关于(M_0)型(LF)—空间的注记

A NOTE ON (LF) - SPACES OF TYPE (M0)

设(E,t)=ind(E_n),t_n)为(M.)型的(LF)—空间,则下述命题为等价: (1)(E,t)为正则; (2)(E,t)为α—正则; (3)(E,t)具一个由Frechet空间序列组成的定义谱(F_a,S_a)a∈N,使对于每个自然数n,(F_a,S_n)具一个ο—邻域基其成员都闭于(E,t); (4)(E,t)具一个由Frechet空间序列组成的定义谱(F_a,S_a)_a∈N,使对于每个自然数n,(F_a,S_a)具一个闭于(E,t)的ο—邻域; (5)(E,t)具一个由Frechet空间序列组成的定义谱(F_a,S_a)_a∈N,使对于每个自然数n及每个k≥n,(F_a,S_a)具一个ο—邻域基其成员都闭于(F_a,S_a); (6)(E,t)具一个由Frechet空间序列组成的定义谱(F_a,S_a)_a∈N,使对于每个自然数n及每个k≥n,(F_a,S_a)具一个闭于(F_h,S_h)的ο—邻域。

Let (E,t) = ind(En,tn) be an (LF) - space of type (Mo). Then the following statements are equivalent :(1)(E,t) is regular;(2)(E,t) is α- regular;(3)(E,t) admits a defining spectrum (Fn,Sn)n∈N of Frechet spaces such that for each n ∈ N, (Fn,sn) has a base of o - neighborhoods whose members are closed in (E,t) ;(4)(E,t) admits a defining spectrum (Fn,sn)n∈N of Frechet spaces such that for each n ∈ N, (Fn,Sn) has a o - neighborhood which is close in (E,t) ;(5)(E,t) admits a defining spectrum (Fn,sn)D∈N of Frechet spaces such that for each n ∈ N, and each k≥n, (Fn,sn) has a base of o - neighborhoods whose members are closed in (Fk,sk);(6)(E,t) admits a defining spectrum (Fn,sn)n∈N of Frechet spaces such that for each n ∈ N and each k ≥ n, (Fn,Sn) has a o - neighborhood which is closed in (Fk,sk);