On the completion of (LF)-spaces (Q1087777)

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 is 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 carry out a thorough study of the completeness of (LF)-spaces stressing upon the rather stable completion properties of (LB)-spaces. Namely, the main result of the paper is as follows: The completion of an \((LB)_ 1\)-space is an \((LB)_ 1\) space and the completion of an \((LB)_ 2\)-space is either an \((LB)_ 2\)-space or a Banach space. We supply examples proving that this result cannot be extended to the class of (LF)-spaces which do not enjoy good properties of completeness. A basic tool for handling this problem is an open mapping theorem for completions of (LF)-spaces, proved as well in the paper.