Spaces of Moscatelli type. A survey (Q2902310)

The paper under review is a survey on the spaces of Moscatelli type. These spaces are obtained through a specific construction (described briefly in Section 2). The method of constructing such spaces allows to find counterexamples to several conjectures (some of them being open for a reasonable amount of time).NEWLINENEWLINEThe article is divided into five parts. Section 1 is the Introduction -- it contains a nice historical review of what has happened in the field. Section 2 deals with Fréchet spaces of Moscatelli type. Among other results, it is characterized here when such spaces are Montel (Prop. 2) or satisfy Vogt's conditions \((\Omega)\) (Prop. 4) and \((DN)\) (Prop. 6). Part 3 gives several examples of Fréchet spaces of smooth functions which also can be obtained via the Moscatelli type construction. Section 4 describes the analogous construction in order to get (LF)-spaces. It is shown that regularity and completeness for these spaces coincide (Prop. 9). It is connected with the last functional analytic question of Grothendieck which still remains open; that, is whether or not every regular (LF)-space is complete. Therefore, for possible counterexamples one must go beyond the construction of Moscatelli type. The last section gives several further results, among them the negative answer to Grothendieck's `problème de topologies' due to Taskinen (who used the Moscatelli type construction).