Weak Property Y .and Regularity of Inductive Limits

An inductive limit E, t.sind En, tn.is regular if and only if it satisfies the weak property Y0.; i.e., each weakly unconditionally Cauchy series in E, t. is contained and is a weakly unconditionally Cauchy series in some En, tn.. In particular, an LF.-space E, t.sind En, tn.is regular if and only if every weakly unconditionally Cauchy series kxk is a C-series; i.e., for any scalar sequence j k.gc0 , the series kjkxk is convergent. Furthermore, for inductive limits of Fr´echet spaces containing no copy of c0 , a number of characteristic conditions of regularity are given