ON ULTRAREGULAR INDUCTIVE LIMITS

Abstract

An inductive limit $(E,t)=\text{ind}\,(E_n,t_n)$ is said to have property (P) if every closed absolutely convex neighborhood in $(E_n,t_n)$ is closed in $(E_{n+1},t_{n+1})$. This property was introduced and investigated by J. Kucera. In this paper we give some equivalent descriptions of property (P) and prove that property (P) implies ultraregularity. Particularly, if all $(E_n,t_n)$ are metrizable locally convex spaces, we have: $(E,t)$ is ultraregular if and only if $(E,t)$ is a strict inductive limit and for each $n\in {\Bbb N}$, there is $m=m(n)\in {\Bbb N}$ such that $\overline E_n^{E}\subset E_m;$ $(E,t)$ has property (P) if and only if $(E,t)$ is a strict inductive limit and each $E_n$ is closed in $(E_{n+1},t_{n+1}).$