On the completion of (LF)-spaces

Once the existence of metrizable (LF)-spaces was discovered, the problem whether the completion of an (LF)-space is or is not an (LF)-space was answered in the negative, because no (LF)-space can be a Fréchet space. However, some (non-metrizable) (LF)-spaces are complete, e.g. the classical Köthe's strict (LF)-spaces. In this paper we will carry out a thorough study of the completeness of (LF)-spaces stressing upon the stable completion properties of (LB)-spaces. A basic tool for handling this problem is an Open Mapping Theorem for completions of (LF)-spaces, which is also proved in the present paper.This research, supported by a Fulbright-MEC fellowship, was carried out during the author's visit at the University of Maryland while on leave from the University of Extremadura. The author is grateful to Professor S. A. Saxon for very helpful conversation and, especially, for calling his attention to the problem of completions of (LF)-spaces.