且(SP[U_n^E],t|SP[U_n^E])为速完备;则(E,t)为完备。 (iii)若更设(E,t)=ind(E_n,t_n)为(LB)-空间,则下述条件为等价: (1)(E,t)为正则; (2)对于任意n∈N,(SP[D_n^E],pv_n^E)为Banach空间; (3)对于任意n∈N,存在m=m(n)≥n,使D_n^E E_m且(SP[D_n^E],pv_n^E)为Banach空间。 这里,D_n为(E_n,t_n)中的单位闭球而Pv_n^E为D_n^E的Minkowski泛函。

Let (E,t)=ind (En,tn)be an (LF)-space. We prove the following results.(1)(E,t) is regular if and only if there is a increasing sequence of absolutely convex0-neighborhoods Unin (En,tn) such that (SP[UnE],ηn) is fast complete, uhere ηn denotesthe locally convex topology whose o - base is {εUnE ∩ U ;ε> 0,U ∈ U}and U is a O-basein-(E,t);(ii)If for any n ∈ N, there is an absolutely convex O-neighborhood Unin (En,tn) and m = m (n) ≥ n such that UnE Em and (sp[UnE] ,t |sp[UnE]) is fast complete , then (E ,t) is regular.(iii)Moreover , let (E,t) = ind(En,tn) be an (LB)-space, then the following conditions are equivalent ;(1)(E,t) is regular;(2)for any n ∈ N, (sp[UnE], PDnE) is a Banach space;(3)for any n ∈ N , there is m = m (n) ≥ n such that DnE Em and (sp[UnE, pDnE) is a Banach space.Here Dn is the closed unit ball of (En,tn) and pDEis the Minkowski functional of DnE.